(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 7488, 239]*) (*NotebookOutlinePosition[ 8133, 261]*) (* CellTagsIndexPosition[ 8089, 257]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[TextData[{ "Solving ", Cell[BoxData[ \(TraditionalForm\`x > 13874 Abs[x\^3 - 3 y\^3]\)]] }], "Subtitle"], Cell["by Ed Pegg Jr.", "Text"], Cell["\<\ Since the days of Fermat, it has been an ongoing challenge to find \ simple-appearing equations with large solutions. The above equation seems \ simple enough, and can be easily solved with continued fractions. It can be \ called a Diophantine Inequality. I'll start with the answer, and then work \ back to how it was gotten.\ \>", "Text"], Cell[BoxData[ \(<< NumberTheory`ContinuedFractions`\)], "Input"], Cell[BoxData[ \(\({x, y}\ = \ With[{k = \ \(Convergents[ ContinuedFraction[3^\((1/3)\), 4000]]\)[\([3432]\)]}, {Numerator[k], Denominator[k]}];\)\)], "Input"], Cell[TextData[{ "We now have the answer! I won't print them out completely, since these are \ huge numbers. ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["can", FontWeight->"Bold"], " print the numbers out easily, but it can also express them in a shortened \ form, with Short[ ]. Each number has 1781 digits. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \({Floor[Log[10, x]], Short[x, .5], Short[y, .5]}\)], "Input"], Cell[BoxData[ RowBox[{"{", RowBox[{"1781", ",", TagBox[\(859039215272799721 \[LeftSkeleton]1746\[RightSkeleton] 640453686806155041\), (Short[ #, .5]&)], ",", TagBox[\(595624525018717634 \[LeftSkeleton]1746\[RightSkeleton] 162512473362587582\), (Short[ #, .5]&)]}], "}"}]], "Output"] }, Open ]], Cell["We can test that they actually work.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(x\ > \ 13874 Abs[x\^3 - \ 3\ y\^3]\)], "Input"], Cell[BoxData[ \(True\)], "Output"] }, Open ]], Cell[TextData[{ "We can use the function below to look for other solutions of the form ", Cell[BoxData[ \(x\ > \ a\ Abs[x\^b - \ c\ y\^b]\)]], ", where ", StyleBox["b", FontSlant->"Italic"], " is the power, and ", StyleBox["c", FontSlant->"Italic"], " is the root." }], "Text"], Cell[BoxData[ \(CFRoots[root_, power_, \ range_] := Take[Reverse[ Sort[Transpose[{Map[ Floor[Numerator[#]/ Abs[root\ Denominator[#]^power\ - \ Numerator[#]^power]] &, Convergents[ContinuedFraction[root^\((1/power)\), range]]], Range[range]}]]], 10]\)], "Input"], Cell[TextData[{ "Can the ", StyleBox["a", FontSlant->"Italic"], " term in ", Cell[BoxData[ \(x\ > \ a\ Abs[x\^b - \ c\ y\^b]\)]], "be greater than a million? Definitely ... but I haven't found an example, \ yet. If you find one, send it to me at ed@mathpuzzle.com. Here's the sort \ of thing that necessary, in the continued fraction. First, here's what the \ cube root of three looks like, as a continued fraction." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ContinuedFractionForm[ContinuedFraction[3^\((1/3)\), 15]]\)], "Input"], Cell[BoxData[ InterpretationBox[ StyleBox[\(1 + 1\/\(2 + 1\/\(3 + 1\/\(1 + 1\/\(4 + 1\/\(1 + 1\/\(5 + 1\/\(1 + \ 1\/\(1 + 1\/\(6 + 1\/\(2 + 1\/\(5 + 1\/\(8 + 1\/\(3 + \ 1\/3\)\)\)\)\)\)\)\)\)\)\)\)\)\), ScriptSizeMultipliers->1], ContinuedFractionForm[ {1, 2, 3, 1, 4, 1, 5, 1, 1, 6, 2, 5, 8, 3, 3}]]], "Output"] }, Open ]], Cell["\<\ Since the important part is the set of numbers, we can look at those only.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ContinuedFraction[3^\((1/3)\), 70]\)], "Input"], Cell[BoxData[ \({1, 2, 3, 1, 4, 1, 5, 1, 1, 6, 2, 5, 8, 3, 3, 4, 2, 6, 4, 4, 1, 3, 2, 3, 4, 1, 4, 9, 1, 8, 4, 3, 1, 3, 2, 6, 1, 6, 1, 3, 1, 1, 1, 1, 12, 3, 1, 3, 1, 1, 4, 1, 6, 1, 5, 1, 2, 1, 3, 3, 11, 8, 1, 139, 8, 2, 8, 5, 1, 2}\)], "Output"] }, Open ]], Cell["\<\ Places where the number is unusually high, in this case 139, indicates a \ place where the continued fraction is more accurate than might be expected. \ \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[ Floor[Numerator[#]/Abs[3\ Denominator[#]^3\ - \ Numerator[#]^3]] &, Convergents[ContinuedFraction[3^\((1/3)\), 70]]]\)], "Input"], Cell[BoxData[ \({0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 2, 2, 0, 32, 1, 0, 1, 1, 0, 0, 0}\)], "Output"] }, Open ]], Cell[BoxData[ \(\({\(,\)\({3369, 1990}\)\(,\)\({2209, 2406}\)\(,\)\({3129, 578}\)\(,\)\({9740, 812}\)\(,\)\({772, 1020}\)\(,\)\({418, 2240}\)\(,\)\(,\)\({1148, 2741}\)\(,\)\({422, 1377}\)\(,\)\({511, 532}\)\(,\)\({1154, 2903}\)\(,\)\({1317, 1063}\)\(,\)\({1865, 677}\)\(,\)\({1103, 126}\)};\)\)], "Input"], Cell[TextData[{ "I'll close with a few puzzles that are easily solved with ", StyleBox["Mathematica", FontSlant->"Italic"], ", using the same techniques. Find the minimal {x,y} for the below \ diophantine inequalities.\n\n1. ", Cell[BoxData[ \(x\ > \ 57 Abs[x\^3 - \ 3\ y\^3]\)]], "\n2. ", Cell[BoxData[ \(x\ > \ 56 Abs[x\^3 - \ 4\ y\^3]\)]], "\n3. ", Cell[BoxData[ \(x\ > \ 595 Abs[x\^3 - \ 5\ y\^3]\)]], "\n4. ", Cell[BoxData[ \(x\ > \ 93 Abs[x\^3 - \ 6\ y\^3]\)]], "\n5. ", Cell[BoxData[ \(x\^2\ > \ 214 Abs[x\^4 - \ 2\ y\^4]\)]], "\n6. ", Cell[BoxData[ \(x\^2\ > \ 3 Abs[x\^5 - \ 11\ y\^5]\)]] }], "Text"] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 723}}, WindowSize->{905, 527}, WindowMargins->{{50, Automatic}, {Automatic, 19}} ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 121, 4, 52, "Subtitle"], Cell[1900, 59, 30, 0, 33, "Text"], Cell[1933, 61, 353, 6, 52, "Text"], Cell[2289, 69, 68, 1, 30, "Input"], Cell[2360, 72, 224, 5, 30, "Input"], Cell[2587, 79, 361, 10, 52, "Text"], Cell[CellGroupData[{ Cell[2973, 93, 82, 1, 30, "Input"], Cell[3058, 96, 357, 8, 29, "Output"] }, Open ]], Cell[3430, 107, 52, 0, 33, "Text"], Cell[CellGroupData[{ Cell[3507, 111, 69, 1, 31, "Input"], Cell[3579, 114, 38, 1, 29, "Output"] }, Open ]], Cell[3632, 118, 306, 11, 33, "Text"], Cell[3941, 131, 390, 8, 130, "Input"], Cell[4334, 141, 453, 11, 52, "Text"], Cell[CellGroupData[{ Cell[4812, 156, 90, 1, 30, "Input"], Cell[4905, 159, 356, 8, 196, "Output"] }, Open ]], Cell[5276, 170, 98, 2, 33, "Text"], Cell[CellGroupData[{ Cell[5399, 176, 67, 1, 30, "Input"], Cell[5469, 179, 269, 4, 48, "Output"] }, Open ]], Cell[5753, 186, 178, 4, 33, "Text"], Cell[CellGroupData[{ Cell[5956, 194, 168, 3, 50, "Input"], Cell[6127, 199, 266, 4, 48, "Output"] }, Open ]], Cell[6408, 206, 353, 5, 50, "Input"], Cell[6764, 213, 708, 23, 166, "Text"] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)