(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 7.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 151898, 3762] NotebookOptionsPosition[ 146704, 3587] NotebookOutlinePosition[ 147049, 3602] CellTagsIndexPosition[ 147006, 3599] WindowFrame->Normal*) (* Beginning of Notebook Content *) Notebook[{ Cell[BoxData[ RowBox[{" ", StyleBox[ RowBox[{ "The", " ", "Theory", " ", "of", " ", "Rational", " ", "Approximations", " ", "to", " ", "Irrational", " ", "Numbers"}], FontFamily->"Mathematica7", FontSize->16, FontWeight->"Bold", FontSlant->"Plain", FontVariations->{"StrikeThrough"->False, "Underline"->False}]}]], "Input", CellChangeTimes->{{3.483056329046875*^9, 3.48305636915625*^9}, { 3.48305642209375*^9, 3.483056472703125*^9}}], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Any", " ", "irrational", " ", "can", " ", "be", " ", "approximated", " ", "by", " ", RowBox[{"fractions", ".", " ", "The"}], " ", "function"}], ",", " ", "Phoenix", ",", " ", RowBox[{ "will", " ", "deliver", " ", "fractional", " ", "approxomations", " ", "with", " ", "virtually", " ", "unlimited", " ", "accuracy", " ", "for", " ", "any", " ", "real", " ", RowBox[{"number", ".", " ", StyleBox["E", FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox[".", FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox["g", FontSlant->"Italic", FontVariations->{"Underline"->True}], StyleBox[".", FontSlant->"Italic", FontVariations->{"Underline"->True}]}]}], ",", " ", RowBox[{ "it", " ", "zeros", " ", "in", " ", "on", " ", "two", " ", "fractions", " ", RowBox[{"(", RowBox[{ "one", " ", "on", " ", "its", " ", "left", " ", "and", " ", "one", " ", "to", " ", "its", " ", "right"}], ")"}], " ", "that", " ", "are", " ", "accurate", " ", "for", " ", "\[Pi]", " ", "to", " ", "over", " ", "11830", " ", "decimal", " ", "places", " ", "in", " ", "8", " ", "minutes", " ", "by", " ", "using", " ", "the", " ", "command", " ", "\"\\"", " ", "defined", " ", "within", "\[IndentingNewLine]", "Mathematica7"}], ",", " ", RowBox[{ RowBox[{"below", ".", " ", "The"}], " ", "earth", " ", "moving", " ", "function", " ", "plows", " ", "through", " ", "8000", " ", "Partial", " ", "Sequences"}], ",", " ", RowBox[{ "cultivating", " ", "78819", " ", "fractions", " ", "to", " ", "uncover", " ", "the", " ", "last", " ", "two", " ", "\[Pi]", " ", RowBox[{"impersonators", ".", " ", "See"}], " ", RowBox[{"Below", "."}]}]}], " ", "\[IndentingNewLine]", " ", "*)"}]], "Input", CellChangeTimes->{{3.48215433*^9, 3.482154543625*^9}, {3.482154604578125*^9, 3.48215465715625*^9}, {3.48216603309375*^9, 3.48216603425*^9}, { 3.482166234703125*^9, 3.48216632765625*^9}, {3.48216663625*^9, 3.482166683421875*^9}, {3.482166736734375*^9, 3.482166739375*^9}, { 3.482597336125*^9, 3.48259735390625*^9}, {3.48259740240625*^9, 3.48259751878125*^9}, {3.4826355510047755`*^9, 3.482635610082901*^9}, { 3.4826356401922755`*^9, 3.482635672489151*^9}, {3.4826357169110255`*^9, 3.4826357177547755`*^9}, {3.4826390148172755`*^9, 3.482639016645401*^9}, { 3.482745416921875*^9, 3.482745481734375*^9}, {3.482862579703125*^9, 3.48286258996875*^9}, {3.4829153100625*^9, 3.482915367671875*^9}, { 3.483057690046875*^9, 3.483057691609375*^9}, {3.483058799796875*^9, 3.483058800515625*^9}, {3.48397528746875*^9, 3.483975365515625*^9}}], Cell[BoxData[ RowBox[{"\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"We", " ", "will", " ", "define", " ", "the", " ", "term", " ", StyleBox["Partial", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["Sequence", FontVariations->{"Underline"->True}], " ", "to", " ", "refer", " ", "to", " ", "a", " ", "finite", " ", "sequence", " ", "of", " ", "2"}], "-", RowBox[{"D", " ", "vectors"}]}], ",", " ", RowBox[{ RowBox[{ "where", " ", "the", " ", "first", " ", "element", " ", "corresponds", " ", "to", " ", "the", " ", "numerator", " ", "of", " ", "a", " ", "given", " ", "fraction", " ", "and", " ", "the", " ", RowBox[{"2", "'"}], "nd", " ", "element", " ", "corresponds", " ", "to", " ", "its", " ", RowBox[{"denominator", ".", " ", "When"}], " ", "each", " ", "2"}], "-", RowBox[{ "D", " ", "vector", " ", "of", " ", "a", " ", "Partial", " ", "sequence", " ", "is", " ", "converted", " ", "to", " ", "a", " ", "fraction"}]}], ",", " ", RowBox[{ "the", " ", "sequence", " ", "of", " ", "fractions", " ", "approaches", " ", "a", " ", "target", " ", "number"}], ",", " ", RowBox[{"usually", " ", RowBox[{"irrational", ".", " ", "To"}], " ", "qualify", " ", "as", " ", "an", " ", "entry", " ", "to", " ", "the", " ", RowBox[{"P", ".", "S", "."}]}], ",", RowBox[{"each", " ", "vector", " ", RowBox[{"(", RowBox[{"as", " ", "a", " ", "fraction"}], ")"}], " ", "must", " ", "be", " ", "closer", " ", "to", " ", "the", " ", "irrational", " ", "target", " ", "than", " ", "the", " ", "previous", " ", RowBox[{"one", ".", " ", "Also"}]}], ",", " ", RowBox[{ RowBox[{ "all", " ", "first", " ", "elements", " ", "of", " ", "the", " ", "2"}], "-", RowBox[{"D", " ", "vectors", " ", RowBox[{"(", RowBox[{"the", " ", StyleBox["yin", FontSlant->"Italic"], " ", "sequence"}], ")"}], " ", "are", " ", "in", " ", "ascending", " ", "order", " ", "likewise", " ", "for", " ", "all", " ", "second", " ", "elements", " ", RowBox[{ RowBox[{"(", RowBox[{"the", " ", StyleBox["yang", FontSlant->"Italic"], " ", "sequence"}], ")"}], ".", " ", "All"}], " ", "three", " ", "sequences", " ", "are", " ", "limited", " ", "for", " ", "a", " ", "rational", " ", "target", " ", "and", " ", "unlimited", " ", "for", " ", "an", " ", "irrational", " ", RowBox[{ "target", ".", "\[IndentingNewLine]", " ", "\[IndentingNewLine]", "The"}], " ", "first", " ", "vector", " ", "of", " ", "each", " ", "new", " ", "Partial", " ", "Sequence", " ", "arises", " ", "from", " ", "operations", " ", "on", " ", "the", " ", StyleBox["last", FontVariations->{"Underline"->True}], " ", "vector", " ", "in", " ", "each", " ", "of", " ", "the", " ", StyleBox["two", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["previous", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["Partial", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], RowBox[{ StyleBox["Sequences", FontVariations->{"Underline"->True}], ".", " ", "In"}], " ", "the", " ", "function", " ", "below", " ", StyleBox["Phoenix", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], "these", " ", "two", " ", "particular", " ", "vectors", " ", "are", " ", "called", " ", StyleBox["CrimsonAsh", FontVariations->{"Underline"->True}], " ", "and", " ", StyleBox["GoldenAsh", FontVariations->{"Underline"->True}]}]}], ",", " ", RowBox[{ RowBox[{ "exploiting", " ", "the", " ", "saga", " ", "of", " ", "the", " ", "God"}], "-", RowBox[{"like", " ", "Bird"}]}], ",", " ", "Phoenix", ",", " ", RowBox[{"from", " ", "Greek", " ", "mythology"}], ",", " ", RowBox[{ "who", " ", "sprang", " ", "up", " ", "again", " ", "and", " ", "again"}], ",", " ", RowBox[{"from", " ", "the", " ", "ashes", " ", "of", " ", "previous", " ", RowBox[{"incarnations", ".", "\[IndentingNewLine]", " ", "Once"}], " ", "the", " ", "first", " ", "vector", " ", "of", " ", "a", " ", "new", " ", RowBox[{"P", ".", "S", ".", " ", "is"}], " ", "found"}], ",", " ", RowBox[{ StyleBox["CrimsonAsh", FontVariations->{"Underline"->True}], " ", "is", " ", "recursively", " ", "added", " ", "until", " ", "the", " ", "\[IndentingNewLine]", "newborn", " ", "vector", " ", RowBox[{"(", RowBox[{"converted", " ", "to", " ", "a", " ", "fraction"}], ")"}], " ", "is", " ", "further", " ", "from", " ", "the", " ", "target", " ", "than", " ", "the", " ", "previous", " ", "converted", " ", "\[IndentingNewLine]", RowBox[{"vector", ".", " ", "At"}], " ", "this", " ", "point"}], ",", " ", RowBox[{"the", " ", "Partial", " ", "Sequence", " ", RowBox[{"dies", ".", " ", "The"}], " ", "lengths", " ", "of", " ", "Partial", " ", "Sequences", " ", "for", " ", "\[Pi]", " ", "vary", " ", "widely"}], ",", " ", "\[IndentingNewLine]", RowBox[{"from", " ", "1", " ", RowBox[{"(", RowBox[{"in", " ", "many", " ", RowBox[{"P", ".", "S", ".", "'"}], "s"}], ")"}], " ", "to", " ", "10"}], ",", RowBox[{ RowBox[{"389", " ", RowBox[{ RowBox[{"(", RowBox[{"in", " ", RowBox[{"P", ".", "S", ".", " ", "292"}]}], ")"}], ".", "\[IndentingNewLine]", "\[IndentingNewLine]", "In"}], " ", "our", " ", "use", " ", "of", " ", "2"}], "-", RowBox[{"D", " ", "vectors"}]}], ",", " ", RowBox[{ "we", " ", "mean", " ", "that", " ", "they", " ", "have", " ", "the", " ", "same", " ", "operations", " ", "of", " ", "addition", " ", "and", " ", "scalar", " ", RowBox[{"multiplication", ".", " ", "\[IndentingNewLine]", "We"}], " ", "say", " ", "that", " ", "the", " ", "vectors", " ", "encountered", " ", "here", " ", StyleBox["correspond", FontVariations->{"Underline"->True}], StyleBox[" ", FontVariations->{"Underline"->True}], StyleBox["to", FontVariations->{"Underline"->True}], " ", RowBox[{"fractions", ".", " ", "Thus"}]}], ",", " ", RowBox[{"for", " ", "all", " ", "integers", " ", "a"}], ",", " ", "b", ",", " ", "c", ",", " ", RowBox[{"d", " ", "and", " ", "k"}], ",", " ", RowBox[{"noting", " ", "that", " ", "for", " ", "our", " ", "purposes"}], ",", " ", RowBox[{"k", " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"-", "1"}], " ", "or", " ", "k"}], " ", ">", " ", "1"}]}], ",", " ", RowBox[{ RowBox[{"we", " ", RowBox[{ "have", ":", "\[IndentingNewLine]", "\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t ", RowBox[{"Phoenix", " ", "-", " ", RowBox[{ StyleBox["Mathematica7", FontVariations->{"Underline"->True}], "\[IndentingNewLine]", "\t\t\t\t", "Vector", " ", RowBox[{"Addition", ":", " ", RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], " ", "+", " ", RowBox[{"[", RowBox[{"c", ",", " ", "d"}], "]"}]}]}]}]}]}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"[", RowBox[{ RowBox[{"a", " ", "+", " ", "c"}], ",", " ", RowBox[{"b", " ", "+", " ", "d"}]}], "]"}], " ", "GoldenAsh"}], " ", "+", " ", RowBox[{ "CrimsonAsh", "\[IndentingNewLine]", " ", "\[IndentingNewLine]", "\t\t\t\t", "Scalar", " ", RowBox[{"Multiplication", ":", " ", RowBox[{"k", "[", RowBox[{"a", ",", " ", "b"}], "]"}]}]}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"ka", ",", " ", "kb"}], "]"}], ".", " ", "n"}], "*", "GoldenAsh"}]}]}], ",", " ", RowBox[{ RowBox[{"sgn", "*", RowBox[{"CrimsonAsh", "\[IndentingNewLine]", "--"}], RowBox[{ RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"-", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{"--", RowBox[{ "--", "\[IndentingNewLine]", "\t\t\t\t\t ", "U"}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}\ ]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}]}], "/", "D"}], " ", RowBox[{"fcns", ".", "\[IndentingNewLine]", "\t\t\t\t\t\t ", RowBox[{"Conversion", ":", " ", RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "'"}]}]}]}], " ", "=", " ", RowBox[{ RowBox[{"a", "/", "b"}], " ", RowBox[{"(", RowBox[{"VP", "[", "v", "]"}], ")"}], "\[IndentingNewLine]", "\[IndentingNewLine]", "\t\t ", "Comparison", " ", "with", " ", "target"}]}], ",", " ", RowBox[{ RowBox[{"t", ":", " ", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "'"}], RowBox[{"(", "t", ")"}]}]}], " ", "=", " ", RowBox[{ RowBox[{"a", "/", "b"}], " ", "-", " ", RowBox[{"t", " ", RowBox[{"(", RowBox[{"VecPrm", "[", RowBox[{"v", ",", "t"}], "]"}], ")"}], " ", "\[IndentingNewLine]", "\[IndentingNewLine]", "\t\t ", "If", " ", "negative"}]}]}], ",", " ", RowBox[{ "fraction", " ", "is", " ", "to", " ", "left", " ", "of", " ", "target"}], ",", " ", RowBox[{"otherwise", " ", "to", " ", RowBox[{ "right", ".", "\[IndentingNewLine]", " ", "\[IndentingNewLine]", "\t\t ", "Distance"}], " ", "to", " ", "the", " ", "target"}], ",", " ", RowBox[{"t", ":", " ", RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "\"\<(t) = |a/b - t|. (VecPrm2[v,t])\n\t\ ---------------------\ -----------------------\n\n\t\t\t\t\t \!\(\* StyleBox[\"The\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"Fundamental\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"Identity\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"in\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"Rational\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"Approximation\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"Theory\", FontFamily->\"Courier New\", FontSize->14, FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\":\", FontFamily->\"Courier New\", FontSize->14]\)\!\(\* StyleBox[\"\\\\n\", FontFamily->\"Courier New\", FontSize->14]\)\!\(\* StyleBox[\"\\\\n\", FontFamily->\"Courier New\", FontSize->14]\)\t\t\t\t\t\t\t\t\t\t\t [ka, kb]' = [a, b]'\t\t\t\t\n\t\n\t\ The FIRAT above involves conversion with scalsr multiplication of any vector.\ \[IndentingNewLine]\n Most of the auxillary functions below are very \ simple ones. For example, the function \!\(\* StyleBox[\"VP\", FontVariations->{\"Underline\"->True}]\)[v] merely returns the fraction \ v[[1]]/v[[2]]. Each Partial Sequence, when their elements are converted to \ fractions represent rational numbers that get closer and closer to the target \ of the function \!\(\* StyleBox[\"Phoenix\", FontVariations->{\"Underline\"->True}]\). \!\(\* StyleBox[\"VecPrm\", FontVariations->{\"Underline\"->True}]\) computes the difference of v with \ the target, while \!\(\* StyleBox[\"VecPrm2\", FontVariations->{\"Underline\"->True}]\) forms the absolute value of this \ difference for comparison purposes. \!\(\* StyleBox[\"FirstVec\", FontVariations->{\"Underline\"->True}]\) returns the first vector of a P.S. \ \!\(\* StyleBox[\"LastVec\", FontVariations->{\"Underline\"->True}]\) returns the last one and \!\(\* StyleBox[\"Victor\", FontVariations->{\"Underline\"->True}]\) returns the n'th vector \n of \ the k'th P.S. \!\(\* StyleBox[\"TestPolRev\", FontVariations->{\"Underline\"->True}]\) tests (T/F) to see if there has been \ a polarity reversal at the end of a given P.S. \!\(\* StyleBox[\"IndexSeqWithLength\", FontVariations->{\"Underline\"->True}]\) searches for the first P.S. that has \ the length specified. \!\(\* StyleBox[\"Phoenix\", FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"[\", FontVariations->{\"Underline\"->True}]\)\[Pi], k, dp] returns the k'th \ sequence that converges to \[Pi], while the \!\(\* StyleBox[\"GoldenFireBird\", FontVariations->{\"Underline\"->True}]\) only flies after the \!\(\* StyleBox[\"PhoenixMatx\", FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\" \", FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"function\", FontVariations->{\"Underline\"->True}]\) has been used. This function keeps \ everything, including the matrix of two dimensional vectors. If the target is \ rational,\n the number of Partial Sequences is finite. The function stores \ the vectors into a (2 x n) matrix, where n is the total number of fractions \ found by the function, approaching the target. It's good practice to execute \ the \!\(\* StyleBox[\"Exit\", FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"[\", FontVariations->{\"Underline\"->True}]\)\!\(\* StyleBox[\"]\", FontVariations->{\"Underline\"->True}]\) command when finishing a session \ using PhoenixMatx in order to clear the memory.\n \ \ \>"}]}]}], "*)"}]}]], "Input", CellChangeTimes->{ 3.48210684778125*^9, {3.482110125828125*^9, 3.482110182484375*^9}, { 3.4821102509375*^9, 3.482110255328125*^9}, {3.4821102945625*^9, 3.482110324734375*^9}, {3.482110397359375*^9, 3.48211042221875*^9}, { 3.48211045825*^9, 3.482110610953125*^9}, {3.48211086790625*^9, 3.48211087359375*^9}, {3.482110943484375*^9, 3.482111014984375*^9}, { 3.482111077140625*^9, 3.482111118140625*^9}, {3.4821111534375*^9, 3.4821112603125*^9}, {3.482111291859375*^9, 3.482111302734375*^9}, { 3.4821115924375*^9, 3.482111935796875*^9}, {3.482112006*^9, 3.48211204784375*^9}, {3.4821121188125*^9, 3.48211219084375*^9}, { 3.48211223478125*^9, 3.482112257625*^9}, {3.482112474421875*^9, 3.48211254840625*^9}, {3.482112630203125*^9, 3.48211263696875*^9}, { 3.48211266765625*^9, 3.4821126746875*^9}, {3.48211276628125*^9, 3.4821127910625*^9}, {3.482112864625*^9, 3.482112887875*^9}, { 3.482112929703125*^9, 3.482113011828125*^9}, {3.482113056765625*^9, 3.4821131513125*^9}, {3.482113253796875*^9, 3.48211326590625*^9}, { 3.482113629328125*^9, 3.482113634*^9}, 3.48211371115625*^9, 3.482113787796875*^9, 3.482114258734375*^9, {3.48214668575*^9, 3.48214669015625*^9}, {3.482146785015625*^9, 3.4821468249375*^9}, { 3.482146950421875*^9, 3.4821471181875*^9}, {3.48214715571875*^9, 3.482147184484375*^9}, {3.482147223265625*^9, 3.482147272796875*^9}, { 3.48214856821875*^9, 3.48214857521875*^9}, {3.48214865709375*^9, 3.48214865746875*^9}, 3.4821487215625*^9, {3.482148822921875*^9, 3.482148848625*^9}, {3.48214906503125*^9, 3.482149085328125*^9}, { 3.482150326453125*^9, 3.482150331453125*^9}, {3.482150600875*^9, 3.482150621328125*^9}, {3.482167518*^9, 3.4821675284375*^9}, { 3.48216756321875*^9, 3.48216763928125*^9}, {3.482597602984375*^9, 3.482597699515625*^9}, {3.482597740328125*^9, 3.482597750828125*^9}, { 3.482597787125*^9, 3.4825979595625*^9}, {3.482598026015625*^9, 3.48259805003125*^9}, {3.482616845207901*^9, 3.4826168923485255`*^9}, { 3.482617129614151*^9, 3.4826171298172755`*^9}, {3.482617268770401*^9, 3.482617273426651*^9}, {3.482617310676651*^9, 3.4826173357235255`*^9}, { 3.4826173859735255`*^9, 3.482617393364151*^9}, {3.482617454489151*^9, 3.4826175287547755`*^9}, {3.4826377733485255`*^9, 3.4826378124735255`*^9}, {3.4826378565985255`*^9, 3.4826379028485255`*^9}, {3.482637939864151*^9, 3.4826380090360255`*^9}, { 3.482638110332901*^9, 3.4826382330672755`*^9}, {3.4826382716922755`*^9, 3.4826382842235255`*^9}, {3.4826384876297755`*^9, 3.4826384917547755`*^9}, {3.482638539301651*^9, 3.482638571832901*^9}, { 3.482638844739151*^9, 3.4826389067860255`*^9}, {3.4826391161297755`*^9, 3.4826391234422755`*^9}, {3.4826396891610255`*^9, 3.4826396894422755`*^9}, 3.482639734020401*^9, {3.482639863301651*^9, 3.4826399318485255`*^9}, { 3.48274551375*^9, 3.482745592*^9}, 3.48274568459375*^9, { 3.482745806234375*^9, 3.482745852390625*^9}, {3.482745909703125*^9, 3.48274591525*^9}, {3.482746132109375*^9, 3.482746165859375*^9}, { 3.482878800859375*^9, 3.482879398046875*^9}, {3.482879428671875*^9, 3.482879524453125*^9}, {3.48287973359375*^9, 3.482880093671875*^9}, { 3.4828802425*^9, 3.482880243546875*^9}, {3.482880524359375*^9, 3.48288056575*^9}, {3.482880597953125*^9, 3.48288060128125*^9}, { 3.482880632015625*^9, 3.482880634*^9}, {3.482880908375*^9, 3.482880980296875*^9}, {3.482881013234375*^9, 3.4828812968125*^9}, { 3.4828814195*^9, 3.482881532359375*^9}, {3.482881675546875*^9, 3.4828816790625*^9}, {3.48289020378125*^9, 3.482890385203125*^9}, { 3.4829154546875*^9, 3.48291550825*^9}, {3.48291561765625*^9, 3.482915696453125*^9}, {3.4829157431875*^9, 3.482915816546875*^9}, { 3.4829880696875*^9, 3.4829880788125*^9}, {3.483039568328125*^9, 3.483039689875*^9}, 3.483040437765625*^9, {3.48304047496875*^9, 3.4830405465625*^9}, {3.483040594953125*^9, 3.48304061265625*^9}, { 3.48304067328125*^9, 3.48304067425*^9}, {3.48304078675*^9, 3.48304078765625*^9}, {3.48304085171875*^9, 3.4830409099375*^9}, { 3.483040940203125*^9, 3.48304098421875*^9}, {3.483047442640625*^9, 3.483047443265625*^9}, {3.483056498265625*^9, 3.483056704859375*^9}, { 3.483056737578125*^9, 3.48305684940625*^9}, {3.4830568995625*^9, 3.483056938078125*^9}, 3.483056991734375*^9, {3.48305705340625*^9, 3.48305708996875*^9}, {3.48305714003125*^9, 3.4830572914375*^9}, { 3.48305737078125*^9, 3.483057418546875*^9}, {3.483057486734375*^9, 3.4830575036875*^9}, 3.48305757328125*^9, 3.4830577645625*^9, { 3.483058009453125*^9, 3.483058014703125*^9}, {3.48305804809375*^9, 3.483058114515625*^9}, {3.483058158*^9, 3.483058167203125*^9}, { 3.48305855525*^9, 3.4830586149375*^9}, {3.483058679328125*^9, 3.483058753015625*^9}, {3.48305892865625*^9, 3.483059039640625*^9}, { 3.48305921725*^9, 3.483059224640625*^9}, {3.483059314578125*^9, 3.48305931978125*^9}, {3.483059358953125*^9, 3.48305937128125*^9}, { 3.483059427390625*^9, 3.483059475125*^9}, {3.483059509203125*^9, 3.483059521421875*^9}, {3.48305965203125*^9, 3.48305968021875*^9}, { 3.483059785421875*^9, 3.483059831984375*^9}, {3.483060020796875*^9, 3.483060069890625*^9}, {3.483060334671875*^9, 3.483060386046875*^9}, { 3.48318916521875*^9, 3.48318921*^9}, {3.483189301*^9, 3.483189333734375*^9}, {3.483189445484375*^9, 3.483189531796875*^9}, { 3.483189673390625*^9, 3.48318973871875*^9}, {3.483189794171875*^9, 3.4831898699375*^9}, {3.483975102796875*^9, 3.48397513446875*^9}, { 3.483975186375*^9, 3.483975258296875*^9}, {3.483975373875*^9, 3.483975499421875*^9}}], Cell[BoxData[ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Living", " ", "in", " ", "India", " ", "and", " ", "migrating", " ", "to", " ", "Egypt", " ", "every", " ", "500", " ", "years"}], ",", " ", RowBox[{ "the", " ", "Crimson", " ", "and", " ", "Golden", " ", "Bird", " ", "flies", " ", "from", " ", "Sunrise", " ", "to", " ", RowBox[{"Sunset", ".", "\[IndentingNewLine]", " ", "His"}], " ", "name", " ", "is", " ", RowBox[{ "legend", ":", "\[IndentingNewLine]", " ", StyleBox["\[CapitalPhi]\[Omicron]\[Iota]\[Nu]\[Iota]\[Xi]", FontFamily->"Symbol", FontSize->18]}]}]}], " ", "*)"}]], "Input", CellChangeTimes->{{3.482149168171875*^9, 3.48214933390625*^9}, { 3.482149455359375*^9, 3.48214951553125*^9}, {3.48214987978125*^9, 3.4821498878125*^9}, {3.4821499343125*^9, 3.482149936703125*^9}, { 3.482150043109375*^9, 3.482150103296875*^9}, {3.4821501545*^9, 3.482150154890625*^9}, {3.482596335984375*^9, 3.48259637390625*^9}, { 3.482638398832901*^9, 3.482638401489151*^9}, {3.4826387297860255`*^9, 3.4826387338485255`*^9}, {3.4826398263797755`*^9, 3.482639826926651*^9}, { 3.482746654796875*^9, 3.482746662859375*^9}, {3.482880290375*^9, 3.482880307125*^9}, {3.483060258578125*^9, 3.483060260015625*^9}}], Cell[BoxData[{ RowBox[{ RowBox[{ RowBox[{ RowBox[{"Phoenix", "[", RowBox[{"RealNo_", ",", "k_", ",", "dp_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Key", " ", "to", " ", "discover", " ", "the", " ", SuperscriptBox[ StyleBox["k", FontVariations->{"Underline"->True}], "th"], " ", RowBox[{"P", ".", "S", ".", " ", "for"}], " ", "any", " ", "real", " ", "number"}], ",", " ", RowBox[{"with", " ", StyleBox["dp", FontVariations->{"Underline"->True}], " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "CrimsonAsh", ",", "GoldenAsh", ",", "Joaquin", ",", "NewLeaf", ",", "JoaquinPhoenix", ",", "LSK", ",", "NumSeq", ",", "target", ",", "best", ",", "test", ",", "n", ",", "sgn", ",", "LenSeq"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"CrimsonAsh", "=", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], ";", RowBox[{"(*", " ", RowBox[{"Let", " ", "the", " ", "Force", " ", RowBox[{"(", RowBox[{"1", "/", "0"}], ")"}], " ", "be", " ", "with", " ", RowBox[{"you", ":", " ", RowBox[{ RowBox[{"Floor", "[", "RealNo", "]"}], " ", "<", " ", "RealNo", " ", "<", " ", RowBox[{"\[Infinity]", "."}]}]}]}], " ", "*)"}], " ", RowBox[{"GoldenAsh", "=", RowBox[{"{", RowBox[{ RowBox[{"Floor", "[", "RealNo", "]"}], ",", "1"}], "}"}]}], ";", RowBox[{"NumSeq", "=", "1"}], ";", " ", RowBox[{"If", "[", RowBox[{ RowBox[{"k", "\[Equal]", "1"}], ",", RowBox[{"Return", "[", RowBox[{"{", RowBox[{"GoldenAsh", ",", "1"}], "}"}], "]"}]}], "]"}], ";", RowBox[{"target", "=", RowBox[{"N", "[", RowBox[{"RealNo", ",", RowBox[{"Max", "[", RowBox[{"dp", ",", RowBox[{"4", "*", "k"}]}], "]"}]}], "]"}]}], ";", RowBox[{"best", "=", RowBox[{"VecPrm2", "[", RowBox[{"GoldenAsh", ",", "target"}], "]"}]}], ";", RowBox[{"LenSeq", "=", RowBox[{"{", "1", "}"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Label", "[", "Ra", "]"}], ";", RowBox[{"LSK", "=", "1"}], ";", RowBox[{"n", "=", "1"}], ";", RowBox[{"sgn", "=", "1"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"CrimsonAsh", "\[NotEqual]", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], ",", RowBox[{"sgn", "=", RowBox[{"-", RowBox[{"Sign", "[", RowBox[{ RowBox[{"VecPrm", "[", RowBox[{"CrimsonAsh", ",", "target"}], "]"}], "*", RowBox[{"VecPrm", "[", RowBox[{"GoldenAsh", ",", "target"}], "]"}]}], "]"}]}]}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Label", "[", "Heliopolis", "]"}], ";", RowBox[{"n", "=", RowBox[{"n", "+", "1"}]}], ";", RowBox[{"Joaquin", "=", RowBox[{ RowBox[{"n", "*", "GoldenAsh"}], "+", RowBox[{"sgn", "*", "CrimsonAsh"}]}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"Joaquin", ",", "target"}], "]"}]}], ";", RowBox[{"(*", " ", RowBox[{"Discovery", " ", "Loop"}], " ", "*)"}], " ", RowBox[{"If", "[", RowBox[{ RowBox[{"test", ">", "best"}], ",", RowBox[{"(*", " ", RowBox[{ RowBox[{"Sojourn", " ", "back", " ", "to", " ", "Sun", " ", RowBox[{"City", ".", " ", "In"}], " ", "order", " ", "to", " ", "move", " ", "forward"}], ",", " ", RowBox[{ "Joaquin", " ", "must", " ", "be", " ", "closer", " ", "to", " ", "his", " ", "target", " ", "than", " ", "he", " ", "was", " ", RowBox[{"before", "."}]}]}], " ", "*)"}], " ", RowBox[{"Goto", "[", "Heliopolis", "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Otherwise", ",", " ", RowBox[{ "Joaquin", " ", "turns", " ", "over", " ", "a", " ", "New", " ", RowBox[{"Leaf", "."}]}]}], " ", "*)"}], " ", RowBox[{"NewLeaf", "=", "Joaquin"}], ";", RowBox[{"CrimsonAsh", "=", "GoldenAsh"}], ";", RowBox[{"GoldenAsh", "=", "NewLeaf"}], ";", RowBox[{"best", "=", "test"}], ";", " ", RowBox[{"Label", "[", "FrankincenseAndMyrrh", "]"}], ";", RowBox[{"Joaquin", "=", RowBox[{"GoldenAsh", "+", "CrimsonAsh"}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"Joaquin", ",", "target"}], "]"}]}], ";", RowBox[{"(*", " ", RowBox[{"Recursive", " ", "Loop"}], " ", "*)"}], RowBox[{"If", "[", RowBox[{ RowBox[{"test", "<", "best"}], ",", RowBox[{ RowBox[{"best", "=", "test"}], ";", RowBox[{"LSK", "=", RowBox[{"LSK", "+", "1"}]}], ";", RowBox[{"GoldenAsh", "=", "Joaquin"}], ";", RowBox[{"Goto", "[", "FrankincenseAndMyrrh", "]"}]}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "The", " ", "Spices", " ", "have", " ", "been", " ", "prepared", " ", "for", " ", "the", " ", "Nest", " ", "of", " ", "the", " ", "Golden", " ", "Fire", " ", RowBox[{"Bird", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"LenSeq", "=", RowBox[{"Append", "[", RowBox[{"LenSeq", ",", "LSK"}], "]"}]}], ";", RowBox[{"NumSeq", "=", RowBox[{"NumSeq", "+", "1"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"NumSeq", "<", "k"}], ",", RowBox[{"(*", " ", RowBox[{ "Return", " ", "to", " ", "Egyptian", " ", "Sun", " ", "God"}], " ", "*)"}], " ", RowBox[{"Goto", "[", "Ra", "]"}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Eureka", ",", " ", RowBox[{ RowBox[{ RowBox[{ "the", " ", "Great", " ", "Bird", " ", "flies", " ", "again", " ", "as", " ", "the", " ", SuperscriptBox[ StyleBox["k", FontVariations->{"Underline"->True}], "th"], " ", RowBox[{"P", ".", "S", "."}]}], ":", " ", SubscriptBox["S", "k"]}], " ", "=", " ", RowBox[{"{", RowBox[{ SubscriptBox["v", "2"], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"N", " ", "-", " ", "1"}], ")"}], SubscriptBox["v", "1"]}]}], "}"}]}]}], " ", "*)"}], " ", RowBox[{"JoaquinPhoenix", "=", RowBox[{"{", "NewLeaf", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"NewLeaf", "=", RowBox[{"NewLeaf", "+", "CrimsonAsh"}]}], ";", RowBox[{"JoaquinPhoenix", "=", RowBox[{"Append", "[", RowBox[{"JoaquinPhoenix", ",", "NewLeaf"}], "]"}]}]}], ",", RowBox[{"{", RowBox[{"var", ",", RowBox[{"LSK", "-", "1"}]}], "}"}]}], "]"}], ";", RowBox[{"Print", "[", "\"\< \!\(\* StyleBox[\"\[CapitalPhi]\[Omicron]\[Iota]\[Nu]\[Iota]\[Xi]\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\"-\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)The Great Crimson and Golden Bird Soars Above\>\"", "]"}], ";", RowBox[{"Return", "[", RowBox[{"Append", "[", RowBox[{"JoaquinPhoenix", ",", "LenSeq"}], "]"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"VP", "[", "v_", "]"}], ":=", RowBox[{"(", RowBox[{ RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], "/", RowBox[{"v", "[", RowBox[{"[", "2", "]"}], "]"}]}], ")"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "'"}], " ", "=", " ", RowBox[{"a", "/", "b"}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{"v_", ",", "target_"}], "]"}], ":=", RowBox[{"(", RowBox[{"N", "[", RowBox[{ RowBox[{"VP", "[", "v", "]"}], "-", "target"}], "]"}], ")"}]}], ";"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", RowBox[{"b", "'"}]}], "]"}], RowBox[{"(", "t", ")"}]}], " ", "=", " ", RowBox[{ RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "'"}], " ", "-", " ", "t"}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{"v_", ",", "target_"}], "]"}], ":=", RowBox[{"(", RowBox[{"Abs", "[", RowBox[{"VecPrm", "[", RowBox[{"v", ",", "target"}], "]"}], "]"}], ")"}]}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"[", RowBox[{"a", ",", " ", "b"}], "]"}], "\"\<(t) = |[a, b]'(t)| \>"}], "*)"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"PhoenixMatx", "[", RowBox[{"RealNumber_", ",", "LimNoSeq_", ",", "dp_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Provide", " ", "the", " ", "target"}], ",", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"no", ".", " ", "of"}], " ", RowBox[{"P", ".", "S", ".", "'"}], "s"}], " ", "&"}], " ", "decimal", " ", "places", " ", RowBox[{"desired", "."}]}]}], " ", "*)"}], RowBox[{"(", RowBox[{ RowBox[{"st", "=", RowBox[{"AbsoluteTime", "[", "]"}]}], ";", RowBox[{"target", "=", RowBox[{"N", "[", RowBox[{"RealNumber", ",", RowBox[{"Max", "[", RowBox[{"dp", ",", RowBox[{"4", "*", "LimNoSeq"}]}], "]"}]}], "]"}]}], ";", RowBox[{"RevPol", "=", "0"}], ";", RowBox[{"LenSeq", "=", RowBox[{"{", "1", "}"}]}], ";", RowBox[{"LSK", "=", "1"}], ";", RowBox[{"NumSeq", "=", "1"}], ";", RowBox[{"RPSeqs", "=", RowBox[{"{", "}"}]}], ";", RowBox[{"it", "=", RowBox[{"Floor", "[", "target", "]"}]}], ";", RowBox[{"V1", "=", RowBox[{"{", RowBox[{"it", ",", "1"}], "}"}]}], ";", RowBox[{"V2", "=", RowBox[{"{", RowBox[{ RowBox[{"it", "+", "1"}], ",", "1"}], "}"}]}], ";", RowBox[{"bestL", "=", RowBox[{"N", "[", RowBox[{ RowBox[{"target", "-", "it"}], ",", "100"}], "]"}]}], ";", RowBox[{"bestR", "=", RowBox[{"N", "[", RowBox[{ RowBox[{"it", "+", "1", "-", "target"}], ",", "100"}], "]"}]}], ";", RowBox[{"best", "=", RowBox[{"Min", "[", RowBox[{"bestL", ",", "bestR"}], "]"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"bestL", "<", "bestR"}], ",", RowBox[{"Matx", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"V1", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"V1", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "}"}]}], ",", RowBox[{"Matx", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"V2", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"V2", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "}"}]}]}], "]"}], ";", RowBox[{"Label", "[", "initialize", "]"}], ";", RowBox[{"V3", "=", RowBox[{"V1", "+", "V2"}]}], ";", RowBox[{"test", "=", RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{"V3", "[", RowBox[{"[", "1", "]"}], "]"}], "/", RowBox[{"V3", "[", RowBox[{"[", "2", "]"}], "]"}]}], "-", "target"}], "]"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"test", "<", "best"}], ",", " ", RowBox[{"(*", " ", "then", " ", "*)"}], " ", RowBox[{"If", "[", RowBox[{ RowBox[{"bestL", "<", "bestR"}], ",", RowBox[{"Matx", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], ",", RowBox[{"V3", "[", RowBox[{"[", "1", "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}], ",", RowBox[{"V3", "[", RowBox[{"[", "2", "]"}], "]"}]}], "}"}]}], "}"}]}], ",", RowBox[{"Matx", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"V3", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}]}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"V3", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}]}], "}"}]}], "}"}]}]}], "]"}], ",", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"bestL", "<", "bestR"}], ",", RowBox[{"V2", "=", "V3"}], ",", RowBox[{"(*", " ", "else", " ", "*)"}], " ", RowBox[{"V1", "=", "V3"}]}], "]"}], ";", RowBox[{"Goto", "[", "initialize", "]"}]}]}], "]"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"bestL", "<", "bestR"}], ",", RowBox[{"V2", "=", "V3"}], ",", RowBox[{"(*", " ", "else", " ", "*)"}], " ", RowBox[{"V1", "=", "V3"}]}], "]"}], ";", RowBox[{"numFrac", "=", "2"}], ";", "\n", RowBox[{"Label", "[", "continue", "]"}], ";", RowBox[{"LSK", "=", "1"}], ";", RowBox[{"n", "=", "1"}], ";", RowBox[{"sgn", "=", "1"}], ";", RowBox[{"Label", "[", "RecursiveLoop", "]"}], ";", RowBox[{"V3", "=", RowBox[{"V1", "+", "V2"}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"V3", ",", "target"}], "]"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"test", "<", "best"}], ",", " ", RowBox[{"(*", " ", "then", " ", "*)"}], " ", RowBox[{ RowBox[{"best", "=", "test"}], ";", RowBox[{"LSK", "=", RowBox[{"LSK", "+", "1"}]}], ";", RowBox[{"Matx", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"V3", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}]}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"V3", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "]"}], "}"}]}], "]"}]}], ";", RowBox[{"numFrac", "=", RowBox[{"numFrac", "+", "1"}]}], ";", RowBox[{"V2", "=", "V3"}], ";", RowBox[{"Goto", "[", "RecursiveLoop", "]"}]}]}], "]"}], ";", RowBox[{"(*", " ", RowBox[{"Otherwise", ",", " ", RowBox[{"the", " ", "partial", " ", "sequence", " ", RowBox[{"ends", "."}]}]}], " ", "*)"}], " ", RowBox[{"LenSeq", "=", RowBox[{"Append", "[", RowBox[{"LenSeq", ",", "LSK"}], "]"}]}], ";", RowBox[{"NumSeq", "=", RowBox[{"NumSeq", "+", "1"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"best", "<", RowBox[{"2", "*", SuperscriptBox["10", RowBox[{"-", "dp"}]]}]}], ",", RowBox[{"Goto", "[", "fin", "]"}]}], "]"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"NumSeq", "\[Equal]", "LimNoSeq"}], ",", RowBox[{"Goto", "[", "fin", "]"}]}], "]"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{"V1", ",", "target"}], "]"}], "*", RowBox[{"VecPrm", "[", RowBox[{"V2", ",", "target"}], "]"}]}], ">", "0"}], ",", RowBox[{ RowBox[{"sgn", "=", RowBox[{"-", "1"}]}], ";", RowBox[{"RevPol", "=", RowBox[{"RevPol", "+", "1"}]}]}]}], "]"}], ";", RowBox[{"Label", "[", "DiscoveryLoop", "]"}], ";", RowBox[{"n", "=", RowBox[{"n", "+", "1"}]}], ";", RowBox[{"V3", "=", RowBox[{ RowBox[{"n", "*", "V2"}], "+", RowBox[{"sgn", "*", "V1"}]}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"V3", ",", "target"}], "]"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"test", ">", "best"}], ",", " ", RowBox[{"(*", " ", RowBox[{"then", " ", "go", " ", "back", " ", "until", " ", "true"}], " ", "*)"}], " ", RowBox[{"Goto", "[", "DiscoveryLoop", "]"}], ",", " ", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ RowBox[{"else", " ", "the", " ", "first", " ", "element"}], ",", " ", "V3", ",", " ", RowBox[{"that", " ", "is"}], ",", " ", RowBox[{ "of", " ", "the", " ", "next", " ", "partial", " ", "sequence", " ", "has", " ", "been", " ", "found"}]}], " ", "*)"}], " ", RowBox[{ RowBox[{"best", "=", "test"}], ";", RowBox[{"numFrac", "=", RowBox[{"numFrac", "+", "1"}]}], ";", RowBox[{"Matx", "=", RowBox[{"Join", "[", RowBox[{ RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"V3", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}]}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"Join", "[", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"V3", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "]"}], "}"}]}], "]"}]}], ";", RowBox[{"V1", "=", "V2"}], ";", RowBox[{"V2", "=", "V3"}], ";", RowBox[{"Goto", "[", "continue", "]"}]}]}], "]"}], ";", RowBox[{"Label", "[", "fin", "]"}], ";", RowBox[{"tt", "=", RowBox[{ RowBox[{"AbsoluteTime", "[", "]"}], "-", "st"}]}], ";", RowBox[{"Print", "[", "\"\< \!\(\* StyleBox[\"\[CapitalPhi]\[Omicron]\[Iota]\[Nu]\[Iota]\[Xi]\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\"-\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)The Great Crimson and Golden Bird Soars Above\>\"", "]"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"NumSeq", "<", "LimNoSeq"}], ",", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", "NumSeq"}], "]"}]}], "]"}]}], ")"}]}], ";"}], "\[IndentingNewLine]", "\[IndentingNewLine]", " ", RowBox[{"(*", " ", RowBox[{"This", " ", "function", " ", "delivers", " ", "the", " ", RowBox[{"k", "'"}], "th", " ", "Partial", " ", "Sequence", " ", "from", " ", "any", " ", StyleBox["Ma", FontVariations->{"Underline"->True}], StyleBox["tx", FontVariations->{"Underline"->True}], " ", "found", " ", "in", " ", "the", " ", "PhoenixMatx", " ", RowBox[{"function", "."}]}], " ", "*)"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"GoldenFireBird", "[", RowBox[{"Matx_", ",", "k_", ",", "LenSeq_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Arguments", " ", "are", " ", "the", " ", "matrix"}], ",", " ", RowBox[{"the", " ", "index", " ", "of", " ", "the", " ", RowBox[{"P", ".", "S", ".", " ", "and"}], " ", "the", " ", "LenSeq", " ", RowBox[{"array", "."}]}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "Vec1", ",", "Vec2", ",", "Vec3", ",", "LSK", ",", "GFB", ",", "NewBird"}], "}"}], ",", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"k", "\[Equal]", "1"}], ",", RowBox[{"Return", "[", RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}]}], "}"}], "]"}]}], "]"}], ";", RowBox[{"Vec1", "=", RowBox[{"LastVec", "[", RowBox[{"Matx", ",", RowBox[{"k", "-", "1"}], ",", "LenSeq"}], "]"}]}], ";", RowBox[{"Vec2", "=", RowBox[{"FirstVec", "[", RowBox[{"Matx", ",", "k", ",", "LenSeq"}], "]"}]}], ";", RowBox[{"Vec3", "=", RowBox[{"LastVec", "[", RowBox[{"Matx", ",", "k", ",", "LenSeq"}], "]"}]}], ";", RowBox[{"LSK", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Vec3", "[", RowBox[{"[", "2", "]"}], "]"}], "-", RowBox[{"Vec2", "[", RowBox[{"[", "2", "]"}], "]"}]}], ")"}], "/", RowBox[{"Vec1", "[", RowBox[{"[", "2", "]"}], "]"}]}], "+", "1"}]}], ";", RowBox[{"GFB", "=", RowBox[{"{", "Vec2", "}"}]}], ";", RowBox[{"NewBird", "=", "Vec2"}], ";", RowBox[{"Print", "[", "\"\< \!\(\* StyleBox[\"\[CapitalPhi]\[Omicron]\[Iota]\[Nu]\[Iota]\[Xi]\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\"-\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)The Great Crimson and Golden Bird Soars Above\>\"", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"NewBird", "=", RowBox[{"NewBird", "+", "Vec1"}]}], ";", RowBox[{"GFB", "=", RowBox[{"Append", "[", RowBox[{"GFB", ",", "NewBird"}], "]"}]}]}], ",", RowBox[{"{", RowBox[{"n", ",", RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "k", "]"}], "]"}], "-", "1"}]}], "}"}]}], "]"}], ";", RowBox[{"Return", "[", "GFB", "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"GFBirdShort", "[", RowBox[{"Matx_", ",", "k_", ",", "LenSeq_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ "The", " ", "output", " ", "shortened", " ", "form", " ", "of", " ", "the", " ", "GoldenFireBird"}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"Vec1", ",", "Vec2", ",", "Vec3", ",", "LSK"}], "}"}], ",", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"k", "\[Equal]", "1"}], ",", RowBox[{"Return", "[", RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "1"}], "]"}], "]"}]}], "}"}], "]"}]}], "]"}], ";", RowBox[{"Vec1", "=", RowBox[{"LastVec", "[", RowBox[{"Matx", ",", RowBox[{"k", "-", "1"}], ",", "LenSeq"}], "]"}]}], ";", RowBox[{"Vec2", "=", RowBox[{"FirstVec", "[", RowBox[{"Matx", ",", "k", ",", "LenSeq"}], "]"}]}], ";", RowBox[{"Vec3", "=", RowBox[{"LastVec", "[", RowBox[{"Matx", ",", "k", ",", "LenSeq"}], "]"}]}], ";", RowBox[{"LSK", "=", RowBox[{ RowBox[{ RowBox[{"(", RowBox[{ RowBox[{"Vec3", "[", RowBox[{"[", "2", "]"}], "]"}], "-", RowBox[{"Vec2", "[", RowBox[{"[", "2", "]"}], "]"}]}], ")"}], "/", RowBox[{"Vec1", "[", RowBox[{"[", "2", "]"}], "]"}]}], "+", "1"}]}], ";", RowBox[{"Print", "[", "\"\< \!\(\* StyleBox[\"\[CapitalPhi]\[Omicron]\[Iota]\[Nu]\[Iota]\[Xi]\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\"-\", FontFamily->\"Symbol\", FontSize->18]\)\!\(\* StyleBox[\" \", FontFamily->\"Symbol\", FontSize->18]\)The Great Crimson and Golden Bird Soars Above\>\"", "]"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"LSK", "\[Equal]", "1"}], ",", RowBox[{ RowBox[{"Print", "[", RowBox[{"\"\< Single Vector = \>\"", ",", "Vec2"}], "]"}], ";", RowBox[{"Return", "[", "]"}]}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\< First Vector = \>\"", ",", "Vec2"}], "]"}], ";", RowBox[{"Print", "[", RowBox[{"\"\< Incrementing by: \>\"", ",", "Vec1"}], "]"}], ";", RowBox[{"Print", "[", RowBox[{"\"\< Last Vector = \>\"", ",", "Vec3"}], "]"}], ";", RowBox[{"Print", "[", RowBox[{ "\"\< Number of Vectors in Partial Sequence = \>\"", ",", "LSK"}], "]"}], ";", RowBox[{"Return", "[", "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"FirstVec", "[", RowBox[{"Matx_", ",", "k_", ",", "LenSeq_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ "Returns", " ", "the", " ", "first", " ", "vector", " ", "in", " ", "Sequence", " ", RowBox[{"No", ".", " ", "k", "."}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "Col", "}"}], ",", RowBox[{ RowBox[{"Col", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"k", "-", "1"}]}], "}"}]}], "]"}]}], ";", RowBox[{"Return", "[", RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", RowBox[{"Col", "+", "1"}]}], "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", RowBox[{"Col", "+", "1"}]}], "]"}], "]"}]}], "}"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"LastVec", "[", RowBox[{"Matx_", ",", "k_", ",", "LenSeq_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ "Returns", " ", "the", " ", "last", " ", "vector", " ", "in", " ", "Sequence", " ", RowBox[{"No", ".", " ", "k", "."}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "Col", "}"}], ",", RowBox[{ RowBox[{"Col", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "k"}], "}"}]}], "]"}]}], ";", RowBox[{"Return", "[", RowBox[{"{", RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "Col"}], "]"}], "]"}], ",", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "Col"}], "]"}], "]"}]}], "}"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Victor", "[", RowBox[{"Matx_", ",", "n_", ",", "k_", ",", "LenSeqMtx_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"What", "'"}], "s", " ", "your", " ", "vector"}], ",", " ", RowBox[{ RowBox[{"Victor", "?", " ", "In"}], " ", "Matx"}], ",", " ", RowBox[{ RowBox[{"the", " ", "n"}], "-", RowBox[{"th", " ", "vector", " ", "in", " ", RowBox[{"P", ".", "S", ".", " ", "k", "."}]}]}]}], " ", "*)"}], " ", RowBox[{"(", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{"n", ">", RowBox[{"LenSeqMtx", "[", RowBox[{"[", "k", "]"}], "]"}]}], ",", RowBox[{"Return", "[", "\"\\"", "]"}]}], "]"}], ";", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"V1", ",", "V2"}], "}"}], ",", RowBox[{ RowBox[{"V1", "=", RowBox[{"LastVec", "[", RowBox[{"Matx", ",", RowBox[{"k", "-", "1"}], ",", "LenSeqMtx"}], "]"}]}], ";", RowBox[{"V2", "=", RowBox[{"FirstVec", "[", RowBox[{"Matx", ",", "k", ",", "LenSeqMtx"}], "]"}]}], ";", RowBox[{"Return", "[", RowBox[{"V2", "+", RowBox[{ RowBox[{"(", RowBox[{"n", "-", "1"}], ")"}], "*", "V1"}]}], "]"}]}]}], "]"}]}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"IndexSeqWithLength", "[", RowBox[{"LenSeq_", ",", "Len_"}], "]"}], ":=", RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "k", "}"}], ",", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{"If", "[", RowBox[{ RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "i", "]"}], "]"}], "\[Equal]", "Len"}], ",", RowBox[{ RowBox[{"k", "=", "i"}], ";", RowBox[{"Goto", "[", "fin", "]"}]}]}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"Length", "[", "LenSeq", "]"}]}], "}"}]}], "]"}], ";", RowBox[{"Return", "[", "\"\\"", "]"}], ";", RowBox[{"Label", "[", "fin", "]"}], ";", RowBox[{"Return", "[", "k", "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"TestPolRev", "[", RowBox[{"Matx_", ",", "IdxLS_", ",", "target_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Tests", " ", "for", " ", "Polarity", " ", RowBox[{"Reversal", ".", " ", "IdxLS"}], " ", "is", " ", "the", " ", "index", " ", "of", " ", "the", " ", "last", " ", "sequence"}], ",", "\[IndentingNewLine]", "thus", ",", " ", RowBox[{"IdxLS", " ", "-", " ", RowBox[{ "1", " ", "is", " ", "the", " ", "index", " ", "of", " ", "the", " ", "sequence", " ", "before", " ", "the", " ", "last", " ", RowBox[{"one", "."}]}]}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"Colm1", ",", "Colm2", ",", "test"}], "}"}], ",", RowBox[{ RowBox[{"Colm2", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", "IdxLS"}], "}"}]}], "]"}]}], ";", RowBox[{"Colm1", "=", RowBox[{"Sum", "[", RowBox[{ RowBox[{"LenSeq", "[", RowBox[{"[", "i", "]"}], "]"}], ",", RowBox[{"{", RowBox[{"i", ",", RowBox[{"IdxLS", "-", "1"}]}], "}"}]}], "]"}]}], ";", RowBox[{"test", "=", RowBox[{"Sign", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "Colm2"}], "]"}], "]"}], "/", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "Colm2"}], "]"}], "]"}]}], "-", "target"}], ")"}], "*", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "Colm1"}], "]"}], "]"}], "/", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "Colm1"}], "]"}], "]"}]}], "-", "target"}], ")"}]}], "]"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"test", "<", "0"}], ",", RowBox[{"Return", "[", "False", "]"}], ",", RowBox[{"Return", "[", "True", "]"}]}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"RatMat", "[", RowBox[{"Matx_", ",", "col_", ",", "dp_"}], "]"}], ":=", RowBox[{"(", RowBox[{"N", "[", RowBox[{ RowBox[{"Abs", "[", RowBox[{ RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "col"}], "]"}], "]"}], "/", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "col"}], "]"}], "]"}]}], "-", "target"}], "]"}], ",", "dp"}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"RatMat2", "[", RowBox[{"Matx_", ",", "col_", ",", "dp_"}], "]"}], ":=", RowBox[{"(*", " ", RowBox[{ RowBox[{ "This", " ", "returns", " ", "the", " ", "ratio", " ", "of", " ", "the", " ", "accuracy", " ", "of", " ", "Matx", " ", "in", " ", "column"}], ",", " ", "col", ",", " ", RowBox[{"to", " ", "its", " ", "previous", " ", "column"}], ",", " ", RowBox[{"col", " ", "-", " ", RowBox[{ "1.", " ", "The", " ", "accuracy", " ", "is", " ", "computed", " ", "to", " ", "dp", " ", "decimal", " ", RowBox[{"places", "."}]}]}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"(", RowBox[{"N", "[", RowBox[{ RowBox[{"Abs", "[", RowBox[{ RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", RowBox[{"col", "-", "1"}]}], "]"}], "]"}], "/", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", RowBox[{"col", "-", "1"}]}], "]"}], "]"}]}], "-", "target"}], ")"}], "/", RowBox[{"(", RowBox[{ RowBox[{ RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"1", ",", "col"}], "]"}], "]"}], "/", RowBox[{"Matx", "[", RowBox[{"[", RowBox[{"2", ",", "col"}], "]"}], "]"}]}], "-", "target"}], ")"}]}], "]"}], ",", "dp"}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{"Test", "[", RowBox[{"u_", ",", "v_", ",", "target_"}], "]"}], ":=", RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", "w", "}"}], ",", RowBox[{ RowBox[{"w", "=", RowBox[{"u", "+", "v"}]}], ";", RowBox[{"VecPrm2", "[", RowBox[{"w", ",", "target"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"PhoenixNoBird", "[", RowBox[{"RealNo_", ",", "k_", ",", "dp_"}], "]"}], ":=", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{"Key", " ", "to", " ", "discover", " ", "the", " ", SuperscriptBox[ StyleBox["k", FontVariations->{"Underline"->True}], "th"], " ", RowBox[{"P", ".", "S", ".", " ", "for"}], " ", "any", " ", "real", " ", "number"}], ",", " ", RowBox[{"with", " ", StyleBox["dp", FontVariations->{"Underline"->True}], " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}], RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "CrimsonAsh", ",", "GoldenAsh", ",", "Joaquin", ",", "NewLeaf", ",", "JoaquinPhoenix", ",", "LSK", ",", "NumSeq", ",", "target", ",", "best", ",", "test", ",", "n", ",", "sgn", ",", "LenSeq"}], "}"}], ",", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{ RowBox[{"CrimsonAsh", "=", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], ";", RowBox[{"(*", " ", RowBox[{"Let", " ", "the", " ", "Force", " ", RowBox[{"(", RowBox[{"1", "/", "0"}], ")"}], " ", "be", " ", "with", " ", RowBox[{"you", ":", " ", RowBox[{ RowBox[{"Floor", "[", "RealNo", "]"}], " ", "<", " ", "RealNo", " ", "<", " ", RowBox[{"\[Infinity]", "."}]}]}]}], " ", "*)"}], " ", RowBox[{"GoldenAsh", "=", RowBox[{"{", RowBox[{ RowBox[{"Floor", "[", "RealNo", "]"}], ",", "1"}], "}"}]}], ";", RowBox[{"NumSeq", "=", "1"}], ";", " ", RowBox[{"If", "[", RowBox[{ RowBox[{"k", "\[Equal]", "1"}], ",", RowBox[{"Return", "[", RowBox[{"{", RowBox[{"GoldenAsh", ",", "1"}], "}"}], "]"}]}], "]"}], ";", RowBox[{"target", "=", RowBox[{"N", "[", RowBox[{"RealNo", ",", RowBox[{"Max", "[", RowBox[{"dp", ",", RowBox[{"4", "*", "k"}]}], "]"}]}], "]"}]}], ";", RowBox[{"best", "=", RowBox[{"VecPrm2", "[", RowBox[{"GoldenAsh", ",", "target"}], "]"}]}], ";", RowBox[{"LenSeq", "=", RowBox[{"{", "1", "}"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Label", "[", "Ra", "]"}], ";", RowBox[{"LSK", "=", "1"}], ";", RowBox[{"n", "=", "1"}], ";", RowBox[{"sgn", "=", "1"}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"CrimsonAsh", "\[NotEqual]", RowBox[{"{", RowBox[{"1", ",", "0"}], "}"}]}], ",", RowBox[{"sgn", "=", RowBox[{"-", RowBox[{"Sign", "[", RowBox[{ RowBox[{"VecPrm", "[", RowBox[{"CrimsonAsh", ",", "target"}], "]"}], "*", RowBox[{"VecPrm", "[", RowBox[{"GoldenAsh", ",", "target"}], "]"}]}], "]"}]}]}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Label", "[", "Heliopolis", "]"}], ";", RowBox[{"n", "=", RowBox[{"n", "+", "1"}]}], ";", RowBox[{"Joaquin", "=", RowBox[{ RowBox[{"n", "*", "GoldenAsh"}], "+", RowBox[{"sgn", "*", "CrimsonAsh"}]}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"Joaquin", ",", "target"}], "]"}]}], ";", RowBox[{"(*", " ", RowBox[{"Discovery", " ", "Loop"}], " ", "*)"}], " ", RowBox[{"If", "[", RowBox[{ RowBox[{"test", ">", "best"}], ",", RowBox[{"(*", " ", RowBox[{ RowBox[{"Sojourn", " ", "back", " ", "to", " ", "Sun", " ", RowBox[{"City", ".", " ", "In"}], " ", "order", " ", "to", " ", "move", " ", "forward"}], ",", " ", RowBox[{ "Joaquin", " ", "must", " ", "be", " ", "closer", " ", "to", " ", "his", " ", "target", " ", "than", " ", "he", " ", "was", " ", RowBox[{"before", "."}]}]}], " ", "*)"}], " ", RowBox[{"Goto", "[", "Heliopolis", "]"}]}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Otherwise", ",", " ", RowBox[{ "Joaquin", " ", "turns", " ", "over", " ", "a", " ", "New", " ", RowBox[{"Leaf", "."}]}]}], " ", "*)"}], " ", RowBox[{"NewLeaf", "=", "Joaquin"}], ";", RowBox[{"CrimsonAsh", "=", "GoldenAsh"}], ";", RowBox[{"GoldenAsh", "=", "NewLeaf"}], ";", RowBox[{"best", "=", "test"}], ";", " ", RowBox[{"Label", "[", "FrankincenseAndMyrrh", "]"}], ";", RowBox[{"Joaquin", "=", RowBox[{"GoldenAsh", "+", "CrimsonAsh"}]}], ";", RowBox[{"test", "=", RowBox[{"VecPrm2", "[", RowBox[{"Joaquin", ",", "target"}], "]"}]}], ";", RowBox[{"(*", " ", RowBox[{"Recursive", " ", "Loop"}], " ", "*)"}], RowBox[{"If", "[", RowBox[{ RowBox[{"test", "<", "best"}], ",", RowBox[{ RowBox[{"best", "=", "test"}], ";", RowBox[{"LSK", "=", RowBox[{"LSK", "+", "1"}]}], ";", RowBox[{"GoldenAsh", "=", "Joaquin"}], ";", RowBox[{"Goto", "[", "FrankincenseAndMyrrh", "]"}]}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{ "The", " ", "Spices", " ", "have", " ", "been", " ", "prepared", " ", "for", " ", "the", " ", "Nest", " ", "of", " ", "the", " ", "Golden", " ", "Fire", " ", RowBox[{"Bird", "."}]}], " ", "*)"}], "\[IndentingNewLine]", RowBox[{"LenSeq", "=", RowBox[{"Append", "[", RowBox[{"LenSeq", ",", "LSK"}], "]"}]}], ";", RowBox[{"NumSeq", "=", RowBox[{"NumSeq", "+", "1"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"NumSeq", "<", "k"}], ",", RowBox[{"(*", " ", RowBox[{ "Return", " ", "to", " ", "Egyptian", " ", "Sun", " ", "God"}], " ", "*)"}], " ", RowBox[{"Goto", "[", "Ra", "]"}]}], "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"(*", " ", RowBox[{"Eureka", ",", " ", RowBox[{ RowBox[{ RowBox[{ "the", " ", "Great", " ", "Bird", " ", "flies", " ", "again", " ", "as", " ", "the", " ", SuperscriptBox[ StyleBox["k", FontVariations->{"Underline"->True}], "th"], " ", RowBox[{"P", ".", "S", "."}]}], ":", " ", SubscriptBox["S", "k"]}], " ", "=", " ", RowBox[{"{", RowBox[{ SubscriptBox["v", "2"], " ", "+", " ", RowBox[{ RowBox[{"(", RowBox[{"N", " ", "-", " ", "1"}], ")"}], SubscriptBox["v", "1"]}]}], "}"}]}]}], " ", "*)"}], " ", RowBox[{"JoaquinPhoenix", "=", RowBox[{"{", "NewLeaf", "}"}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"NewLeaf", "=", RowBox[{"NewLeaf", "+", "CrimsonAsh"}]}], ";", RowBox[{"JoaquinPhoenix", "=", RowBox[{"Append", "[", RowBox[{"JoaquinPhoenix", ",", "NewLeaf"}], "]"}]}]}], ",", RowBox[{"{", RowBox[{"var", ",", RowBox[{"LSK", "-", "1"}]}], "}"}]}], "]"}], ";", RowBox[{"Return", "[", RowBox[{"Append", "[", RowBox[{"JoaquinPhoenix", ",", "LenSeq"}], "]"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"VishnusVecs", "[", RowBox[{ "Seq1_", ",", "Indx1_", ",", "Seq2_", ",", "Indx2_", ",", "target_"}], "]"}], ":=", RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{ "u", ",", "v", ",", "m", ",", "z", ",", "LenPS1", ",", "LenPS2", ",", "PartSeq1", ",", "PartSeq2", ",", "OL"}], "}"}], ",", RowBox[{ RowBox[{"PartSeq1", "=", RowBox[{"PhoenixNoBird", "[", RowBox[{"target", ",", "Seq1", ",", "1000"}], "]"}]}], ";", RowBox[{"LenPS1", "=", RowBox[{ RowBox[{"Length", "[", "PartSeq1", "]"}], "-", "1"}]}], ";", RowBox[{ "OL", "=", "\"\\""}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"Indx1", ">", "LenPS1"}], ",", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", "Seq1", ",", "\"\<, only has \>\"", ",", "LenPS1", ",", "\"\< vectors.\>\""}], "]"}], ";", RowBox[{"Return", "[", "OL", "]"}]}]}], "]"}], ";", RowBox[{"PartSeq2", "=", RowBox[{"PhoenixNoBird", "[", RowBox[{"target", ",", "Seq2", ",", "1000"}], "]"}]}], ";", RowBox[{"LenPS2", "=", RowBox[{ RowBox[{"Length", "[", "PartSeq2", "]"}], "-", "1"}]}], ";", RowBox[{"If", "[", RowBox[{ RowBox[{"Indx2", ">", "LenPS2"}], ",", RowBox[{ RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", "Seq2", ",", "\"\<, only has \>\"", ",", "LenPS2", ",", "\"\< vectors.\>\""}], "]"}], ";", RowBox[{"Return", "[", "OL", "]"}]}]}], "]"}], ";", RowBox[{"u", "=", RowBox[{"PartSeq1", "[", RowBox[{"[", "Indx1", "]"}], "]"}]}], ";", RowBox[{"v", "=", RowBox[{"PartSeq2", "[", RowBox[{"[", "Indx2", "]"}], "]"}]}], ";", RowBox[{"m", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"u", ",", "v"}], "}"}], "]"}]}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"uu", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"u", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"u", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "}"}]}], ";", RowBox[{"vv", "=", RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], "}"}], ",", RowBox[{"{", RowBox[{"v", "[", RowBox[{"[", "2", "]"}], "]"}], "}"}]}], "}"}]}], ";", RowBox[{ "Print", "[", "\"\\"", "]"}], ";", "\[IndentingNewLine]", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"MatrixForm", "[", "uu", "]"}], ",", "\"\< and \>\"", ",", RowBox[{"MatrixForm", "[", "vv", "]"}]}], "]"}], ";", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"MatrixForm", "[", "m", "]"}]}], "]"}], ";", RowBox[{"Print", "[", RowBox[{ "\"\\"", ",", RowBox[{"Det", "[", "m", "]"}]}], "]"}], ";", RowBox[{"z", "=", RowBox[{"Eigenvalues", "[", "m", "]"}]}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"N", "[", RowBox[{ RowBox[{ RowBox[{"z", "[", RowBox[{"[", "1", "]"}], "]"}], "*", RowBox[{"z", "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", "55"}], "]"}]}], "]"}], ";", RowBox[{ "Print", "[", "\"\\"", "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"N", "[", RowBox[{"z", ",", "33"}], "]"}], "]"}], ";", "\[IndentingNewLine]", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{"N", "[", RowBox[{ RowBox[{ RowBox[{"z", "[", RowBox[{"[", "1", "]"}], "]"}], "+", RowBox[{"z", "[", RowBox[{"[", "2", "]"}], "]"}]}], ",", "33"}], "]"}]}], "]"}], ";", RowBox[{"Print", "[", RowBox[{"\"\\"", ",", RowBox[{ RowBox[{"m", "[", RowBox[{"[", RowBox[{"1", ",", "1"}], "]"}], "]"}], "+", RowBox[{"m", "[", RowBox[{"[", RowBox[{"2", ",", "2"}], "]"}], "]"}]}]}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{ RowBox[{ RowBox[{"VishnusDet", "[", RowBox[{ "Seq1_", ",", "Indx1_", ",", "Seq2_", ",", "Indx2_", ",", "target_"}], "]"}], ":=", RowBox[{"(", RowBox[{"Module", "[", RowBox[{ RowBox[{"{", RowBox[{"u", ",", "v", ",", "m"}], "}"}], ",", RowBox[{ RowBox[{"u", "=", RowBox[{ RowBox[{"PhoenixNoBird", "[", RowBox[{"target", ",", "Seq1", ",", "1000"}], "]"}], "[", RowBox[{"[", "Indx1", "]"}], "]"}]}], ";", RowBox[{"v", "=", RowBox[{ RowBox[{"PhoenixNoBird", "[", RowBox[{"target", ",", "Seq2", ",", "1000"}], "]"}], "[", RowBox[{"[", "Indx2", "]"}], "]"}]}], ";", RowBox[{"m", "=", RowBox[{"Transpose", "[", RowBox[{"{", RowBox[{"u", ",", "v"}], "}"}], "]"}]}], ";", RowBox[{"Return", "[", RowBox[{"Det", "[", "m", "]"}], "]"}]}]}], "]"}], ")"}]}], ";"}], "\[IndentingNewLine]"}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Two", " ", "irrational", " ", "constants", " ", "are", " ", "set"}], ",", " ", RowBox[{"first", " ", RowBox[{"Euler", "'"}], " ", "little", " ", "gamma"}]}], " ", "*)"}], " ", RowBox[{ RowBox[{"\[Gamma]", "=", "EulerGamma"}], ";"}]}], "\[IndentingNewLine]", RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{"\"\\"", " ", "Edwards"}], ",", " ", RowBox[{ RowBox[{"p", ".67"}], ":", " ", RowBox[{"The", " ", "infinite", " ", "sum"}]}], ",", " ", RowBox[{ "where", " ", "\[Rho]", " ", "represents", " ", "an", " ", "imaginary", " ", "zero", "\[IndentingNewLine]", " ", "of", " ", "the", " ", "Zeta", " ", "Fcn"}], ",", " ", RowBox[{ RowBox[{"\[CapitalSigma]", " ", RowBox[{"1", "/", "\[Rho]"}]}], " ", "="}]}], " ", "*)"}], " ", RowBox[{ RowBox[{"RiemannsGarden", "=", RowBox[{"1", "+", " ", RowBox[{ FractionBox["1", "2"], "\[Gamma]"}], " ", "+", " ", RowBox[{"Log", "[", " ", RowBox[{ FractionBox["1", "2"], FractionBox["1", RowBox[{"Gamma", "[", FractionBox["1", "2"], "]"}]]}], " ", "]"}]}]}], ";", " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "The", " ", "dueces", " ", "are", " ", "wild", " ", "in", " ", "this", " ", RowBox[{"constant", ".", " ", "RH"}], " ", "is", " ", "half", " ", "full", " ", RowBox[{"of", " ", "'"}], "em"}], ",", " ", RowBox[{"by", " ", RowBox[{"Gamma", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]"}]}]}], "Input", CellChangeTimes->{{3.47979829340625*^9, 3.47979830109375*^9}, { 3.47979835090625*^9, 3.479798357328125*^9}, {3.479798416234375*^9, 3.4797984408125*^9}, {3.4797986314375*^9, 3.479798671765625*^9}, { 3.479798810203125*^9, 3.479798894578125*^9}, {3.479798929390625*^9, 3.47979898775*^9}, {3.479799087359375*^9, 3.479799182484375*^9}, { 3.479799252796875*^9, 3.4797993815625*^9}, {3.479799414375*^9, 3.47979948853125*^9}, {3.47979968865625*^9, 3.47979971978125*^9}, 3.4798000083125*^9, {3.47980022*^9, 3.479800239890625*^9}, { 3.479810451921875*^9, 3.47981049803125*^9}, {3.47981091259375*^9, 3.479810942765625*^9}, {3.48015761665625*^9, 3.480157647734375*^9}, { 3.480160350890625*^9, 3.480160392859375*^9}, {3.4801604660625*^9, 3.48016047284375*^9}, {3.480160524640625*^9, 3.480160573015625*^9}, { 3.480160608734375*^9, 3.480160673*^9}, {3.48016071515625*^9, 3.48016079053125*^9}, {3.4801608273125*^9, 3.480160922*^9}, { 3.480160962328125*^9, 3.480161062296875*^9}, {3.480161146*^9, 3.480161152609375*^9}, {3.480161849953125*^9, 3.480161850125*^9}, { 3.480161951875*^9, 3.480161952390625*^9}, {3.480766463515625*^9, 3.48076660525*^9}, {3.4807747931875*^9, 3.4807748245*^9}, { 3.48077534015625*^9, 3.48077534471875*^9}, {3.480777739609375*^9, 3.480777742953125*^9}, {3.480892894515625*^9, 3.480892899359375*^9}, { 3.480968945234375*^9, 3.480968945296875*^9}, {3.4809696099375*^9, 3.480969669109375*^9}, {3.480969958484375*^9, 3.480969960640625*^9}, { 3.4809699938125*^9, 3.480970003015625*^9}, {3.48097003371875*^9, 3.480970043078125*^9}, {3.480971257171875*^9, 3.48097131921875*^9}, { 3.48097137409375*^9, 3.480971398453125*^9}, {3.480971473*^9, 3.480971497421875*^9}, {3.480971620453125*^9, 3.4809716371875*^9}, { 3.480971678921875*^9, 3.4809718385625*^9}, {3.480971875828125*^9, 3.480971958484375*^9}, {3.480972067515625*^9, 3.4809720770625*^9}, { 3.480972115265625*^9, 3.48097211590625*^9}, {3.4809721715625*^9, 3.48097217603125*^9}, {3.48097261928125*^9, 3.480972785875*^9}, { 3.480972818609375*^9, 3.4809728194375*^9}, {3.48097287621875*^9, 3.480972916*^9}, {3.480972959984375*^9, 3.480973050234375*^9}, { 3.4809731580625*^9, 3.480973172765625*^9}, {3.48097320684375*^9, 3.480973220671875*^9}, 3.480973321765625*^9, {3.48097339953125*^9, 3.480973403203125*^9}, 3.480973441375*^9, {3.48097353853125*^9, 3.48097356259375*^9}, {3.48097377334375*^9, 3.4809737754375*^9}, { 3.480973971328125*^9, 3.480973975328125*^9}, {3.480974223015625*^9, 3.48097423296875*^9}, {3.48097441571875*^9, 3.480974423734375*^9}, { 3.480974569796875*^9, 3.4809746773125*^9}, {3.480974747203125*^9, 3.480974747921875*^9}, {3.480974911640625*^9, 3.480974945046875*^9}, { 3.480974995421875*^9, 3.480975005921875*^9}, {3.480975037796875*^9, 3.480975222359375*^9}, 3.48097546853125*^9, {3.48097551678125*^9, 3.480975548625*^9}, 3.48097577853125*^9, {3.48097611503125*^9, 3.48097613821875*^9}, {3.48129206665625*^9, 3.481292116625*^9}, { 3.481292182359375*^9, 3.481292191984375*^9}, {3.481292365171875*^9, 3.48129244421875*^9}, {3.481293088625*^9, 3.4812930898125*^9}, { 3.481334931921875*^9, 3.481334932546875*^9}, {3.481341583765625*^9, 3.481341587796875*^9}, {3.48134471796875*^9, 3.481344809484375*^9}, 3.48134545078125*^9, {3.481345837484375*^9, 3.48134583965625*^9}, { 3.481405086859375*^9, 3.481405088828125*^9}, {3.481405167890625*^9, 3.481405176046875*^9}, {3.48140524790625*^9, 3.48140526825*^9}, { 3.4814056444375*^9, 3.481405654109375*^9}, {3.481405704015625*^9, 3.48140573659375*^9}, {3.481405790765625*^9, 3.481405803875*^9}, 3.481406911703125*^9, {3.4814069455625*^9, 3.48140706046875*^9}, { 3.48140714928125*^9, 3.481407351265625*^9}, {3.481407405546875*^9, 3.48140741065625*^9}, {3.481407529890625*^9, 3.4814075595625*^9}, { 3.4814076316875*^9, 3.48140763315625*^9}, {3.481415051984375*^9, 3.48141506296875*^9}, {3.481417660484375*^9, 3.48141770565625*^9}, { 3.4814177391875*^9, 3.48141782103125*^9}, {3.481418072390625*^9, 3.481418099765625*^9}, {3.48141814521875*^9, 3.48141817428125*^9}, { 3.481418332640625*^9, 3.481418346828125*^9}, {3.481418471890625*^9, 3.48141849334375*^9}, {3.481418523515625*^9, 3.48141862584375*^9}, { 3.48141898575*^9, 3.48141903584375*^9}, {3.481450383453125*^9, 3.481450390203125*^9}, {3.48145275046875*^9, 3.48145277265625*^9}, { 3.48145313753125*^9, 3.4814531483125*^9}, {3.48145329765625*^9, 3.481453298125*^9}, {3.481455770359375*^9, 3.481455790921875*^9}, { 3.48145591496875*^9, 3.4814559293125*^9}, {3.48145597540625*^9, 3.48145603775*^9}, {3.4814561331875*^9, 3.481456146234375*^9}, { 3.48149665990625*^9, 3.481496667625*^9}, {3.481501426171875*^9, 3.48150145503125*^9}, {3.48150157475*^9, 3.481501584140625*^9}, { 3.48150200846875*^9, 3.48150202671875*^9}, {3.4815054363125*^9, 3.481505438765625*^9}, {3.48154095478125*^9, 3.48154097415625*^9}, { 3.48154102196875*^9, 3.481541051859375*^9}, 3.481541544453125*^9, { 3.481541826*^9, 3.48154182840625*^9}, {3.481541942078125*^9, 3.481541988984375*^9}, {3.481542021171875*^9, 3.48154211309375*^9}, { 3.481542179234375*^9, 3.481542210515625*^9}, {3.481577352109375*^9, 3.4815773973125*^9}, {3.481577435390625*^9, 3.48157744209375*^9}, { 3.481599860453125*^9, 3.481599963984375*^9}, {3.481628143875*^9, 3.481628160125*^9}, {3.481629847703125*^9, 3.481629864828125*^9}, { 3.4816298996875*^9, 3.48162994803125*^9}, {3.4816300655625*^9, 3.481630082640625*^9}, {3.4816314189375*^9, 3.481631419390625*^9}, { 3.4816320375625*^9, 3.481632037734375*^9}, {3.4816331784375*^9, 3.481633208515625*^9}, {3.48163336209375*^9, 3.481633382234375*^9}, { 3.481633452140625*^9, 3.48163345559375*^9}, {3.48163743175*^9, 3.481637502234375*^9}, {3.481637558140625*^9, 3.481637655921875*^9}, { 3.481637862421875*^9, 3.481637874359375*^9}, {3.4816379175625*^9, 3.481637969921875*^9}, {3.481638001859375*^9, 3.48163809584375*^9}, { 3.48163812665625*^9, 3.481638144953125*^9}, {3.481638177578125*^9, 3.481638193*^9}, {3.481638398234375*^9, 3.481638405390625*^9}, { 3.481638496453125*^9, 3.481638504359375*^9}, {3.48163914309375*^9, 3.481639203984375*^9}, {3.4816392535625*^9, 3.48163926053125*^9}, { 3.481639351578125*^9, 3.48163943240625*^9}, {3.481639531875*^9, 3.481639542625*^9}, {3.48163994159375*^9, 3.48164003109375*^9}, { 3.481640125953125*^9, 3.481640454609375*^9}, 3.481641378421875*^9, { 3.481641426015625*^9, 3.481641428609375*^9}, 3.48164288240625*^9, { 3.4816726605625*^9, 3.481672681390625*^9}, {3.481674297*^9, 3.481674304703125*^9}, {3.481677583140625*^9, 3.48167767125*^9}, { 3.48167774628125*^9, 3.48167774846875*^9}, {3.48167789996875*^9, 3.481677922890625*^9}, {3.481678015453125*^9, 3.48167802553125*^9}, { 3.481678093796875*^9, 3.48167813884375*^9}, {3.48167866759375*^9, 3.481678727546875*^9}, {3.481678765875*^9, 3.481678787625*^9}, { 3.48167897396875*^9, 3.481679004296875*^9}, {3.481679098921875*^9, 3.481679134390625*^9}, 3.481679165375*^9, {3.481685114359375*^9, 3.481685140515625*^9}, {3.481822043953125*^9, 3.481822069703125*^9}, { 3.481822828015625*^9, 3.481822839796875*^9}, {3.48182552221875*^9, 3.4818258034375*^9}, {3.481825835671875*^9, 3.481826031609375*^9}, { 3.481850459875*^9, 3.48185048878125*^9}, {3.48185054946875*^9, 3.481850550078125*^9}, {3.4818573234375*^9, 3.481857385046875*^9}, { 3.48185759559375*^9, 3.4818575963125*^9}, {3.481904131078125*^9, 3.481904189578125*^9}, {3.48190422515625*^9, 3.481904269265625*^9}, { 3.48190441259375*^9, 3.48190442325*^9}, {3.481925106421875*^9, 3.4819251631875*^9}, {3.481925209296875*^9, 3.4819252150625*^9}, { 3.48193682903125*^9, 3.48193694775*^9}, 3.481937299890625*^9, { 3.4820535023125*^9, 3.482053535109375*^9}, {3.482054193859375*^9, 3.482054215890625*^9}, 3.482054672953125*^9, {3.482056188453125*^9, 3.482056194046875*^9}, {3.482056284296875*^9, 3.482056297265625*^9}, { 3.48205666378125*^9, 3.48205668603125*^9}, {3.482056941203125*^9, 3.482057008234375*^9}, {3.48205704265625*^9, 3.482057046*^9}, { 3.482057167125*^9, 3.48205718725*^9}, {3.48205721815625*^9, 3.48205722903125*^9}, {3.482058151546875*^9, 3.482058155578125*^9}, { 3.482105518046875*^9, 3.482105696265625*^9}, {3.482105765484375*^9, 3.482105766671875*^9}, {3.48210581571875*^9, 3.482105897859375*^9}, { 3.482105953453125*^9, 3.48210610309375*^9}, {3.482106173609375*^9, 3.4821062168125*^9}, {3.4821062865*^9, 3.482106306015625*^9}, { 3.482106421078125*^9, 3.48210649396875*^9}, 3.482106725796875*^9, { 3.48210678815625*^9, 3.4821067915625*^9}, {3.482106828078125*^9, 3.4821068340625*^9}, 3.48211076115625*^9, {3.48211385328125*^9, 3.482113871296875*^9}, {3.482114015140625*^9, 3.48211402615625*^9}, { 3.482147580328125*^9, 3.482147676765625*^9}, {3.482147789875*^9, 3.482147800875*^9}, {3.482147874125*^9, 3.482147898*^9}, { 3.48214812915625*^9, 3.482148142296875*^9}, {3.482148206734375*^9, 3.4821482319375*^9}, 3.482148336359375*^9, {3.482149762*^9, 3.48214976275*^9}, {3.482167857109375*^9, 3.482167875765625*^9}, { 3.4821680260625*^9, 3.4821680960625*^9}, {3.4821684865*^9, 3.482168506546875*^9}, {3.48217284109375*^9, 3.482172841828125*^9}, { 3.482172936484375*^9, 3.482172943171875*^9}, {3.4821748469375*^9, 3.48217489134375*^9}, {3.482175137296875*^9, 3.482175161640625*^9}, { 3.48217885440625*^9, 3.4821790559375*^9}, {3.482179523375*^9, 3.48217953146875*^9}, {3.4821797969375*^9, 3.482179813578125*^9}, { 3.48218406975*^9, 3.482184070921875*^9}, {3.482184101078125*^9, 3.482184114796875*^9}, {3.4821844439375*^9, 3.482184535609375*^9}, { 3.4821879178125*^9, 3.482187918546875*^9}, {3.482195781546875*^9, 3.482195824203125*^9}, {3.482195874734375*^9, 3.482196026796875*^9}, { 3.482196086765625*^9, 3.48219609053125*^9}, {3.482196176828125*^9, 3.482196201671875*^9}, {3.4822344388125*^9, 3.4822344525*^9}, { 3.482234580453125*^9, 3.482234696265625*^9}, {3.482234729078125*^9, 3.482234753734375*^9}, {3.48223480640625*^9, 3.482234808140625*^9}, { 3.48223484034375*^9, 3.482234905671875*^9}, {3.482234949734375*^9, 3.482234957234375*^9}, {3.48223504078125*^9, 3.482235055890625*^9}, { 3.48223607278125*^9, 3.482236091453125*^9}, {3.482236184984375*^9, 3.482236279546875*^9}, {3.4822363215*^9, 3.482236349890625*^9}, { 3.48223641028125*^9, 3.48223644446875*^9}, {3.482236531296875*^9, 3.48223656525*^9}, {3.482236639796875*^9, 3.48223674759375*^9}, { 3.4822368193125*^9, 3.48223682715625*^9}, {3.48223691709375*^9, 3.482236925390625*^9}, {3.482238210171875*^9, 3.48223823878125*^9}, { 3.4822382741875*^9, 3.482238277328125*^9}, {3.48245026434375*^9, 3.482450287921875*^9}, {3.4824834255625*^9, 3.482483435203125*^9}, { 3.482483504859375*^9, 3.4824835321875*^9}, {3.48248369659375*^9, 3.482483704671875*^9}, {3.482596275671875*^9, 3.4825962806875*^9}, 3.482596320078125*^9, {3.48259692565625*^9, 3.482596945078125*^9}, { 3.482597167609375*^9, 3.482597168953125*^9}, {3.482597237703125*^9, 3.482597268296875*^9}, {3.4826177931610255`*^9, 3.4826177986610255`*^9}, { 3.482618547707901*^9, 3.4826185523485255`*^9}, {3.482618588989151*^9, 3.482618594895401*^9}, {3.4826249793485255`*^9, 3.4826249807860255`*^9}, { 3.4826250885047755`*^9, 3.482625119207901*^9}, {3.482625613395401*^9, 3.4826256691610255`*^9}, {3.482625748551651*^9, 3.4826257542860255`*^9}, { 3.482625916770401*^9, 3.482625936239151*^9}, 3.4826278377860255`*^9, { 3.4826282230047755`*^9, 3.4826282369735255`*^9}, {3.4826283296297755`*^9, 3.4826283587860255`*^9}, {3.482628392707901*^9, 3.4826283960985255`*^9}, { 3.4826343912547755`*^9, 3.482634467707901*^9}, {3.4826358879735255`*^9, 3.4826358934110255`*^9}, {3.4826396896610255`*^9, 3.482639690895401*^9}, { 3.482639734239151*^9, 3.4826397357235255`*^9}, {3.48270895628125*^9, 3.482708994140625*^9}, {3.482746365125*^9, 3.4827463663125*^9}, { 3.482746972078125*^9, 3.482746973296875*^9}, {3.48274701615625*^9, 3.482747028328125*^9}, {3.482799870140625*^9, 3.482800029296875*^9}, 3.482800060046875*^9, {3.4828002190625*^9, 3.482800219671875*^9}, 3.482800582234375*^9, {3.482800627671875*^9, 3.482800776890625*^9}, { 3.482800880703125*^9, 3.482800890578125*^9}, {3.482800988578125*^9, 3.4828009910625*^9}, {3.482803390359375*^9, 3.482803395625*^9}, { 3.4828585130625*^9, 3.48285852484375*^9}, {3.482858652796875*^9, 3.482858698015625*^9}, {3.4828588084375*^9, 3.482858825671875*^9}, { 3.4829161141875*^9, 3.482916120640625*^9}, {3.4830338243125*^9, 3.483033858484375*^9}, 3.48318887775*^9, 3.48318893865625*^9, { 3.483188968890625*^9, 3.48318900096875*^9}, 3.483189089296875*^9, { 3.48318912503125*^9, 3.483189133828125*^9}, {3.483743109984375*^9, 3.483743194703125*^9}, 3.483743503984375*^9, {3.4837435968125*^9, 3.48374359890625*^9}, {3.4837436308125*^9, 3.483743647828125*^9}, { 3.48374368221875*^9, 3.483743709296875*^9}, {3.483743879140625*^9, 3.48374396565625*^9}, {3.483963408734375*^9, 3.4839634126875*^9}, { 3.483975684109375*^9, 3.483975701875*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"st", "=", RowBox[{"AbsoluteTime", "[", "]"}]}], ";", RowBox[{"v", "=", RowBox[{"Phoenix", "[", RowBox[{"\[Pi]", ",", "8000", ",", "8000"}], "]"}]}], ";", RowBox[{ RowBox[{"AbsoluteTime", "[", "]"}], "-", "st"}]}]], "Input", CellChangeTimes->{{3.482692446421875*^9, 3.48269252825*^9}}], Cell[BoxData["484.1875`10.136558566806771"], "Output", CellChangeTimes->{3.482692530109375*^9, 3.48269304565625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "/", "60"}]], "Input", CellChangeTimes->{{3.482708549671875*^9, 3.48270855278125*^9}}], Cell[BoxData["8.0697916666666666667`10.136558566806771"], "Output", CellChangeTimes->{3.482708554546875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Length", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.482708570109375*^9, 3.48270857584375*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.4827085770625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "15000"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "fraction", " ", "to", " ", "the", " ", "right", " ", "of", " ", "\[Pi]"}], ",", " ", RowBox[{"accurate", " ", "to", " ", "11830", " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482708685*^9, 3.48270876121875*^9}, { 3.482708798703125*^9, 3.482708821890625*^9}, {3.48274708290625*^9, 3.482747156390625*^9}}], Cell[BoxData["9.60559359261444190518582588`15.954589770191005*^-11832"], \ "Output", CellChangeTimes->{ 3.4827087019375*^9, {3.48270874984375*^9, 3.4827087619375*^9}, { 3.48270881121875*^9, 3.482708828765625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"StringLength", "[", RowBox[{"ToString", "[", RowBox[{"VP", "[", RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "]"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"This", " ", "is", " ", "a", " ", "boat"}], "-", RowBox[{"load", " ", "of", " ", "digits"}]}], ",", " ", RowBox[{"top", " ", "and", " ", RowBox[{"bottom", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482747748421875*^9, 3.482747755*^9}, { 3.48274783603125*^9, 3.482747853265625*^9}, {3.482881794625*^9, 3.48288182471875*^9}}], Cell[BoxData["17750"], "Output", CellChangeTimes->{3.482747855328125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "15000"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "fraction", " ", "to", " ", "the", " ", "left", " ", "of", " ", "\[Pi]"}], ",", " ", RowBox[{"accurate", " ", "to", " ", "11831", " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482708685*^9, 3.48270876121875*^9}, { 3.482708798703125*^9, 3.482708847265625*^9}, {3.48274717128125*^9, 3.482747185203125*^9}}], Cell[BoxData[ RowBox[{"-", "1.3199838201819569269528474`15.954589770191005*^-11832"}]], "Output", CellChangeTimes->{3.482708848265625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"{", RowBox[{ RowBox[{"Mean", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"StandardDeviation", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"Max", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"Median", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.482708579015625*^9, 3.482708581375*^9}, { 3.48274728328125*^9, 3.48274738915625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "9.852625`", ",", "135.92768094621235`", ",", "10389.`", ",", "3.`"}], "}"}]], "Output", CellChangeTimes->{3.482747393421875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"v", "=", RowBox[{"Phoenix", "[", RowBox[{"\[Pi]", ",", "292", ",", "200"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.4826364721922755`*^9, 3.4826364785672755`*^9}, { 3.482636567489151*^9, 3.482636595176651*^9}, {3.4826366902235255`*^9, 3.4826367115047755`*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.482803590890625*^9, 3.48285892390625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"v", "[", RowBox[{"[", "10390", "]"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "The", " ", "last", " ", "vector", " ", "contains", " ", "the", " ", "lengths", " ", "of", " ", "the", " ", "first", " ", "292", " ", RowBox[{"P", ".", "S", ".", "'"}], "s"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.4826367875047755`*^9, 3.482636807614151*^9}, { 3.482881907703125*^9, 3.482881942671875*^9}, {3.4828820326875*^9, 3.482882060328125*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "43", ",", "2", ",", "1", ",", "8", ",", "2", ",", "8", ",", "4", ",", "1", ",", "4", ",", "4", ",", "51", ",", "3", ",", "1", ",", "4", ",", "2", ",", "4", ",", "1", ",", "4", ",", "5", ",", "5", ",", "2", ",", "2", ",", "5", ",", "3", ",", "1", ",", "7", ",", "2", ",", "3", ",", "1", ",", "5", ",", "1", ",", "2", ",", "5", ",", "1", ",", "3", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "3", ",", "9", ",", "81", ",", "24", ",", "13", ",", "2", ",", "2", ",", "4", ",", "2", ",", "13", ",", "3", ",", "3", ",", "3", ",", "1", ",", "6", ",", "1", ",", "3", ",", "3", ",", "3", ",", "1", ",", "5", ",", "3", ",", "2", ",", "1", ",", "14", ",", "4", ",", "1", ",", "6", ",", "1", ",", "22", ",", "2", ",", "4", ",", "2", ",", "3", ",", "1", ",", "4", ",", "1", ",", "3", ",", "5", ",", "4", ",", "1", ",", "3", ",", "30", ",", "11", ",", "5", ",", "11", ",", "2", ",", "10", ",", "2", ",", "4", ",", "16", ",", "2", ",", "2", ",", "2", ",", "3", ",", "2", ",", "5", ",", "1", ",", "2", ",", "9", ",", "2", ",", "8", ",", "3", ",", "3", ",", "8", ",", "1", ",", "2", ",", "2", ",", "15", ",", "6", ",", "3", ",", "1", ",", "11", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "4", ",", "3", ",", "61", ",", "2", ",", "1", ",", "3", ",", "13", ",", "9", ",", "2", ",", "5", ",", "10", ",", "2", ",", "11", ",", "2", ",", "2", ",", "1", ",", "6", ",", "4", ",", "3", ",", "64", ",", "7", ",", "4", ",", "2", ",", "7", ",", "4", ",", "6", ",", "2", ",", "2", ",", "3", ",", "3", ",", "1", ",", "3", ",", "1", ",", "15", ",", "3", ",", "2", ",", "1", ",", "2", ",", "3", ",", "2", ",", "3", ",", "1", ",", "6", ",", "3", ",", "3", ",", "2", ",", "7", ",", "4", ",", "2", ",", "1", ",", "5", ",", "1", ",", "2", ",", "6", ",", "3", ",", "5", ",", "3", ",", "25", ",", "9", ",", "3", ",", "3", ",", "2", ",", "1", ",", "3", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "2", ",", "3", ",", "11", ",", "1", ",", "3", ",", "3", ",", "219", ",", "5", ",", "2", ",", "2", ",", "2", ",", "2", ",", "4", ",", "3", ",", "2", ",", "4", ",", "6", ",", "2", ",", "3", ",", "2", ",", "2", ",", "3", ",", "3", ",", "4", ",", "6", ",", "4", ",", "3", ",", "9", ",", "6", ",", "12", ",", "2", ",", "3", ",", "3", ",", "2", ",", "3", ",", "5", ",", "2", ",", "3", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "30", ",", "4", ",", "3", ",", "3", ",", "2", ",", "1", ",", "2", ",", "5", ",", "9", ",", "3", ",", "5", ",", "1", ",", "3", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "6", ",", "3", ",", "2", ",", "9", ",", "3", ",", "8", ",", "2", ",", "2", ",", "13", ",", "2", ",", "3", ",", "6", ",", "3", ",", "6", ",", "2", ",", "2", ",", "12", ",", "2", ",", "19", ",", "2", ",", "3", ",", "9", ",", "3", ",", "23", ",", "3", ",", "10389"}], "}"}]], "Output", CellChangeTimes->{3.482636810457901*^9, 3.48280360184375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v2", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "2", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48367663709375*^9, 3.48396344790625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"13", ",", "4"}], "}"}], ",", RowBox[{"{", RowBox[{"16", ",", "5"}], "}"}], ",", RowBox[{"{", RowBox[{"19", ",", "6"}], "}"}], ",", RowBox[{"{", RowBox[{"22", ",", "7"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483676637109375*^9, 3.483963447921875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v3", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "3", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963459828125*^9, 3.483963463515625*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48396346409375*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"179", ",", "57"}], "}"}], ",", RowBox[{"{", RowBox[{"201", ",", "64"}], "}"}], ",", RowBox[{"{", RowBox[{"223", ",", "71"}], "}"}], ",", RowBox[{"{", RowBox[{"245", ",", "78"}], "}"}], ",", RowBox[{"{", RowBox[{"267", ",", "85"}], "}"}], ",", RowBox[{"{", RowBox[{"289", ",", "92"}], "}"}], ",", RowBox[{"{", RowBox[{"311", ",", "99"}], "}"}], ",", RowBox[{"{", RowBox[{"333", ",", "106"}], "}"}], ",", RowBox[{"{", RowBox[{"355", ",", "113"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4", ",", "9"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483963464109375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v4", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "4", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963469625*^9, 3.483963473453125*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.483963474140625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"52163", ",", "16604"}], "}"}], ",", RowBox[{"{", RowBox[{"52518", ",", "16717"}], "}"}], ",", RowBox[{"{", RowBox[{"52873", ",", "16830"}], "}"}], ",", RowBox[{"{", RowBox[{"53228", ",", "16943"}], "}"}], ",", RowBox[{"{", RowBox[{"53583", ",", "17056"}], "}"}], ",", RowBox[{"{", RowBox[{"53938", ",", "17169"}], "}"}], ",", RowBox[{"{", RowBox[{"54293", ",", "17282"}], "}"}], ",", RowBox[{"{", RowBox[{"54648", ",", "17395"}], "}"}], ",", RowBox[{"{", RowBox[{"55003", ",", "17508"}], "}"}], ",", RowBox[{"{", RowBox[{"55358", ",", "17621"}], "}"}], ",", RowBox[{"{", RowBox[{"55713", ",", "17734"}], "}"}], ",", RowBox[{"{", RowBox[{"56068", ",", "17847"}], "}"}], ",", RowBox[{"{", RowBox[{"56423", ",", "17960"}], "}"}], ",", RowBox[{"{", RowBox[{"56778", ",", "18073"}], "}"}], ",", RowBox[{"{", RowBox[{"57133", ",", "18186"}], "}"}], ",", RowBox[{"{", RowBox[{"57488", ",", "18299"}], "}"}], ",", RowBox[{"{", RowBox[{"57843", ",", "18412"}], "}"}], ",", RowBox[{"{", RowBox[{"58198", ",", "18525"}], "}"}], ",", RowBox[{"{", RowBox[{"58553", ",", "18638"}], "}"}], ",", RowBox[{"{", RowBox[{"58908", ",", "18751"}], "}"}], ",", RowBox[{"{", RowBox[{"59263", ",", "18864"}], "}"}], ",", RowBox[{"{", RowBox[{"59618", ",", "18977"}], "}"}], ",", RowBox[{"{", RowBox[{"59973", ",", "19090"}], "}"}], ",", RowBox[{"{", RowBox[{"60328", ",", "19203"}], "}"}], ",", RowBox[{"{", RowBox[{"60683", ",", "19316"}], "}"}], ",", RowBox[{"{", RowBox[{"61038", ",", "19429"}], "}"}], ",", RowBox[{"{", RowBox[{"61393", ",", "19542"}], "}"}], ",", RowBox[{"{", RowBox[{"61748", ",", "19655"}], "}"}], ",", RowBox[{"{", RowBox[{"62103", ",", "19768"}], "}"}], ",", RowBox[{"{", RowBox[{"62458", ",", "19881"}], "}"}], ",", RowBox[{"{", RowBox[{"62813", ",", "19994"}], "}"}], ",", RowBox[{"{", RowBox[{"63168", ",", "20107"}], "}"}], ",", RowBox[{"{", RowBox[{"63523", ",", "20220"}], "}"}], ",", RowBox[{"{", RowBox[{"63878", ",", "20333"}], "}"}], ",", RowBox[{"{", RowBox[{"64233", ",", "20446"}], "}"}], ",", RowBox[{"{", RowBox[{"64588", ",", "20559"}], "}"}], ",", RowBox[{"{", RowBox[{"64943", ",", "20672"}], "}"}], ",", RowBox[{"{", RowBox[{"65298", ",", "20785"}], "}"}], ",", RowBox[{"{", RowBox[{"65653", ",", "20898"}], "}"}], ",", RowBox[{"{", RowBox[{"66008", ",", "21011"}], "}"}], ",", RowBox[{"{", RowBox[{"66363", ",", "21124"}], "}"}], ",", RowBox[{"{", RowBox[{"66718", ",", "21237"}], "}"}], ",", RowBox[{"{", RowBox[{"67073", ",", "21350"}], "}"}], ",", RowBox[{"{", RowBox[{"67428", ",", "21463"}], "}"}], ",", RowBox[{"{", RowBox[{"67783", ",", "21576"}], "}"}], ",", RowBox[{"{", RowBox[{"68138", ",", "21689"}], "}"}], ",", RowBox[{"{", RowBox[{"68493", ",", "21802"}], "}"}], ",", RowBox[{"{", RowBox[{"68848", ",", "21915"}], "}"}], ",", RowBox[{"{", RowBox[{"69203", ",", "22028"}], "}"}], ",", RowBox[{"{", RowBox[{"69558", ",", "22141"}], "}"}], ",", RowBox[{"{", RowBox[{"69913", ",", "22254"}], "}"}], ",", RowBox[{"{", RowBox[{"70268", ",", "22367"}], "}"}], ",", RowBox[{"{", RowBox[{"70623", ",", "22480"}], "}"}], ",", RowBox[{"{", RowBox[{"70978", ",", "22593"}], "}"}], ",", RowBox[{"{", RowBox[{"71333", ",", "22706"}], "}"}], ",", RowBox[{"{", RowBox[{"71688", ",", "22819"}], "}"}], ",", RowBox[{"{", RowBox[{"72043", ",", "22932"}], "}"}], ",", RowBox[{"{", RowBox[{"72398", ",", "23045"}], "}"}], ",", RowBox[{"{", RowBox[{"72753", ",", "23158"}], "}"}], ",", RowBox[{"{", RowBox[{"73108", ",", "23271"}], "}"}], ",", RowBox[{"{", RowBox[{"73463", ",", "23384"}], "}"}], ",", RowBox[{"{", RowBox[{"73818", ",", "23497"}], "}"}], ",", RowBox[{"{", RowBox[{"74173", ",", "23610"}], "}"}], ",", RowBox[{"{", RowBox[{"74528", ",", "23723"}], "}"}], ",", RowBox[{"{", RowBox[{"74883", ",", "23836"}], "}"}], ",", RowBox[{"{", RowBox[{"75238", ",", "23949"}], "}"}], ",", RowBox[{"{", RowBox[{"75593", ",", "24062"}], "}"}], ",", RowBox[{"{", RowBox[{"75948", ",", "24175"}], "}"}], ",", RowBox[{"{", RowBox[{"76303", ",", "24288"}], "}"}], ",", RowBox[{"{", RowBox[{"76658", ",", "24401"}], "}"}], ",", RowBox[{"{", RowBox[{"77013", ",", "24514"}], "}"}], ",", RowBox[{"{", RowBox[{"77368", ",", "24627"}], "}"}], ",", RowBox[{"{", RowBox[{"77723", ",", "24740"}], "}"}], ",", RowBox[{"{", RowBox[{"78078", ",", "24853"}], "}"}], ",", RowBox[{"{", RowBox[{"78433", ",", "24966"}], "}"}], ",", RowBox[{"{", RowBox[{"78788", ",", "25079"}], "}"}], ",", RowBox[{"{", RowBox[{"79143", ",", "25192"}], "}"}], ",", RowBox[{"{", RowBox[{"79498", ",", "25305"}], "}"}], ",", RowBox[{"{", RowBox[{"79853", ",", "25418"}], "}"}], ",", RowBox[{"{", RowBox[{"80208", ",", "25531"}], "}"}], ",", RowBox[{"{", RowBox[{"80563", ",", "25644"}], "}"}], ",", RowBox[{"{", RowBox[{"80918", ",", "25757"}], "}"}], ",", RowBox[{"{", RowBox[{"81273", ",", "25870"}], "}"}], ",", RowBox[{"{", RowBox[{"81628", ",", "25983"}], "}"}], ",", RowBox[{"{", RowBox[{"81983", ",", "26096"}], "}"}], ",", RowBox[{"{", RowBox[{"82338", ",", "26209"}], "}"}], ",", RowBox[{"{", RowBox[{"82693", ",", "26322"}], "}"}], ",", RowBox[{"{", RowBox[{"83048", ",", "26435"}], "}"}], ",", RowBox[{"{", RowBox[{"83403", ",", "26548"}], "}"}], ",", RowBox[{"{", RowBox[{"83758", ",", "26661"}], "}"}], ",", RowBox[{"{", RowBox[{"84113", ",", "26774"}], "}"}], ",", RowBox[{"{", RowBox[{"84468", ",", "26887"}], "}"}], ",", RowBox[{"{", RowBox[{"84823", ",", "27000"}], "}"}], ",", RowBox[{"{", RowBox[{"85178", ",", "27113"}], "}"}], ",", RowBox[{"{", RowBox[{"85533", ",", "27226"}], "}"}], ",", RowBox[{"{", RowBox[{"85888", ",", "27339"}], "}"}], ",", RowBox[{"{", RowBox[{"86243", ",", "27452"}], "}"}], ",", RowBox[{"{", RowBox[{"86598", ",", "27565"}], "}"}], ",", RowBox[{"{", RowBox[{"86953", ",", "27678"}], "}"}], ",", RowBox[{"{", RowBox[{"87308", ",", "27791"}], "}"}], ",", RowBox[{"{", RowBox[{"87663", ",", "27904"}], "}"}], ",", RowBox[{"{", RowBox[{"88018", ",", "28017"}], "}"}], ",", RowBox[{"{", RowBox[{"88373", ",", "28130"}], "}"}], ",", RowBox[{"{", RowBox[{"88728", ",", "28243"}], "}"}], ",", RowBox[{"{", RowBox[{"89083", ",", "28356"}], "}"}], ",", RowBox[{"{", RowBox[{"89438", ",", "28469"}], "}"}], ",", RowBox[{"{", RowBox[{"89793", ",", "28582"}], "}"}], ",", RowBox[{"{", RowBox[{"90148", ",", "28695"}], "}"}], ",", RowBox[{"{", RowBox[{"90503", ",", "28808"}], "}"}], ",", RowBox[{"{", RowBox[{"90858", ",", "28921"}], "}"}], ",", RowBox[{"{", RowBox[{"91213", ",", "29034"}], "}"}], ",", RowBox[{"{", RowBox[{"91568", ",", "29147"}], "}"}], ",", RowBox[{"{", RowBox[{"91923", ",", "29260"}], "}"}], ",", RowBox[{"{", RowBox[{"92278", ",", "29373"}], "}"}], ",", RowBox[{"{", RowBox[{"92633", ",", "29486"}], "}"}], ",", RowBox[{"{", RowBox[{"92988", ",", "29599"}], "}"}], ",", RowBox[{"{", RowBox[{"93343", ",", "29712"}], "}"}], ",", RowBox[{"{", RowBox[{"93698", ",", "29825"}], "}"}], ",", RowBox[{"{", RowBox[{"94053", ",", "29938"}], "}"}], ",", RowBox[{"{", RowBox[{"94408", ",", "30051"}], "}"}], ",", RowBox[{"{", RowBox[{"94763", ",", "30164"}], "}"}], ",", RowBox[{"{", RowBox[{"95118", ",", "30277"}], "}"}], ",", RowBox[{"{", RowBox[{"95473", ",", "30390"}], "}"}], ",", RowBox[{"{", RowBox[{"95828", ",", "30503"}], "}"}], ",", RowBox[{"{", RowBox[{"96183", ",", "30616"}], "}"}], ",", RowBox[{"{", RowBox[{"96538", ",", "30729"}], "}"}], ",", RowBox[{"{", RowBox[{"96893", ",", "30842"}], "}"}], ",", RowBox[{"{", RowBox[{"97248", ",", "30955"}], "}"}], ",", RowBox[{"{", RowBox[{"97603", ",", "31068"}], "}"}], ",", RowBox[{"{", RowBox[{"97958", ",", "31181"}], "}"}], ",", RowBox[{"{", RowBox[{"98313", ",", "31294"}], "}"}], ",", RowBox[{"{", RowBox[{"98668", ",", "31407"}], "}"}], ",", RowBox[{"{", RowBox[{"99023", ",", "31520"}], "}"}], ",", RowBox[{"{", RowBox[{"99378", ",", "31633"}], "}"}], ",", RowBox[{"{", RowBox[{"99733", ",", "31746"}], "}"}], ",", RowBox[{"{", RowBox[{"100088", ",", "31859"}], "}"}], ",", RowBox[{"{", RowBox[{"100443", ",", "31972"}], "}"}], ",", RowBox[{"{", RowBox[{"100798", ",", "32085"}], "}"}], ",", RowBox[{"{", RowBox[{"101153", ",", "32198"}], "}"}], ",", RowBox[{"{", RowBox[{"101508", ",", "32311"}], "}"}], ",", RowBox[{"{", RowBox[{"101863", ",", "32424"}], "}"}], ",", RowBox[{"{", RowBox[{"102218", ",", "32537"}], "}"}], ",", RowBox[{"{", RowBox[{"102573", ",", "32650"}], "}"}], ",", RowBox[{"{", RowBox[{"102928", ",", "32763"}], "}"}], ",", RowBox[{"{", RowBox[{"103283", ",", "32876"}], "}"}], ",", RowBox[{"{", RowBox[{"103638", ",", "32989"}], "}"}], ",", RowBox[{"{", RowBox[{"103993", ",", "33102"}], "}"}], ",", RowBox[{"{", RowBox[{"104348", ",", "33215"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4", ",", "9", ",", "148"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483963474171875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v5", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "5", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963479015625*^9, 3.4839634829375*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48396348390625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"208341", ",", "66317"}], "}"}], ",", RowBox[{"{", RowBox[{"312689", ",", "99532"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4", ",", "9", ",", "148", ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.4839634839375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v6", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "6", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963489828125*^9, 3.483963494046875*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48396349471875*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"833719", ",", "265381"}], "}"}], ",", RowBox[{"{", RowBox[{"1146408", ",", "364913"}], "}"}], ",", RowBox[{"{", RowBox[{"1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.48396349475*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v7", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "7", ",", "20"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Calculator", " ", "accuracy"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963500078125*^9, 3.483963547390625*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.483963507515625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"3126535", ",", "995207"}], "}"}], ",", RowBox[{"{", RowBox[{"4272943", ",", "1360120"}], "}"}], ",", RowBox[{"{", RowBox[{"5419351", ",", "1725033"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.48396350753125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v8", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "8", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963514484375*^9, 3.48396351821875*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48396351903125*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"42208400", ",", "13435351"}], "}"}], ",", RowBox[{"{", RowBox[{"47627751", ",", "15160384"}], "}"}], ",", RowBox[{"{", RowBox[{"53047102", ",", "16885417"}], "}"}], ",", RowBox[{"{", RowBox[{"58466453", ",", "18610450"}], "}"}], ",", RowBox[{"{", RowBox[{"63885804", ",", "20335483"}], "}"}], ",", RowBox[{"{", RowBox[{"69305155", ",", "22060516"}], "}"}], ",", RowBox[{"{", RowBox[{"74724506", ",", "23785549"}], "}"}], ",", RowBox[{"{", RowBox[{"80143857", ",", "25510582"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483963519046875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v9", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "9", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.48396359053125*^9, 3.48396359465625*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.483963595625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"165707065", ",", "52746197"}], "}"}], ",", RowBox[{"{", RowBox[{"245850922", ",", "78256779"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483963595625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v10", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "10", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.48396360421875*^9, 3.4839636098125*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.483963610875*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"411557987", ",", "131002976"}], "}"}], ",", RowBox[{"{", RowBox[{"657408909", ",", "209259755"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483963610890625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{ RowBox[{"v10", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "25"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Accurate", " ", "to", " ", "16", " ", "decimal", " ", "places"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.48396362796875*^9, 3.483963663640625*^9}, { 3.48396377025*^9, 3.483963797359375*^9}}], Cell[BoxData["1.7114372187989227`*^-17"], "Output", CellChangeTimes->{{3.483963652734375*^9, 3.48396366475*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v20", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "20", ",", "20"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963692421875*^9, 3.4839637088125*^9}, {3.483963743734375*^9, 3.483963744734375*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{{3.4839636975625*^9, 3.4839637095625*^9}, 3.483963745390625*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{"3137327371971917", ",", "998642318693672"}], "}"}], ",", RowBox[{"{", RowBox[{"3565551965321221", ",", "1134950440263789"}], "}"}], ",", RowBox[{"{", RowBox[{"3993776558670525", ",", "1271258561833906"}], "}"}], ",", RowBox[{"{", RowBox[{"4422001152019829", ",", "1407566683404023"}], "}"}], ",", RowBox[{"{", RowBox[{"4850225745369133", ",", "1543874804974140"}], "}"}], ",", RowBox[{"{", RowBox[{"5278450338718437", ",", "1680182926544257"}], "}"}], ",", RowBox[{"{", RowBox[{"5706674932067741", ",", "1816491048114374"}], "}"}], ",", RowBox[{"{", RowBox[{"6134899525417045", ",", "1952799169684491"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "43", ",", "2", ",", "1", ",", "8", ",", "2", ",", "8"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{{3.48396369759375*^9, 3.48396370959375*^9}, 3.48396374540625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{ RowBox[{"v20", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "35"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Accurate", " ", "to", " ", "29", " ", "decimal", " ", "places"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.48396362796875*^9, 3.483963663640625*^9}, { 3.4839637296875*^9, 3.48396373515625*^9}, {3.48396380296875*^9, 3.483963822546875*^9}}], Cell[BoxData["2.6585505940994753`*^-30"], "Output", CellChangeTimes->{3.48396374903125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v50", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "50", ",", "100"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963692421875*^9, 3.4839637088125*^9}, {3.483963743734375*^9, 3.483963744734375*^9}, {3.483963837*^9, 3.483963844484375*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.4839638458125*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ "463534480895187976369779868172446660", ",", "147547607856009776231472393883246517"}], "}"}], ",", RowBox[{"{", RowBox[{ "676098659066651648790638346602690933", ",", "215208887216519379303904173950722446"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "43", ",", "2", ",", "1", ",", "8", ",", "2", ",", "8", ",", "4", ",", "1", ",", "4", ",", "4", ",", "51", ",", "3", ",", "1", ",", "4", ",", "2", ",", "4", ",", "1", ",", "4", ",", "5", ",", "5", ",", "2", ",", "2", ",", "5", ",", "3", ",", "1", ",", "7", ",", "2", ",", "3", ",", "1", ",", "5", ",", "1", ",", "2", ",", "5", ",", "1", ",", "3", ",", "2"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.48396384584375*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{ RowBox[{"v50", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "99"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Accurate", " ", "to", " ", "70", " ", "decimal", " ", "places"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.48396362796875*^9, 3.483963663640625*^9}, { 3.4839637296875*^9, 3.48396373515625*^9}, {3.48396380296875*^9, 3.483963822546875*^9}, {3.483963869359375*^9, 3.48396388590625*^9}}], Cell[BoxData["1.4663801973816771`*^-71"], "Output", CellChangeTimes->{3.48396387871875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"v150", "=", RowBox[{"Phoenix", "[", RowBox[{"Pi", ",", "150", ",", "200"}], "]"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963692421875*^9, 3.4839637088125*^9}, {3.483963743734375*^9, 3.483963744734375*^9}, {3.483963837*^9, 3.483963844484375*^9}, { 3.48396390265625*^9, 3.483963909234375*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.4839638458125*^9, 3.483963911046875*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ "3473379989928271870414323997727789146747679301542441716041715230843786059\ 3876848962023853237265443440464685529912", ",", "110561118926712420810335154004707002119796365417602451678696426351635824\ 53573544166066847139981538314197445228485"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "43", ",", "2", ",", "1", ",", "8", ",", "2", ",", "8", ",", "4", ",", "1", ",", "4", ",", "4", ",", "51", ",", "3", ",", "1", ",", "4", ",", "2", ",", "4", ",", "1", ",", "4", ",", "5", ",", "5", ",", "2", ",", "2", ",", "5", ",", "3", ",", "1", ",", "7", ",", "2", ",", "3", ",", "1", ",", "5", ",", "1", ",", "2", ",", "5", ",", "1", ",", "3", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "3", ",", "9", ",", "81", ",", "24", ",", "13", ",", "2", ",", "2", ",", "4", ",", "2", ",", "13", ",", "3", ",", "3", ",", "3", ",", "1", ",", "6", ",", "1", ",", "3", ",", "3", ",", "3", ",", "1", ",", "5", ",", "3", ",", "2", ",", "1", ",", "14", ",", "4", ",", "1", ",", "6", ",", "1", ",", "22", ",", "2", ",", "4", ",", "2", ",", "3", ",", "1", ",", "4", ",", "1", ",", "3", ",", "5", ",", "4", ",", "1", ",", "3", ",", "30", ",", "11", ",", "5", ",", "11", ",", "2", ",", "10", ",", "2", ",", "4", ",", "16", ",", "2", ",", "2", ",", "2", ",", "3", ",", "2", ",", "5", ",", "1", ",", "2", ",", "9", ",", "2", ",", "8", ",", "3", ",", "3", ",", "8", ",", "1", ",", "2", ",", "2", ",", "15", ",", "6", ",", "3", ",", "1", ",", "11", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "4", ",", "3", ",", "61", ",", "2", ",", "1", ",", "3", ",", "13", ",", "9", ",", "2", ",", "5", ",", "10", ",", "2", ",", "11", ",", "2", ",", "2", ",", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.48396384584375*^9, 3.483963911078125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{ RowBox[{"v150", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "300"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Accurate", " ", "to", " ", "225", " ", "decimal", " ", "places"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.48396362796875*^9, 3.483963663640625*^9}, { 3.4839637296875*^9, 3.48396373515625*^9}, {3.48396380296875*^9, 3.483963822546875*^9}, {3.483963869359375*^9, 3.48396388590625*^9}, { 3.483963918453125*^9, 3.483963959484375*^9}}], Cell[BoxData["7.983638343525108`*^-226"], "Output", CellChangeTimes->{ 3.48396387871875*^9, {3.483963925390625*^9, 3.483963952578125*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"v292", "=", RowBox[{"Phoenix", "[", RowBox[{"\[Pi]", ",", "292", ",", "200"}], "]"}]}], ";"}]], "Input", CellChangeTimes->{{3.4826364721922755`*^9, 3.4826364785672755`*^9}, { 3.482636567489151*^9, 3.482636595176651*^9}, {3.4826366902235255`*^9, 3.4826367115047755`*^9}, {3.483964053859375*^9, 3.483964054578125*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.482803590890625*^9, 3.48285892390625*^9, 3.483964078203125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"v292", "[", RowBox[{"[", "10390", "]"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "The", " ", "last", " ", "vector", " ", "contains", " ", "the", " ", "lengths", " ", "of", " ", "the", " ", "first", " ", "292", " ", RowBox[{"P", ".", "S", ".", "'"}], "s"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.4826367875047755`*^9, 3.482636807614151*^9}, { 3.482881907703125*^9, 3.482881942671875*^9}, {3.4828820326875*^9, 3.482882060328125*^9}, {3.483964064515625*^9, 3.483964065265625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "1", ",", "4", ",", "9", ",", "148", ",", "2", ",", "2", ",", "3", ",", "8", ",", "2", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "43", ",", "2", ",", "1", ",", "8", ",", "2", ",", "8", ",", "4", ",", "1", ",", "4", ",", "4", ",", "51", ",", "3", ",", "1", ",", "4", ",", "2", ",", "4", ",", "1", ",", "4", ",", "5", ",", "5", ",", "2", ",", "2", ",", "5", ",", "3", ",", "1", ",", "7", ",", "2", ",", "3", ",", "1", ",", "5", ",", "1", ",", "2", ",", "5", ",", "1", ",", "3", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "3", ",", "9", ",", "81", ",", "24", ",", "13", ",", "2", ",", "2", ",", "4", ",", "2", ",", "13", ",", "3", ",", "3", ",", "3", ",", "1", ",", "6", ",", "1", ",", "3", ",", "3", ",", "3", ",", "1", ",", "5", ",", "3", ",", "2", ",", "1", ",", "14", ",", "4", ",", "1", ",", "6", ",", "1", ",", "22", ",", "2", ",", "4", ",", "2", ",", "3", ",", "1", ",", "4", ",", "1", ",", "3", ",", "5", ",", "4", ",", "1", ",", "3", ",", "30", ",", "11", ",", "5", ",", "11", ",", "2", ",", "10", ",", "2", ",", "4", ",", "16", ",", "2", ",", "2", ",", "2", ",", "3", ",", "2", ",", "5", ",", "1", ",", "2", ",", "9", ",", "2", ",", "8", ",", "3", ",", "3", ",", "8", ",", "1", ",", "2", ",", "2", ",", "15", ",", "6", ",", "3", ",", "1", ",", "11", ",", "2", ",", "1", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "4", ",", "3", ",", "61", ",", "2", ",", "1", ",", "3", ",", "13", ",", "9", ",", "2", ",", "5", ",", "10", ",", "2", ",", "11", ",", "2", ",", "2", ",", "1", ",", "6", ",", "4", ",", "3", ",", "64", ",", "7", ",", "4", ",", "2", ",", "7", ",", "4", ",", "6", ",", "2", ",", "2", ",", "3", ",", "3", ",", "1", ",", "3", ",", "1", ",", "15", ",", "3", ",", "2", ",", "1", ",", "2", ",", "3", ",", "2", ",", "3", ",", "1", ",", "6", ",", "3", ",", "3", ",", "2", ",", "7", ",", "4", ",", "2", ",", "1", ",", "5", ",", "1", ",", "2", ",", "6", ",", "3", ",", "5", ",", "3", ",", "25", ",", "9", ",", "3", ",", "3", ",", "2", ",", "1", ",", "3", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "2", ",", "3", ",", "11", ",", "1", ",", "3", ",", "3", ",", "219", ",", "5", ",", "2", ",", "2", ",", "2", ",", "2", ",", "4", ",", "3", ",", "2", ",", "4", ",", "6", ",", "2", ",", "3", ",", "2", ",", "2", ",", "3", ",", "3", ",", "4", ",", "6", ",", "4", ",", "3", ",", "9", ",", "6", ",", "12", ",", "2", ",", "3", ",", "3", ",", "2", ",", "3", ",", "5", ",", "2", ",", "3", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "30", ",", "4", ",", "3", ",", "3", ",", "2", ",", "1", ",", "2", ",", "5", ",", "9", ",", "3", ",", "5", ",", "1", ",", "3", ",", "2", ",", "2", ",", "3", ",", "2", ",", "3", ",", "2", ",", "2", ",", "6", ",", "3", ",", "2", ",", "9", ",", "3", ",", "8", ",", "2", ",", "2", ",", "13", ",", "2", ",", "3", ",", "6", ",", "3", ",", "6", ",", "2", ",", "2", ",", "12", ",", "2", ",", "19", ",", "2", ",", "3", ",", "9", ",", "3", ",", "23", ",", "3", ",", "10389"}], "}"}]], "Output", CellChangeTimes->{3.482636810457901*^9, 3.48280360184375*^9, 3.483964084640625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm2", "[", RowBox[{ RowBox[{"v292", "[", RowBox[{"[", "10389", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "500"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{"Accurate", " ", "to", " ", "440", " ", "decimal", " ", "places"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.48396362796875*^9, 3.483963663640625*^9}, { 3.4839637296875*^9, 3.48396373515625*^9}, {3.48396380296875*^9, 3.483963822546875*^9}, {3.483963869359375*^9, 3.48396388590625*^9}, { 3.483963918453125*^9, 3.483963959484375*^9}, {3.483964111921875*^9, 3.48396413696875*^9}}], Cell[BoxData["2.4085066017286195686069632865311`15.954589770191005*^-441"], \ "Output", CellChangeTimes->{3.483964130078125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"st", "=", RowBox[{"AbsoluteTime", "[", "]"}]}], ";", RowBox[{"v", "=", RowBox[{"Phoenix", "[", RowBox[{"\[Pi]", ",", "8000", ",", "8000"}], "]"}]}], ";", RowBox[{ RowBox[{"AbsoluteTime", "[", "]"}], "-", "st"}]}]], "Input", CellChangeTimes->{{3.482692446421875*^9, 3.48269252825*^9}, { 3.48396418678125*^9, 3.483964193421875*^9}}], Cell[BoxData["484.1875`10.136558566806771"], "Output", CellChangeTimes->{3.482692530109375*^9, 3.48269304565625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"%", "/", "60", " ", RowBox[{"(*", " ", RowBox[{"8", " ", "minutes"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482708549671875*^9, 3.48270855278125*^9}, { 3.483964154796875*^9, 3.4839641691875*^9}}], Cell[BoxData["8.0697916666666666667`10.136558566806771"], "Output", CellChangeTimes->{3.482708554546875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Length", "[", "v", "]"}]], "Input", CellChangeTimes->{{3.482708570109375*^9, 3.48270857584375*^9}}], Cell[BoxData["3"], "Output", CellChangeTimes->{3.4827085770625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "15000"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "fraction", " ", "to", " ", "the", " ", "right", " ", "of", " ", "\[Pi]"}], ",", " ", RowBox[{"accurate", " ", "to", " ", "11830", " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482708685*^9, 3.48270876121875*^9}, { 3.482708798703125*^9, 3.482708821890625*^9}, {3.48274708290625*^9, 3.482747156390625*^9}}], Cell[BoxData["9.60559359261444190518582588`15.954589770191005*^-11832"], \ "Output", CellChangeTimes->{ 3.4827087019375*^9, {3.48270874984375*^9, 3.4827087619375*^9}, { 3.48270881121875*^9, 3.482708828765625*^9}}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"StringLength", "[", RowBox[{"ToString", "[", RowBox[{"VP", "[", RowBox[{"v", "[", RowBox[{"[", "1", "]"}], "]"}], "]"}], "]"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ RowBox[{"This", " ", "is", " ", "a", " ", "boat"}], "-", RowBox[{"load", " ", "of", " ", "digits"}]}], ",", " ", RowBox[{"top", " ", "and", " ", RowBox[{"bottom", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482747748421875*^9, 3.482747755*^9}, { 3.48274783603125*^9, 3.482747853265625*^9}, {3.482881794625*^9, 3.48288182471875*^9}}], Cell[BoxData["17750"], "Output", CellChangeTimes->{3.482747855328125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"VecPrm", "[", RowBox[{ RowBox[{"v", "[", RowBox[{"[", "2", "]"}], "]"}], ",", RowBox[{"N", "[", RowBox[{"Pi", ",", "15000"}], "]"}]}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ RowBox[{ "fraction", " ", "to", " ", "the", " ", "left", " ", "of", " ", "\[Pi]"}], ",", " ", RowBox[{"accurate", " ", "to", " ", "11831", " ", "decimal", " ", RowBox[{"places", "."}]}]}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.482708685*^9, 3.48270876121875*^9}, { 3.482708798703125*^9, 3.482708847265625*^9}, {3.48274717128125*^9, 3.482747185203125*^9}}], Cell[BoxData[ RowBox[{"-", "1.3199838201819569269528474`15.954589770191005*^-11832"}]], "Output", CellChangeTimes->{3.482708848265625*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"N", "[", RowBox[{"{", RowBox[{ RowBox[{"Mean", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"StandardDeviation", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"Max", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}], ",", RowBox[{"Median", "[", RowBox[{"v", "[", RowBox[{"[", "3", "]"}], "]"}], "]"}]}], "}"}], "]"}]], "Input", CellChangeTimes->{{3.482708579015625*^9, 3.482708581375*^9}, { 3.48274728328125*^9, 3.48274738915625*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "9.852625`", ",", "135.92768094621235`", ",", "10389.`", ",", "3.`"}], "}"}]], "Output", CellChangeTimes->{3.482747393421875*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"vE50", "=", RowBox[{"Phoenix", "[", RowBox[{"E", ",", "50", ",", "200"}], "]"}], " ", RowBox[{"(*", " ", RowBox[{ "Unlike", " ", "other", " ", "trancendentals", " ", "LenSeq", " ", "for", " ", "E", " ", "exhibits", " ", "a", " ", "pattern"}], " ", "*)"}]}]], "Input", CellChangeTimes->{{3.483676597859375*^9, 3.483676635484375*^9}, { 3.483963692421875*^9, 3.4839637088125*^9}, {3.483963743734375*^9, 3.483963744734375*^9}, {3.483963837*^9, 3.483963844484375*^9}, { 3.48396390265625*^9, 3.483963909234375*^9}, {3.483964268109375*^9, 3.483964387203125*^9}}], Cell[BoxData["\<\" \\!\\(\\*\\nStyleBox[\\\"\[CapitalPhi]\[Omicron]\[Iota]\ \[Nu]\[Iota]\[Xi]\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\\(\ \\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\\!\ \\(\\*\\nStyleBox[\\\"-\\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\\)\ \\!\\(\\*\\nStyleBox[\\\" \\\",\\nFontFamily->\\\"Symbol\\\",\\nFontSize->18]\ \\)The Great Crimson and Golden Bird Soars Above\"\>"], "Print", CellChangeTimes->{3.48396430259375*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ "3224546177070375504839670260260400797251", ",", "1186244245652160441069426743288856160849"}], "}"}], ",", RowBox[{"{", RowBox[{ "1", ",", "1", ",", "1", ",", "2", ",", "4", ",", "1", ",", "5", ",", "1", ",", "6", ",", "1", ",", "7", ",", "1", ",", "8", ",", "1", ",", "9", ",", "1", ",", "10", ",", "1", ",", "11", ",", "1", ",", "12", ",", "1", ",", "13", ",", "1", ",", "14", ",", "1", ",", "15", ",", "1", ",", "16", ",", "1", ",", "17", ",", "1", ",", "18", ",", "1", ",", "19", ",", "1", ",", "20", ",", "1", ",", "21", ",", "1", ",", "22", ",", "1", ",", "23", ",", "1", ",", "24", ",", "1", ",", "25", ",", "1", ",", "26", ",", "1"}], "}"}]}], "}"}]], "Output", CellChangeTimes->{3.483964302625*^9}] }, Open ]], Cell[BoxData[ RowBox[{ RowBox[{"(*", " ", RowBox[{ RowBox[{ "Place", " ", "any", " ", "real", " ", "number", " ", "in", " ", "the", " ", "first", " ", "argument", " ", "of", " ", "Phoenix"}], ",", " ", RowBox[{ "the", " ", "index", " ", "of", " ", "the", " ", "Partial", " ", "Sequence", " ", "desired", "\[IndentingNewLine]", "into", " ", "the", " ", "second", " ", RowBox[{"argument", ".", " ", "If"}], " ", "you", " ", "get", " ", "funny", " ", "results"}], ",", " ", RowBox[{"like", " ", "all", " ", RowBox[{"1", "'"}], "s", " ", "after", " ", "a", " ", "certain", " ", "point", " ", "in", " ", "LenSeq"}], ",", " ", RowBox[{ "just", "\[IndentingNewLine]", "enlarge", " ", "the", " ", "decimal", " ", "place", " ", "accuracy", " ", "with", " ", "the", " ", "third", " ", "argument", " ", "of", " ", RowBox[{"Phoenix", "."}]}]}], " ", "*)"}], "\[IndentingNewLine]"}]], "Input", CellChangeTimes->{{3.483964445359375*^9, 3.483964675578125*^9}}] }, WindowSize->{919, 556}, WindowMargins->{{24, Automatic}, {Automatic, 30}}, FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[545, 20, 506, 13, 31, "Input"], Cell[1054, 35, 2846, 58, 152, "Input"], Cell[3903, 95, 22206, 493, 1192, "Input"], Cell[26112, 590, 1356, 25, 92, "Input"], Cell[27471, 617, 66653, 1591, 3021, "Input"], Cell[CellGroupData[{ Cell[94149, 2212, 342, 9, 31, "Input"], Cell[94494, 2223, 117, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[94648, 2229, 115, 2, 31, "Input"], Cell[94766, 2233, 109, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[94912, 2239, 124, 2, 31, "Input"], Cell[95039, 2243, 68, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[95144, 2249, 642, 17, 52, "Input"], Cell[95789, 2268, 219, 4, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[96045, 2277, 623, 16, 31, "Input"], Cell[96671, 2295, 74, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[96782, 2301, 641, 17, 52, "Input"], Cell[97426, 2320, 143, 3, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[97606, 2328, 609, 17, 31, "Input"], Cell[98218, 2347, 175, 5, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[98430, 2357, 317, 7, 31, "Input"], Cell[98750, 2366, 519, 6, 25, "Print"] }, Open ]], Cell[CellGroupData[{ Cell[99306, 2377, 508, 11, 31, "Input"], Cell[99817, 2390, 3232, 43, 164, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[103086, 2438, 186, 4, 31, "Input"], Cell[103275, 2444, 518, 6, 25, "Print"], Cell[103796, 2452, 416, 13, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[104249, 2470, 235, 5, 31, "Input"], Cell[104487, 2477, 497, 6, 25, "Print"], Cell[104987, 2485, 724, 23, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[105748, 2513, 232, 5, 31, "Input"], Cell[105983, 2520, 498, 6, 25, "Print"], Cell[106484, 2528, 10105, 301, 468, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[116626, 2834, 233, 5, 31, "Input"], Cell[116862, 2841, 497, 6, 25, "Print"], Cell[117362, 2849, 323, 10, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[117722, 2864, 235, 5, 31, "Input"], Cell[117960, 2871, 497, 6, 25, "Print"], Cell[118460, 2879, 336, 10, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[118833, 2894, 328, 7, 31, "Input"], Cell[119164, 2903, 498, 6, 25, "Print"], Cell[119665, 2911, 427, 13, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[120129, 2929, 234, 5, 31, "Input"], Cell[120366, 2936, 497, 6, 25, "Print"], Cell[120866, 2944, 811, 23, 50, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[121714, 2972, 233, 5, 31, "Input"], Cell[121950, 2979, 495, 6, 25, "Print"], Cell[122448, 2987, 382, 11, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[122867, 3003, 234, 5, 31, "Input"], Cell[123104, 3010, 495, 6, 25, "Print"], Cell[123602, 3018, 397, 11, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[124036, 3034, 452, 12, 31, "Input"], Cell[124491, 3048, 113, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[124641, 3054, 284, 6, 31, "Input"], Cell[124928, 3062, 544, 7, 25, "Print"], Cell[125475, 3071, 1115, 26, 107, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[126627, 3102, 501, 13, 31, "Input"], Cell[127131, 3117, 92, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[127260, 3123, 325, 6, 31, "Input"], Cell[127588, 3131, 496, 6, 25, "Print"], Cell[128087, 3139, 964, 21, 69, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[129088, 3165, 546, 13, 31, "Input"], Cell[129637, 3180, 92, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[129766, 3186, 375, 7, 31, "Input"], Cell[130144, 3195, 518, 6, 25, "Print"], Cell[130665, 3203, 2105, 33, 145, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[132807, 3241, 598, 14, 31, "Input"], Cell[133408, 3257, 141, 2, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[133586, 3264, 366, 7, 31, "Input"], Cell[133955, 3273, 544, 7, 25, "Print"] }, Open ]], Cell[CellGroupData[{ Cell[134536, 3285, 557, 11, 31, "Input"], Cell[135096, 3298, 3257, 44, 164, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[138390, 3347, 650, 15, 31, "Input"], Cell[139043, 3364, 129, 2, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[139209, 3371, 390, 10, 31, "Input"], Cell[139602, 3383, 117, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[139756, 3389, 245, 5, 31, "Input"], Cell[140004, 3396, 109, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[140150, 3402, 124, 2, 31, "Input"], Cell[140277, 3406, 68, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[140382, 3412, 642, 17, 52, "Input"], Cell[141027, 3431, 219, 4, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[141283, 3440, 623, 16, 31, "Input"], Cell[141909, 3458, 74, 1, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[142020, 3464, 641, 17, 52, "Input"], Cell[142664, 3483, 143, 3, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[142844, 3491, 609, 17, 31, "Input"], Cell[143456, 3510, 175, 5, 30, "Output"] }, Open ]], Cell[CellGroupData[{ Cell[143668, 3520, 622, 13, 31, "Input"], Cell[144293, 3535, 497, 6, 25, "Print"], Cell[144793, 3543, 843, 17, 69, "Output"] }, Open ]], Cell[145651, 3563, 1049, 22, 92, "Input"] } ] *) (* End of internal cache information *)