Nancy Street - Hobbies - Mathematica

	E-Root-Pi

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In 1975, rumours were passing around Monash University that was an integer, causing a bit of head scratching about this strange "coincidence". At the time we had no calculators or software capable of giving the exact value with suitable precision to verify the claim. Up in the maths labs they had some huge desk calculators that had about 16 digits (I can't remember how many digits exactly), but they weren't accurate enough either. Even now in 2002 my brand new Sharp EL-5120 calculatoronly gives the value as: 2.625374125e17. Mathematica shows us the answer is not an integer, but it's amazingly close.

It turns out the rumour of the value being an integer was sparked by an April Fools joke from Martin Gardner writing in Scientific American. For more information see:

If you can wade through some of the mathematics in the first link you will see that our mystery 163 related value is close to:

And the reasons this is so are tangled up in Heegner Numbers, Class Numbers and Imaginary Quadratic Fields and other higher mathematics.

The last link is to a book titled Pi Unleashed. This fabulous book discusses the 4000 year history of Pi, covering every imaginable aspect of the number at various levels of difficulty. This book also discusses the phenomena where strings of 9's appear in the decimal expansion of Pi, which I think is related to the E-Root-Pi phenomena via the j-function and the work of the famous mathematician Ramanujan.

I see that many people have calculated higher values to look for more "coincidences" in the E-Root-Pi phenomena, so I just had to have a bash at it myself. I used Mathematica to blindly search up through the powers to 10000 looking for values that are within 0.0005 of an integer.

$MaxExtraPrecision = 200;
upper = 1000;
tolerance = 0.0005;
Clear[lstplot]; lstplot = {};
Clear[lstsel]; lstsel = {};
Do[
   val = Exp[Pi Sqrt[n]];
   len = Floor[Log[10, ipart] + 1];
   ipart = IntegerPart[val];
   fpart = N[FractionalPart[val], 20];
   If[fpart > tolerance && fpart < 1 - tolerance, Continue[]];
   nf = NumberForm[N[val, len + 15],
      ExponentFunction -> (If[Abs[#] < 100, Null, #] &)];
   plotdiff = If[fpart <= tolerance, fpart, fpart - 1];
   lstsel = Append[lstsel, {n, Floor[Log[10, ipart] + 1], NumberForm[N[fpart, 18],
       ExponentFunction -> (If[Abs[#] < 140, Null, #] &)], Floor[Log[10, Abs[plotdiff]]]}];
   lstplot = Append[lstplot, {n, plotdiff}];
   Print[{n, nf}], {n, 1, upper}]

{  37, 199148647.999978046551857}
{  43, 884736743.999777466034907}
{  58, 24591257751.999999822213241}
{  67, 147197952743.999998662454225}
{ 163, 262537412640768743.999999999999250}
{ 232, 604729957825300084759.999992171526856}
{ 268, 21667237292024856735768.000292038842413}
{ 522, 14871070263238043663567627879007.999848726482795}
{ 652, 68925893036109279891085639286943768.000000000163739}
{ 719, 3842614373539548891490294277805829192.999987249566012}
{1169, 44555719382988281777368496770130045948309444044.999960802863868}
{1194, 139661526073504116557581973059759277212070858620.000390060318804}
{1467, 18095625621654510801615355531263454706630064771074975.999999990123694}
{1519, 149858811115653723490615404030746848884530532908161003.000334908620251}
{1850, 48310987197300327887464627364483701432184761367510399635913.999859502992241}
{2086, 206451827879257016814581919876113403019895777062127238001987714.999638099591867}
{2608, 4750778730825177725463920948909726618214491718039471366318747406368792.000000308464322}
{3368, 15165684121539667152189321546204272264948042335714455797675844105845908998393075.999812089087555}
{4075, 1247257156019637304856107520018074552566824585862995272173368815794085495792299621093743.999993654187469}
{5773, 4.6309587632860353087565367053742331250287153098248715578209888177688338779879045292937243508078581367989999915568853814×10^103}
{5774, 4.7276902786418157286141596960874283191339041811828552065002280342087376838477704381388918066542704829123000314593384839×10^103}
{5868, 3.27451666639079200503292535866541250265248788274691526825971156747731856100971255480468836963064283775072000097175254163×10^104}
{7942, 3.8936301432067213757267567003279536256382650708761582766848885427711285563811259082729255749693482335240556580390666283873999745090020369×10^121}
{9058, 7.120459468060721603518969580597318101378057175923683991559410705137623562390661799382568239069778563505991817785232498957426610310000449005922486×10^129}

To show the near-integers a bit more clearly we can reformat the results into columns showing the power, the number of integer digits in the value, the fraction part of the value and the exponent of the "distance" from an integer.

TableForm[lstsel, TableAlignments -> {Right}]
ListPlot[lstplot, PlotJoined -> True, AspectRatio -> 1/3,
    AxesOrigin -> {0, 0}, 
    PlotRange -> {{1, upper}, {-1 tolerance, tolerance}}, Frame -> True];

  37   9  0.999978046551856767          -5
  43   9  0.999777466034906662          -4
  58  11  0.999999822213241470          -7
  67  12  0.999998662454224507          -6
 163  18  0.99999999999925007           -13
 232  21  0.999992171526856430          -6
 268  23  0.000292038842412959454       -4
 522  32  0.999848726482794815          -4
 652  35  0.000000000163738644209234608 -10
 719  37  0.999987249566012188          -5
1169  47  0.999960802863868462          -5
1194  48  0.000390060318803553269       -4
1467  53  0.999999990123693671          -9
1519  54  0.000334908620251054552       -4
1850  59  0.999859502992240728          -4
2086  63  0.999638099591866692          -4
2608  70  0.000000308464322129981180    -7
3368  80  0.999812089087554565          -4
4075  88  0.999993654187468972          -6
5773 104  0.999915568853814166          -5
5774 104  0.000314593384838513470       -4
5868 105  0.0000971752541625920841      -5
7942 122  0.999745090020369282          -4
9058 130  0.000449005922486202180       -4

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