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Mexico jrangel@sunserver.uaq.mx Copyright \[Copyright]August, 2000, UAQ, Facultad de Inform\[AAcute]tica\ \>", "Caption", TextAlignment->Left, TextJustification->0], Cell[TextData[StyleBox["Permission is hereby granted to reproduce part or all \ of this file for any purpose other than direct profit providing that the \ source is acknowledged.\n", FontColor->RGBColor[0, 0, 1]]], "Quote", TextAlignment->Left, TextJustification->0], Cell[BoxData[ \(Off[General::"\", General::"\"]\)], "Input", InitializationCell->True], Cell["\<\ Two numbers (not necessarily different) are chosen from the range of positive numbers greater than 1 and not greater than 20. Only the sum of the two numbers is given to mathematician S. Only the product of the two is given to mathematician P. On the telephone S says to P, \"I see no way you can determine my sum.\" An hour later P calls him back to say, \"I know your sum.\" Later S calls P again to report, \"now I know your product.\" What are the two numbers? 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The author has used this material to \ introduce students to a variety of concepts in number theory, combinatorics \ and algorithm-design throughout interactive sessions of ", StyleBox["Mathematica", FontSlant->"Italic"], ". Because the nature of the subject offers the teacher ample margin to \ exercise his/her particular teaching style and personal taste in enhancing \ or omitting particular sections of this material; for instance, the author \ always includes a few proofs, discussion sessions, individual problem-solving \ sessions, collective problem-solving session (these sessions being held \ simultaneously to cater for continuity), a written assignment, homework and \ an exam to adequately cover a theme such as this one. Thus, no attempt has \ been made to include abundant comments in order to provide a suitable \ framework on which to base a lecture. The proof of the results mentioned \ throughout the work will be omitted as they all can be found in the \ references." }], "Text"] }, Open ]], Cell[CellGroupData[{ Cell["Partitions ", "Section"], Cell[CellGroupData[{ Cell["INTRODUCTION", "Subsection"], Cell["\<\ First, we design an auxiliary function to display lists of numbers as sums of \ powers\ \>", "Text"], Cell[BoxData[{ \(asSum[x_] := asSum[x, 1]\[IndentingNewLine]\), "\[IndentingNewLine]", \(asSum[{a_}, 1] := ToString[a]\), "\[IndentingNewLine]", \(asSum[{a_}, p_Integer?Positive] := ToString[a]\^p\), "\n", \(asSum[{a_, b__}, p_] := DisplayForm[ RowBox[{asSum[{a}, p], "\<+\>", asSum[{b}, p]}]]\)}], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(asSum[Range[5], 2]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{\("1"\^2\), "+", TagBox[ RowBox[{\("2"\^2\), "+", TagBox[ RowBox[{\("3"\^2\), "+", TagBox[\("4"\^2 + "5"\^2\), DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]], "Output"] }, Open ]], 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The first ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(n + r - 2\)}, {"n"} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], " start with 0. In general, there are ", Cell[BoxData[ FormBox[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(n + r - i - 2\)}, {\(n - i\)} }], "\[NegativeThinSpace]", ")"}], TraditionalForm]]], " partitions starting with i. " }], "Text"], Cell[BoxData[{ \(part\_\(n_, 1\) := {{n}}\), "\n", \(part\_\(n_, r_\) := Module[{ans = {}, i}, \[IndentingNewLine]Do[\[IndentingNewLine]AppendTo[ ans, \((Join[{i}, #] &)\) /@ part\_\(n - i, r - 1\)], {i, 0, n}]; \[IndentingNewLine]Flatten[ans, 1]]\)}], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[BoxData[ \(asSum\ /@ \ part\_\(5, 3\)\)], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"0", "+", TagBox[\(0 + 5\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(1 + 4\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(2 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(3 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(4 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(5 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(0 + 4\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(1 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(2 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(3 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(4 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(0 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(1 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(2 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(3 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(0 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(1 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(2 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"4", "+", TagBox[\(0 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"4", "+", TagBox[\(1 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"5", "+", TagBox[\(0 + 0\), DisplayForm]}], DisplayForm]}], "}"}]], "Output"] }, Open ]], Cell["To obtain the i-th partition", "Text"], Cell[BoxData[{ \(numPart[n_, r_] := Binomial[n + r - 1, n]\n\), "\n", \(ithPart\_\(n_, r_\)[i_] := "\"\ /; \ i > numPart[n, r]\), "\n", \(ithPart\_\(n_, 1\)[1] := {n}\), "\n", \(ithPart\_\(n_, r_\)[i_] := Module[{j, k, t}, \[IndentingNewLine]t = Table[0, {n + 1}]; \[IndentingNewLine]t\[LeftDoubleBracket]1\ \[RightDoubleBracket] = numPart[n, r - 1]; \[IndentingNewLine]Do[\[IndentingNewLine]t\ \[LeftDoubleBracket]j + 1\[RightDoubleBracket] = t\[LeftDoubleBracket]j\[RightDoubleBracket] + numPart[n - j, r - 1], {j, n}]; \[IndentingNewLine]PrependTo[t, 0]; \[IndentingNewLine]j = 1; \[IndentingNewLine]While[\ t\[LeftDoubleBracket]j\[RightDoubleBracket] < i, \(j++\)]; \[IndentingNewLine]Join[{j - 2}, ithPart\_\(n - j + 2, r - 1\)[ i - t\[LeftDoubleBracket]j - 1\[RightDoubleBracket]]]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(asSum\ /@ \ Table[ithPart\_\(5, 3\)[i], {i, numPart[5, 3]}]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"0", "+", TagBox[\(0 + 5\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(1 + 4\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(2 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(3 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(4 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"0", "+", TagBox[\(5 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(0 + 4\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(1 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(2 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(3 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(4 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(0 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(1 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(2 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"2", "+", TagBox[\(3 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(0 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(1 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"3", "+", TagBox[\(2 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"4", "+", TagBox[\(0 + 1\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"4", "+", TagBox[\(1 + 0\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"5", "+", TagBox[\(0 + 0\), DisplayForm]}], DisplayForm]}], "}"}]], "Output"] }, Open ]], Cell["\<\ For example, to find the millionth partition of 100 of size 6.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(asSum[ithPart\_\(100, 6\)[1000000]]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"0", "+", TagBox[ RowBox[{"6", "+", TagBox[ RowBox[{"3", "+", TagBox[ RowBox[{"8", "+", TagBox[\(49 + 34\), DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]], "Output"] }, Open ]], Cell["A random partition:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(asSum[ ithPart\_\(Random[Integer, {500, 800}], Random[Integer, {5, 10}]\)[ Random[Integer, {10\^8, 10\^10}]]]\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"0", "+", TagBox[ RowBox[{"0", "+", TagBox[ RowBox[{"0", "+", TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"145", "+", TagBox[ RowBox[{"63", "+", TagBox[ RowBox[{"285", "+", TagBox[\(43 + 13\), DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]}], DisplayForm]], "Output"] }, Open ]], Cell["\<\ These random partitions are not really of the required size. We would like to \ disregard the 0 and consider the restricted partitions of n, that is, \ partitions of n into at most r parts each one being at most m.\ \>", "Text"], Cell[BoxData[ \(restrictedPartitions[m_, r_, n_] := Select[partH[Range[m], n], \((Length[#] \[LessEqual] r)\) &]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(asSum\ /@ \ restrictedPartitions[5, 3, 5]\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[ RowBox[{"1", "+", TagBox[\(1 + 3\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{"1", "+", TagBox[\(2 + 2\), DisplayForm]}], DisplayForm], ",", TagBox[\(1 + 4\), DisplayForm], ",", TagBox[\(2 + 3\), DisplayForm], ",", "\<\"5\"\>"}], "}"}]], "Output"] }, Open ]], Cell["\<\ The number of partitions of n into at most 4 parts each one being at most 5 \ is\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Length[restrictedPartitions[5, 4, n]], {n, 25}]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ In general, from the (mr+1)-th element on all of them are zero.\ \>", "Text"], Cell[BoxData[ \(G[m_, r_] := \(\[Product]\+\(i = r + 1\)\%\(r + m\)\((1 - x\^i)\)\)\/\(\ \[Product]\+\(i = 1\)\%m\((1 - x\^i)\)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(G[5, 4] + O[x]\^26\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{ "1", "+", "x", "+", \(2\ x\^2\), "+", \(3\ x\^3\), "+", \(5\ x\^4\), "+", \(6\ x\^5\), "+", \(8\ x\^6\), "+", \(9\ x\^7\), "+", \(11\ x\^8\), "+", \(11\ x\^9\), "+", \(12\ x\^10\), "+", \(11\ x\^11\), "+", \(11\ x\^12\), "+", \(9\ x\^13\), "+", \(8\ x\^14\), "+", \(6\ x\^15\), "+", \(5\ x\^16\), "+", \(3\ x\^17\), "+", \(2\ x\^18\), "+", \(x\^19\), "+", \(x\^20\), "+", InterpretationBox[\(O[x]\^26\), SeriesData[ x, 0, {}, 0, 26, 1]]}], SeriesData[ x, 0, {1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1}, 0, 26, 1]]], "Output"] }, Open ]], Cell["\<\ In general, the number of partitions of n into two parts is equal to \ \[LeftFloor]n/2\[RightFloor]+1 \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Length[restrictedPartitions[n, 2, n]], {n, 10}]\)], "Input"], Cell[BoxData[ \({1, 2, 2, 3, 3, 4, 4, 5, 5, 6}\)], "Output"] }, Open ]], Cell[TextData[{ "The number of partitions of n into three parts is equal to the nearest \ integer to ", Cell[BoxData[ \(TraditionalForm\`\((n + 3)\)\^2\)]], "/12." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Length[restrictedPartitions[n, 3, n]], {n, 10}]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 7, 8, 10, 12, 14}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Round[\((n + 3)\)\^2\/12], {n, 10}]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 4, 5, 7, 8, 10, 12, 14}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ \(\(p = restrictedPartitions[15, 3, 15];\)\), "\[IndentingNewLine]", \(\(m = Max[Length\ /@ \ p];\)\), "\[IndentingNewLine]", \(\(u = \(PadRight[#, m] &\)\ /@ \ p;\)\), "\n", 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0.155515}}] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["EHRENFEST TECHNIQUES IN NUMBER THEORY", "Subsection"], Cell["\<\ The motivation for this section was to illustrate the following result: \ \>", "Text"], Cell["\<\ For every pair of numbers n and k, each partition of n!k into parts not \ exceeding n is actually a sum of k partitions of n!\ \>", "Commentary"], Cell[TextData[{ "Let us first solve the following problem: Given a list of positive \ integers, how do you partition it into k parts with equal sums? Providing \ that the problem has a solution, we can assume that each part forms a \ non-decreasing sequence.\n\nWe apply a technique akin to the solution of the \ Ehrenfest urn problem. After constructing an approximately optimum starting \ configuration we proceed to interchange numbers from one partition to another \ facilitating the convergence to a stable configuration, which is the \ (non-unique) solution we seek. The choice of which urns are going to be the \ \"sender\" and the \"recipient\" in an interchange is based on their relative \ sizes. As this method is probabilistic, the time to reach convergence might \ be relatively large and for that reason, a safety net is provided as an \ optional parameter in function ", StyleBox["Ehrenfest", "InlineInput"], "; after our experience with urn problems though, we expect a fast \ convergence. The following output was timed on a 500-MHz pentium pc." }], "Text"], Cell[BoxData[ \(Ehrenfest[li_, k_, safetyNet_: 1000] := Module[{s, t = Table[{}, {k}], i = 1, r, from, to, tot, j = 1}, \[IndentingNewLine]Map[\((AppendTo[ t\[LeftDoubleBracket]i\[RightDoubleBracket], #]; \ i = 1 + Mod[i, k])\) &, Sort[li]]; \[IndentingNewLine]s = \(Plus\ @@ \ # &\)\ /@ \ t; \[IndentingNewLine]tot = Plus\ @@ s; \[IndentingNewLine]While[\((j < safetyNet)\) && \((Length[Union[s]] > 1)\), \[IndentingNewLine]r = Flatten[Table[ Table[i, {s\[LeftDoubleBracket]i\[RightDoubleBracket]}], {i, Length[s]}]]; \[IndentingNewLine]While[\(\((from = r\[LeftDoubleBracket]Random[ Integer, {1, Length[r]}]\[RightDoubleBracket])\); Length[t\[LeftDoubleBracket]from\[RightDoubleBracket]] \[Equal] 1\)\(,\)]; \[IndentingNewLine]s = tot - s; \[IndentingNewLine]r = Flatten[Table[ Table[i, {s\[LeftDoubleBracket]i\[RightDoubleBracket]}], {i, Length[s]}]]; \[IndentingNewLine]While[\(\((to = r\[LeftDoubleBracket]Random[ Integer, {1, Length[r]}]\[RightDoubleBracket])\) == from\)\(,\)]; \[IndentingNewLine]i = Random[Integer, {1, Length[t\[LeftDoubleBracket] from\[RightDoubleBracket]]}]; \[IndentingNewLine]AppendTo[ t\[LeftDoubleBracket]to\[RightDoubleBracket], t\[LeftDoubleBracket]from, i\[RightDoubleBracket]]; \[IndentingNewLine]t = Delete[t, {from, i}]; \[IndentingNewLine]s = \(Plus\ @@ \ # &\)\ /@ \ t; \[IndentingNewLine]\(j++\)\[IndentingNewLine]]; \ \[IndentingNewLine]If[j < safetyNet, Sort[Sort\ /@ \ t], {}]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Ehrenfest[{1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3}, 3] // Timing\)], "Input"], Cell[BoxData[ \({0.`\ Second, {{1, 2, 3}, {1, 1, 2, 2}, {1, 1, 1, 1, 2}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ehrenfest[{1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3}, 3] // Timing\)], "Input"], Cell[BoxData[ \({0.05999999999994543`\ Second, {{1, 1, 1, 3}, {1, 1, 2, 2}, {1, 1, 2, 2}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ehrenfest[{1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 6}, 4] // Timing\)], "Input"], Cell[BoxData[ \({0.10999999999967258`\ Second, {{6}, {1, 2, 3}, {1, 1, 2, 2}, {1, 1, 1, 1, 2}}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Ehrenfest[{2, 2, 2, 3, 3, 3, 3}, 3] // Timing\)], "Input"], Cell[BoxData[ \({0.17000000000007276`\ Second, {{3, 3}, {3, 3}, {2, 2, 2}}}\)], "Output"] }, Open ]], Cell["\<\ The following illustrates the problem that motivated this section.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 3;\)\), "\[IndentingNewLine]", \(\(k = 3;\)\), "\[IndentingNewLine]", \(\(p = partH[Range[n], \(n!\) k];\)\), "\[IndentingNewLine]", \(asSum\ /@ \ p // \ TableForm\), "\[IndentingNewLine]", \(\(asSum\ /@ \ \((Ehrenfest[#, k])\) &\)\ /@ \ p // TableForm\)}], "Input"], Cell[BoxData[ InterpretationBox[GridBox[{ { TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"1", "+", TagBox[ RowBox[{"1", "+", TagBox[ 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More generally, the number of compositions of n into exactly m parts \ each one at least k is equal to ", Cell[BoxData[ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(n - \((k - 1)\) m - 1\)}, {\(m - 1\)} }], "\[NegativeThinSpace]", ")"}]]], ". For instance, there are ", Cell[BoxData[ RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(5 - 1\)}, {\(3 - 1\)} }], "\[NegativeThinSpace]", ")"}], "=", "6"}]]], " compositions of 5 into three parts and ", Cell[BoxData[ RowBox[{ RowBox[{"(", "\[NegativeThinSpace]", GridBox[{ {\(5 - \((2 - 1)\) 2 - 1\)}, {\(2 - 1\)} }], "\[NegativeThinSpace]", ")"}], "=", "2"}]]], " compositions of 5 into two parts each one at least equal to 2 as shown in \ the previous output. Also, there are ", Cell[BoxData[ \(2\^\(n - 1\)\)]], " unrestricted compositions of n." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Length[compositions[n]], {n, 10}]\)], "Input"], Cell[BoxData[ \({1, 2, 4, 8, 16, 32, 64, 128, 256, 512}\)], "Output"] }, Open ]], Cell["\<\ There is a one-to-one correspondence between subsets, binary numbers, \ positive integer numbers (their decimal equivalents) and compositions.\ \>", "Text"], Cell[BoxData[{ \(subSets[0] := {{}}\), "\n", \(subSets[n_] := Module[{s = subSets[n - 1]}, \[IndentingNewLine]Join[s, Map[Append[#, n] &, s]]]\[IndentingNewLine]\), "\[IndentingNewLine]", \(binary[n_, s_] := Module[{bn = Table[0, {n}]}, \[IndentingNewLine]bn\[LeftDoubleBracket] s\[RightDoubleBracket] = 1; \[IndentingNewLine]bn]\[IndentingNewLine]\), \ "\[IndentingNewLine]", \(dec[{}] := 0\), "\n", \(dec[{x___, t_}] := \(\(2\ dec[{x}]\)\(+\)\(t\)\(\[IndentingNewLine]\)\)\), "\ \[IndentingNewLine]", \(composit[u_] := \ Module[{v}, \[IndentingNewLine]v = Append[u, 0]\ //. \ \ {{a___, 1, 0, c___} \[Rule] {a, 2, "\<+\>", c}, \[IndentingNewLine]{a___, 1, b_, "\<+\>", c___} \[Rule] {a, b + 1, "\<+\>", c}, \[IndentingNewLine]{a___, 0, c___} \[Rule] {a, 1, "\<+\>", c}}; \[IndentingNewLine]StringJoin[ ToString\ /@ \ Drop[v, \(-1\)]]]\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[{#, u = binary[4, #]; \ StringJoin[ToString\ /@ \ u], dec[u], composit[u]} &, subSets[4]]\ // \ MatrixForm\)], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({}\), "\<\"0000\"\>", "0", "\<\"1+1+1+1+1\"\>"}, {\({1}\), "\<\"1000\"\>", "8", "\<\"2+1+1+1\"\>"}, {\({2}\), "\<\"0100\"\>", "4", "\<\"1+2+1+1\"\>"}, {\({1, 2}\), "\<\"1100\"\>", "12", "\<\"3+1+1\"\>"}, {\({3}\), "\<\"0010\"\>", "2", "\<\"1+1+2+1\"\>"}, {\({1, 3}\), "\<\"1010\"\>", "10", "\<\"2+2+1\"\>"}, {\({2, 3}\), "\<\"0110\"\>", "6", "\<\"1+3+1\"\>"}, {\({1, 2, 3}\), "\<\"1110\"\>", "14", "\<\"4+1\"\>"}, {\({4}\), "\<\"0001\"\>", "1", "\<\"1+1+1+2\"\>"}, {\({1, 4}\), "\<\"1001\"\>", "9", "\<\"2+1+2\"\>"}, {\({2, 4}\), "\<\"0101\"\>", "5", "\<\"1+2+2\"\>"}, {\({1, 2, 4}\), "\<\"1101\"\>", "13", "\<\"3+2\"\>"}, {\({3, 4}\), "\<\"0011\"\>", "3", "\<\"1+1+3\"\>"}, {\({1, 3, 4}\), "\<\"1011\"\>", "11", "\<\"2+3\"\>"}, {\({2, 3, 4}\), "\<\"0111\"\>", "7", "\<\"1+4\"\>"}, {\({1, 2, 3, 4}\), "\<\"1111\"\>", "15", "\<\"5\"\>"} }], "\[NoBreak]", ")"}], (MatrixForm[ #]&)]], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["GOLDBACH'S CONJECTURE", "Subsection"], Cell["\<\ Goldbach's conjecture claims that every odd number greater than 7 is equal to \ the sum of three primes. Euler modified it to: every even number greater than \ 3 is equal to the sum of two primes. The latter implies the former but not \ vice-versa.\ \>", "Text"], Cell[BoxData[ \(Goldbach[n_] := Module[{p, a}, \[IndentingNewLine]p = Select[Range[2, n], PrimeQ]; \[IndentingNewLine]a = First[Select[n - p, PrimeQ, 1]]; \[IndentingNewLine]{n - a, a}]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[{n, asSum[Goldbach[n]]}, {n, 4, 50, 2}] // TableForm\)], "Input"], Cell[BoxData[ TagBox[GridBox[{ {"4", TagBox[\(2 + 2\), DisplayForm]}, {"6", TagBox[\(3 + 3\), DisplayForm]}, {"8", TagBox[\(3 + 5\), DisplayForm]}, {"10", TagBox[\(3 + 7\), DisplayForm]}, {"12", TagBox[\(5 + 7\), DisplayForm]}, {"14", TagBox[\(3 + 11\), DisplayForm]}, {"16", TagBox[\(3 + 13\), DisplayForm]}, {"18", TagBox[\(5 + 13\), DisplayForm]}, {"20", TagBox[\(3 + 17\), DisplayForm]}, {"22", TagBox[\(3 + 19\), DisplayForm]}, {"24", TagBox[\(5 + 19\), DisplayForm]}, {"26", TagBox[\(3 + 23\), DisplayForm]}, {"28", TagBox[\(5 + 23\), DisplayForm]}, {"30", TagBox[\(7 + 23\), DisplayForm]}, {"32", TagBox[\(3 + 29\), DisplayForm]}, {"34", TagBox[\(3 + 31\), DisplayForm]}, {"36", TagBox[\(5 + 31\), DisplayForm]}, {"38", TagBox[\(7 + 31\), DisplayForm]}, {"40", TagBox[\(3 + 37\), DisplayForm]}, {"42", TagBox[\(5 + 37\), DisplayForm]}, {"44", TagBox[\(3 + 41\), DisplayForm]}, {"46", TagBox[\(3 + 43\), DisplayForm]}, {"48", TagBox[\(5 + 43\), DisplayForm]}, {"50", TagBox[\(3 + 47\), DisplayForm]} }, RowSpacings->1, ColumnSpacings->3, RowAlignments->Baseline, ColumnAlignments->{Left}], (TableForm[ #]&)]], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Sums of Squares", "Section"], Cell[CellGroupData[{ Cell["SUMS OF TWO SQUARES", "Subsection"], Cell[TextData[{ "If N is the sum of two unequal positive square numbers, then every power \ of N is the sum of two squares. For instance, if N=", Cell[BoxData[ \(TraditionalForm\`n\^2\)]], "+", Cell[BoxData[ \(TraditionalForm\`m\^2\)]], " then ", Cell[BoxData[ \(TraditionalForm\`N\^2\)]], "=", Cell[BoxData[ FormBox[ SuperscriptBox[ RowBox[{"(", RowBox[{ FormBox[\(n\^2\), "TraditionalForm"], "-", FormBox[\(m\^2\), "TraditionalForm"]}], ")"}], "2"], TraditionalForm]]], "+", Cell[BoxData[ \(TraditionalForm\`\((2 mn)\)\^2\)]], ".\n\nIf N exceeds the sum of two unequal positive squares by k>0, then \ every power of N is the sum of k+2 positive squares.\n\nTheorem (Fermat). \ Write the canonical factorization of n in the form" }], "Text"], Cell[BoxData[ RowBox[{"n", "=", RowBox[{ FormBox[\(2\^\[Alpha]\), "TraditionalForm"], " ", FormBox[\(\[Product]\+\(p \[Congruent] 1\ \((mod\ 4)\)\)p\^\[Beta]\), "TraditionalForm"], " ", FormBox[\(\[Product]\+\(q\ \[Congruent] 3\ \((mod\ 4)\)\)q\^\[Gamma]\ \), "TraditionalForm"]}]}]], "Text", TextAlignment->Center, TextJustification->0], Cell["\<\ The n can be expressed as a sum of two squares of integers iff all exponents \ \[Gamma] are even. If p is an odd prime then p is representable as a sum of two squares if and \ only if p\[Congruent] 1 (mod 4). The set of positive integers which are sums \ of two squares is closed under multiplication. \ \>", "Text"], Cell[BoxData[ \(sum2Squares[n_] := Module[{f = FactorInteger[n]}, \[IndentingNewLine]And\ @@ \ EvenQ[Last\ /@ \ Select[f, \((Mod[First[#], 4] == 3)\) &]]]\)], "Input", InitializationCell->True], Cell[CellGroupData[{ Cell[BoxData[ \(Select[Range[100], sum2Squares]\)], "Input"], Cell[BoxData[ \({1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100}\)], "Output"] }, Open ]], Cell[BoxData[ \(sumTwoSquares[n_] := Module[{t, i, j}, \[IndentingNewLine]t = Table[\[IndentingNewLine]j = \@\(n - i\^2\); \[IndentingNewLine]If[ IntegerQ[ j], {{i, j}, {i, \(-j\)}}, {}], {i, \[LeftCeiling]\(-\@n\)\ \[RightCeiling], \[LeftFloor]\@n\[RightFloor]}]; \[IndentingNewLine]Union[ Sort\ /@ \ Partition[Flatten[t], 2]]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[n, "\< \>", \(asSum[#, 2] &\)\ /@ \ sumTwoSquares[n]], {n, 10}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"1", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-1"\^2 + "0"\^2\), DisplayForm], ",", TagBox[\("0"\^2 + "1"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 1, " ", { DisplayForm[ RowBox[ { Power[ "-1", 2], "+", Power[ "0", 2]}]], DisplayForm[ RowBox[ { Power[ "0", 2], "+", Power[ "1", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"2", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-1"\^2 + "-1"\^2\), DisplayForm], ",", TagBox[\("-1"\^2 + "1"\^2\), DisplayForm], ",", TagBox[\("1"\^2 + "1"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 2, " ", { DisplayForm[ RowBox[ { Power[ "-1", 2], "+", Power[ "-1", 2]}]], DisplayForm[ RowBox[ { Power[ "-1", 2], "+", Power[ "1", 2]}]], DisplayForm[ RowBox[ { Power[ "1", 2], "+", Power[ "1", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(3\[InvisibleSpace]" "\[InvisibleSpace]{}\), SequenceForm[ 3, " ", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"4", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-2"\^2 + "0"\^2\), DisplayForm], ",", TagBox[\("0"\^2 + "2"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 4, " ", { DisplayForm[ RowBox[ { Power[ "-2", 2], "+", Power[ "0", 2]}]], DisplayForm[ RowBox[ { Power[ "0", 2], "+", Power[ "2", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"5", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-2"\^2 + "-1"\^2\), DisplayForm], ",", TagBox[\("-2"\^2 + "1"\^2\), DisplayForm], ",", TagBox[\("-1"\^2 + "2"\^2\), DisplayForm], ",", TagBox[\("1"\^2 + "2"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 5, " ", { DisplayForm[ RowBox[ { Power[ "-2", 2], "+", Power[ "-1", 2]}]], DisplayForm[ RowBox[ { Power[ "-2", 2], "+", Power[ "1", 2]}]], DisplayForm[ RowBox[ { Power[ "-1", 2], "+", Power[ "2", 2]}]], DisplayForm[ RowBox[ { Power[ "1", 2], "+", Power[ "2", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(6\[InvisibleSpace]" "\[InvisibleSpace]{}\), SequenceForm[ 6, " ", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(7\[InvisibleSpace]" "\[InvisibleSpace]{}\), SequenceForm[ 7, " ", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"8", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-2"\^2 + "-2"\^2\), DisplayForm], ",", TagBox[\("-2"\^2 + "2"\^2\), DisplayForm], ",", TagBox[\("2"\^2 + "2"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 8, " ", { DisplayForm[ RowBox[ { Power[ "-2", 2], "+", Power[ "-2", 2]}]], DisplayForm[ RowBox[ { Power[ "-2", 2], "+", Power[ "2", 2]}]], DisplayForm[ RowBox[ { Power[ "2", 2], "+", Power[ "2", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"9", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-3"\^2 + "0"\^2\), DisplayForm], ",", TagBox[\("0"\^2 + "3"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 9, " ", { DisplayForm[ RowBox[ { Power[ "-3", 2], "+", Power[ "0", 2]}]], DisplayForm[ RowBox[ { Power[ "0", 2], "+", Power[ "3", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"10", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[\("-3"\^2 + "-1"\^2\), DisplayForm], ",", TagBox[\("-3"\^2 + "1"\^2\), DisplayForm], ",", TagBox[\("-1"\^2 + "3"\^2\), DisplayForm], ",", TagBox[\("1"\^2 + "3"\^2\), DisplayForm]}], "}"}]}], SequenceForm[ 10, " ", { DisplayForm[ RowBox[ { Power[ "-3", 2], "+", Power[ "-1", 2]}]], DisplayForm[ RowBox[ { Power[ "-3", 2], "+", Power[ "1", 2]}]], DisplayForm[ RowBox[ { Power[ "-1", 2], "+", Power[ "3", 2]}]], DisplayForm[ RowBox[ { Power[ "1", 2], "+", Power[ "3", 2]}]]}], Editable->False]], "Print"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Length[sumTwoSquares[n]], {n, 35}]\)], "Input"], Cell[BoxData[ \({2, 3, 0, 2, 4, 0, 0, 3, 2, 4, 0, 0, 4, 0, 0, 2, 4, 3, 0, 4, 0, 0, 0, 0, 6, 4, 0, 0, 4, 0, 0, 3, 0, 4, 0}\)], "Output"], Cell[BoxData[ \({4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "One surprising technique using the Gaussian integer divisors of n is the \ following (Solution by Stan Wagon Macalester College - see ", StyleBox["MathSource", FontSlant->"Italic"], " item # 0207-616)" }], "Text"], Cell[BoxData[ \(\(\(SumSquares[ n_]\)\(\ \)\(:=\)\(\ \)\(Union[\(Sort[{Re[#], \ Im[#]}]\ &\)\ /@ Select[Divisors[n, \ GaussianIntegers \[Rule] True], \ Abs[#]\^2\ == \ n\ &]]\)\(\ \)\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[n, "\< \>", \(asSum[#, 2] &\)\ /@ \ SumSquares[n]], {n, 10}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"1", 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"\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", TagBox[\("1"\^2 + "2"\^2\), DisplayForm], "}"}]}], SequenceForm[ 5, " ", { DisplayForm[ RowBox[ { Power[ "1", 2], "+", Power[ "2", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(6\[InvisibleSpace]" "\[InvisibleSpace]{}\), SequenceForm[ 6, " ", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[\(7\[InvisibleSpace]" "\[InvisibleSpace]{}\), SequenceForm[ 7, " ", {}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"8", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", TagBox[\("2"\^2 + "2"\^2\), DisplayForm], "}"}]}], SequenceForm[ 8, " ", { DisplayForm[ RowBox[ { Power[ "2", 2], "+", Power[ "2", 2]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"9", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", TagBox[\("0"\^2 + "3"\^2\), DisplayForm], "}"}]}], SequenceForm[ 9, " ", { DisplayForm[ RowBox[ { 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The number of representations of the number n \ \[GreaterEqual] 1 as the sum of two squares is " }], "Text"], Cell[TextData[{ "4( ", Cell[BoxData[ \(TraditionalForm\`\(d\_\(1, \ 4\)\)(n)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`d\_\(3, \ 4\)\)]], "(n) )" }], "Text", TextAlignment->Center, TextJustification->0], Cell[BoxData[ \(d\_\(r_, m_\)[n_] := Length[Select[Divisors[n], \((Mod[#, m] \[Equal] r)\) &]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(t = Table[4 \((d\_\(1, 4\)[n] - d\_\(3, 4\)[n])\), {n, 30}]\)], "Input"], Cell[BoxData[ \({4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0}\)], "Output"] }, Open ]], Cell["Note that negative squares are taken into account.", "Text"], Cell[TextData[{ Cell[BoxData[ \(TraditionalForm\`lim\+\(n \[RightArrow] \[Proportional] \)\)]], Cell[BoxData[ \(TraditionalForm\`1\/n\)]], Cell[BoxData[ \(TraditionalForm\`\[Sum]\+\(i = 0\)\%n t\ [i]\)]], " = \[Pi]" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Plus\ @@ \ t\/30 // N\)], "Input"], Cell[BoxData[ \(3.2`\)], "Output"] }, Open ]], Cell["\<\ t represents the number of lattice points that lie on a circle centered at \ the origin an integral radius.(Gauss) (Dirichlet, 1840) The number of representations of n \[GreaterEqual] 1 as the \ sum of a square and twice a square is \ \>", "Text"], Cell[TextData[{ "2( ", Cell[BoxData[ \(TraditionalForm\`\(d\_\(1, \ 8\)\)(n)\)]], " + ", Cell[BoxData[ \(TraditionalForm\`d\_\(3, \ 8\)\)]], "(n) - ", Cell[BoxData[ \(TraditionalForm\`\(d\_\(5, \ 8\)\)(n)\)]], " - ", Cell[BoxData[ \(TraditionalForm\`d\_\(7, \ 8\)\)]], "(n) )" }], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 9;\)\), "\[IndentingNewLine]", \(\(u = Map[{{#, \@\(n - 2 #\^2\)}, {#, \(-\@\(n - 2 #\^2\)\)}} &, Select[Range[\[LeftCeiling]\(-n\)\/2\[RightCeiling], n\/2], IntegerQ[\@\(n - 2 #\^2\)] &]];\)\), "\[IndentingNewLine]", \(\(asSum[#, 2] &\) /@ \ Map[{"\<2\[Times]\>" <> ToString[First[#]], Last[#]} &, Union[Flatten[u, 1]]]\)}], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[\("2\[Times]-2"\^2 + "-1"\^2\), DisplayForm], ",", TagBox[\("2\[Times]-2"\^2 + "1"\^2\), DisplayForm], ",", TagBox[\("2\[Times]0"\^2 + "-3"\^2\), DisplayForm], ",", TagBox[\("2\[Times]0"\^2 + "3"\^2\), DisplayForm], ",", TagBox[\("2\[Times]2"\^2 + "-1"\^2\), DisplayForm], ",", TagBox[\("2\[Times]2"\^2 + "1"\^2\), DisplayForm]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ 2 \((d\_\(1, 8\)[n] + d\_\(3, 8\)[n] - d\_\(5, 8\)[n] - d\_\(7, 8\)[n])\), {n, 20}]\)], "Input"], Cell[BoxData[ \({2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, 0, 0, 0, 2, 4, 6, 4, 0}\)], "Output"] }, Open ]], Cell["\<\ (Lorenz,1829) The number of representations of n\[GreaterEqual]1 as the sum \ of a square and three times a square is\ \>", "Text"], Cell[TextData[{ "2( ", Cell[BoxData[ \(TraditionalForm\`d\_\(1, \ 3\)\)]], "(n) - ", Cell[BoxData[ \(TraditionalForm\`d\_\(2, \ 3\)\)]], "(n) ) + 4( ", Cell[BoxData[ \(TraditionalForm\`d\_\(4, \ 12\)\)]], "(n) - ", Cell[BoxData[ \(TraditionalForm\`d\_\(8, \ 12\)\)]], "(n) )" }], "Text", TextAlignment->Center, TextJustification->0], Cell[CellGroupData[{ Cell[BoxData[{ \(\(n = 12;\)\), "\[IndentingNewLine]", \(\(u = Map[{{#, \@\(n - 3 #\^2\)}, {#, \(-\@\(n - 3 #\^2\)\)}} &, Select[Range[\[LeftCeiling]\(-n\)\/3\[RightCeiling], n\/3], IntegerQ[\@\(n - 3 #\^2\)] &]];\)\), "\[IndentingNewLine]", \(\(asSum[#, 2] &\) /@ \ Map[{"\<3\[Times]\>" <> ToString[First[#]], Last[#]} &, Union[Flatten[u, 1]]]\)}], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{ TagBox[\("3\[Times]-2"\^2 + "0"\^2\), DisplayForm], ",", TagBox[\("3\[Times]-1"\^2 + "-3"\^2\), DisplayForm], ",", TagBox[\("3\[Times]-1"\^2 + "3"\^2\), DisplayForm], ",", TagBox[\("3\[Times]1"\^2 + "-3"\^2\), DisplayForm], ",", TagBox[\("3\[Times]1"\^2 + "3"\^2\), DisplayForm], ",", TagBox[\("3\[Times]2"\^2 + "0"\^2\), DisplayForm]}], "}"}]], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ 2 \((d\_\(1, 3\)[n] - d\_\(2, 3\)[n])\) + 4 \((d\_\(4, 12\)[n] - d\_\(8, 12\)[n])\), {n, 20}]\)], "Input"], Cell[BoxData[ \({2, 0, 2, 6, 0, 0, 4, 0, 2, 0, 0, 6, 4, 0, 0, 6, 0, 0, 4, 0}\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["SUMS OF THREE SQUARES", "Subsection"], Cell[TextData[{ "An integer is a sum of 3 squares if it is not of the form ", Cell[BoxData[ \(TraditionalForm\`4\^n\)]], "(8m+7), n, m \[GreaterEqual] 0. We include 0 among the powers." }], "Text"], Cell[BoxData[ \(sumThreeSquares[n_] := Module[{t, i, j, k}, \[IndentingNewLine]t = Table[\[IndentingNewLine]k = \@\(n - i\^2 - j\^2\); \ \[IndentingNewLine]If[ IntegerQ[k], {i, j, k}, {}], {i, \[LeftCeiling]\(-\@n\)\[RightCeiling], \ \[LeftFloor]\@n\[RightFloor]}, {j, \[LeftCeiling]\(-\@\(n - i\^2\)\)\ \[RightCeiling], \[LeftFloor]\@\(n - i\^2\)\[RightFloor]}]; \ \[IndentingNewLine]Union[Sort\ /@ \ Partition[Flatten[t], 3]]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[n, "\< \>", \(asSum[#, 2] &\)\ /@ \ sumThreeSquares[n]], {n, 10}]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"1", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", RowBox[{ TagBox[ RowBox[{\("-1"\^2\), "+", TagBox[\("0"\^2 + "0"\^2\), DisplayForm]}], DisplayForm], ",", TagBox[ RowBox[{\("0"\^2\), "+", TagBox[\("0"\^2 + "1"\^2\), DisplayForm]}], DisplayForm]}], "}"}]}], SequenceForm[ 1, " ", { DisplayForm[ RowBox[ { Power[ "-1", 2], 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One solution is to let S be the set of all strings of 0's and 1's such \ that the number of 1's is a sum of two squares. Lagrange's theorem \ establishes that every number is a sum of four squares (of non-negative \ integers), so every string of 0's and 1's is a concatenation of two strings \ chosen from S.\ \>", "Commentary"] }, Closed]] }, Open ]], Cell[CellGroupData[{ Cell["Sums of Powers", "Section"], Cell[CellGroupData[{ Cell["SUMS OF CUBES", "Subsection"], Cell[CellGroupData[{ Cell[BoxData[ \(Do[Print[n, "\< \>", \(asSum[#, 3] &\)\ /@ \ powersSum[n, 3]], {n, 10}]\)], "Input"], Cell[BoxData[ InterpretationBox[\(1\[InvisibleSpace]" "\[InvisibleSpace]{"1"\^3}\), SequenceForm[ 1, " ", { Power[ "1", 3]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"2", "\[InvisibleSpace]", "\<\" \"\>", "\[InvisibleSpace]", RowBox[{"{", TagBox[\("1"\^3 + "1"\^3\), DisplayForm], "}"}]}], SequenceForm[ 2, " ", { DisplayForm[ RowBox[ { Power[ "1", 3], "+", Power[ "1", 3]}]]}], Editable->False]], "Print"], Cell[BoxData[ InterpretationBox[ RowBox[{"3", "\[InvisibleSpace]", "\<\" \"\>", 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He was curious that the cost was the same as the store name, \ so he inquired as to how the figure was derived. The clerk said that he had simply multiplied the prices of the four \ individual items. The customer protested that the four prices should \ have been ADDED, not MULTIPLIED. The clerk said that that was OK with him, \ but, the result was still the same: exactly $7.11. What were the costs \ involved?\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(u = \@7.11\%4;\)\ \ \), "\n", \(\(Do[\[IndentingNewLine]l = 7.11 - i - j - k; \[IndentingNewLine]If[\((l > 0)\)\ && \ \((i\ j\ k\ l == 7.11)\), \ Print[{i, j, k, l}]], {i, 0.01, u, 0.01}, {j, i, u, 0.01}, {k, j, u, 0.01}];\)\ // \ Timing\)}], "Input"], Cell[BoxData[ \({1.2`, 1.25`, 1.5`, 3.16`}\)], "Print"], Cell[BoxData[ \({47.789999999999964`\ Second, Null}\)], "Output"] }, Open ]], Cell["\<\ So, the items were 1.2, 1.25, 1.5 and 3.16 worth. However, in order to generalize we need to dwelve into factorizing a \ number.\ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell["FACTORING A NUMBER INTO UNORDERED PRODUCTS", "Subsection"], Cell["\<\ To obtain all possible lists of r elements multiplying up to n.\ \>", "Text"], Cell[BoxData[{ \(products[n_, r_] := products[n, r, 1]\n\t\t\), "\n", \(products[n_, r_, from_] := {}\ /; \ from\^r > n\), "\n", \(products[n_, r_, from_] := {Table[from, {r}]}\ /; \ \ from\^r == n\), "\n", \(products[n_, 1, from_] := {{n}}\ \), "\n", \(products[n_, r_, from_] := Module[{a, p, ans = {}}, \[IndentingNewLine]Do[\[IndentingNewLine]If[ Mod[n, a] == 0, \[IndentingNewLine]p = products[n/a, r - 1, a]; \[IndentingNewLine]If[p != {}, AppendTo[ans, Map[Join[{a}, #] &, p]]]], {a, from, \@n\%r}]; \[IndentingNewLine]Flatten[ans, 1]]\t\t\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(products[12, 5]\)], "Input"], Cell[BoxData[ \({{1, 1, 1, 1, 12}, {1, 1, 1, 2, 6}, {1, 1, 1, 3, 4}, {1, 1, 2, 2, 3}}\)], "Output"] }, Open ]], Cell["\<\ To obtain all possible lists of r elements multiplying up to n and adding up \ to s.\ \>", "Text"], Cell[BoxData[{ \(prodSum[p_, s_, r_] := prodSum[p, s, r, 1]\n\t\t\), "\n", \(prodSum[p_, s_, r_, from_] := {}\ /; \ \((from\^r > p)\)\ || \ \((p > \((s\/r)\)\^r)\)\), "\n", \(prodSum[p_, s_, r_, from_] := {Table[ from, {r}]}\ /; \ \ \((from\^r == p)\)\ && \ \((r\ from == s)\)\), "\n", \(prodSum[p_, _, r_, from_] := {}\ /; \ \ \(\(from\^r\)\(==\)\(p\)\(\t\)\)\), "\n", \(prodSum[p_, p_, 1, from_] := {{p}}\ \), "\n", \(prodSum[_, _, 1, _] := {}\ \), "\n", \(prodSum[p_, s_, r_, from_] := Module[{a, q, ans = {}}, \[IndentingNewLine]Do[\[IndentingNewLine]If[ Mod[p, a] == 0, \[IndentingNewLine]q = prodSum[p/a, s - a, r - 1, a]; \[IndentingNewLine]If[q != {}, AppendTo[ans, Map[Join[{a}, #] &, q]]]], {a, from, \@p\%r}]; \[IndentingNewLine]Flatten[ans, 1]]\t\t\)}], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(prodSum[12, 10, 5]\)], "Input"], Cell[BoxData[ \({{1, 1, 1, 3, 4}}\)], "Output"] }, Open ]], Cell["Regarding our problem:", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(prodSum[711000000, 711, 4]\ // \ Timing\)], "Input"], Cell[BoxData[ \({0.160000000000025`\ Second, {{120, 125, 150, 316}}}\)], "Output"] }, Open ]], Cell["Given a product of r numbers, what are their possible sums ?", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(t = Table[Table[Length[prodSum[p, s, r]], {p, 25}, {s, 25}], {r, 25}];\)\), "\n", \(t\[LeftDoubleBracket]5\[RightDoubleBracket] // MatrixForm\)}], "Input"], Cell[BoxData[ TagBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {"0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", "0", "0", "1", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0", "0"}, {"0", "0", "0", "0", 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Hence, there are 4 ways of \ finding numbers adding to 10 and multiplying to 16, namely:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(prodSum[16, 10, 2]\), "\n", \(prodSum[16, 10, 4]\), "\n", \(prodSum[16, 10, 5]\), "\n", \(prodSum[16, 10, 6]\)}], "Input"], Cell[BoxData[ \({{2, 8}}\)], "Output"], Cell[BoxData[ \({{1, 1, 4, 4}}\)], "Output"], Cell[BoxData[ \({{1, 1, 2, 2, 4}}\)], "Output"], Cell[BoxData[ \({{1, 1, 2, 2, 2, 2}}\)], "Output"] }, Open ]], Cell["\<\ Note also that, if the product p is prime, the above matrix correctly asserts \ that there is only one solution of the equation s = p + r - 1 for r>0 and no \ solution for any other value of s. 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Burn \nCambridge University Press\n\n\ An Introduction to the Theory of Numbers\nI. Niven, H.S. Zuckerman, H.L. \ Montgomery\nJohn Wiley & Sons\n\nAn Introduction to the Theory of Numbers\n\ G.H. Hardy\nOxford Science Publications\n\nElementary Number Theory\nJ. \ Roberts\nThe MIT Press\n\nThe Theory of Partitions\nG. E. Andrews\nCambridge \ ", StyleBox["Mathematica", FontSlant->"Italic"], "l Library\n\nEric Weisstein's World of Mathematics\n\ http://mathworld.wolfram.com/\n\nPartial Fractions and Four Classical \ Theorems of Number Theory\nM.D. Hirschhorn\nAmerican Mathematical Monthly \ 107,3, March 2000, pp. 260-264\n\nHandbook of Number Theory\nD.S. Mitrinovic, \ J. Sandor, B. Crstici\nKluwer Academic Publishers, 1996\n\nA Combinatorial \ Approach to Sums of Integer Powers\nG. Mackiw\nMathematics Magazine 73, 1, \ February 2000\n\nThe Impossible Problem\nL. Sallows\nThe ", StyleBox["Mathematica", FontSlant->"Italic"], "l Intelligencer 17, 1, 1995 pp. 27-33\n" }], "Reference"] }, Closed]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 979}}, AutoGeneratedPackage->None, WindowToolbars->"EditBar", WindowSize->{811, 807}, WindowMargins->{{109, Automatic}, {Automatic, 72}}, Visible->True, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrivateNotebookOptions->{"ColorPalette"->{RGBColor, -1}}, ShowCellLabel->True, ShowCellTags->False, InputAliases->{"notation"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation>"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongRightArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "notation<"->RowBox[ {"Notation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], " ", "\[DoubleLongLeftArrow]", " ", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"]}], "]"}], "symb"->RowBox[ {"Symbolize", "[", TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], "]"}], "infixnotation"->RowBox[ {"InfixNotation", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "addia"->RowBox[ {"AddInputAlias", "[", RowBox[ { TagBox[ "\[Placeholder]", NotationBoxTag, TagStyle -> "NotationTemplateStyle"], ",", "\[Placeholder]"}], "]"}], "pattwraper"->TagBox[ "\[Placeholder]", NotationPatternTag, TagStyle -> "NotationPatternWrapperStyle"], "madeboxeswraper"->TagBox[ "\[Placeholder]", NotationMadeBoxesTag, TagStyle -> "NotationMadeBoxesWrapperStyle"]}, RenderingOptions->{"ObjectDithering"->True, "RasterDithering"->False}, Magnification->1.25, StyleDefinitions -> "Report.nb" ] (*********************************************************************** Cached data follows. 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