3

If we define a $PPTCountingFunction(n)$ as a function that returns the number of PPF with $c < n$ and $a>b$, then up to first $n=100,000$ it is near linear and

$\dfrac{n}{PPTCountingFunction(n)}=2\pi$

I have several questions (third question is the most interesting to me):

(1) Is this also an asymptotic behavior of this function, or does it have some other slowly changing factors that are not showing up when n is small?

(2) Is there a clear reasoning for frequencies of PPT?

(3) Can we apply similar reasoning to estimate the frequency of primitive counterexamples to Euler's hypothesis for $n=4$ (solution s of $A^{4}+B^{4}+C^{4}=D^{4}$)?

Regarding (3). First solution appears at $95800^{4} + 414560^{4} + 217519^{4} = 422481^{4}$. This is the only solution with $D<2000000$. Another known solution (not necessarily second) is $2682440^{4} + 15365639^{4} + 18796760^{4} = 20615673^{4}$. I am curious if there is a point to look for a solution between these two.

anomaly
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sixtytrees
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    This question (http://math.stackexchange.com/questions/1270902/congruence-properties-of-x-16x-26x-36x-46x-56-z6) quotes a solution of $A^4+B^4+C^4=D^4$ with $D=2813001$. – Adam Bailey Jul 10 '16 at 21:38
  • Please [edit] your post to include the definition of a PPF and a PPT in the body of your post. – D.W. Feb 15 '24 at 04:13

4 Answers4

4

The list of hypotenuses is A020882 in OEIS. The following analysis follows the third comment on that sequence, and provides a reasonability argument, though not a proof.

Counting primitive triples with hypotenuse at most $n$ is the same as counting pairs $(a,b)$ with $\gcd(a,b)=1$, $a$ and $b$ not both odd, and $a>b>0$ inside the circle of radius $\sqrt{n}$. The total number of pairs $(a,b)$ inside such a circle is $\approx \pi n$ (see here, for example). Only $\frac{1}{8}$ of these have $a>b>0$; of these, only $\frac{6}{\pi^2}$ of them are relatively prime. Finally, asking that $a$ and $b$ be not both odd reduces by another factor of $\frac{2}{3}$ (note that $a$, $b$ both even was excluded by the $\gcd$). So altogether, the number of qualifying points is $$n\cdot\pi\cdot\frac{1}{8}\cdot\frac{6}{\pi^2}\cdot\frac{2}{3} = \frac{n}{2\pi}.$$ Thus $\text{ppt}(n)\approx \frac{n}{2\pi}$, and your result follows.

rogerl
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Here is the list of 27 known solutions found before 2010 and below a bound. (Update: As of 2023, there are 27+4 = 31 known solutions below $10^{28}$, the 4 found in 2015 by Bremner and all higher than #20 in this list.)

The list was brute-forced only up to $2\times10^9$ . Other cases were discovered using elliptic curves, which doesn't guarantee completeness of the list above 2 billion. We can notice that the pattern is less regular than that of PPT. It is neither linear not exponential. Conclusions derived from 30 examples may not be accurate. Table below is of the form $(d;c,b,a)$ for $d^4=c^4+b^4+a^4$.

1] 422481; 414560, 217519, 95800 (Roger Frye, 1988)
2] 2813001; 2767624, 1390400, 673865 (Allan MacLeod 1997)
3] 8707481; 8332208, 5507880, 1705575 (D.J. Bernstein, 2001)
4] 12197457; 11289040, 8282543, 5870000 (D.J. Bernstein, 2001)
5] 16003017; 14173720, 12552200, 4479031 (D.J. Bernstein, 2001)
6] 16430513; 16281009, 7028600, 3642840 (D.J. Bernstein, 2001)
7] 20615673; 18796760, 15365639, 2682440 (Noam Elkies, 1986)
8] 44310257; 41084175, 31669120, 2164632 (Robert Gerbicz, 11/08/2006)
9] 68711097; 65932985, 42878560, 10409096 (Robert Gerbicz, 11/08/2006)
10] 117112081; 106161120, 87865617, 34918520 (Robert Gerbicz, 11/02/2006)
11] 145087793; 122055375, 121952168, 1841160 (Juergen Rathmann, 5/31/2007)
12] 156646737; 146627384, 108644015, 27450160 (Juergen Rathmann, 6/1/2007)
13] 589845921; 582665296, 260052385, 186668000 (Seiji Tomita, 03/13/2006)
14] 638523249; 630662624, 275156240, 219076465 (Allan MacLeod,1998)
15] 873822121; 769321280, 606710871, 558424440 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov 11/2/2007)
16] 1259768473; 1166705840, 859396455, 588903336 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov 01/25/2008)
17] 1679142729; 1670617271, 632671960, 50237800 (Seiji Tomita, 03/13/2006)
18] 1787882337; 1662997663, 1237796960, 686398000 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov 11/2/2007)
19] 1871713857; 1593513080, 1553556440, 92622401 (Robert Gerbicz, Leonid Durman, Yuri Radaev, Alexey Zubkov 10/31/2007)
20] 3393603777; 3134081336, 2448718655, 664793200 (Seiji Tomita, 01/28/2007)
21] 12558554489; 11988496761, 7813353720, 4707813440 (Andrew Bremner, 2015)
22] 15434547801; 15355831360, 5821981400, 140976551 (Seiji Tomita, 10/24/2007)
23] 1367141947873; 1226022682752, 1047978087905, 408600530760 (Andrew Bremner, 2015)
24] 5062297699257; 4987588419655, 2480452675600, 502038853976 (Seiji Tomita, 05/15/2008)
25] 29999857938609; 27239791692640, 22495595284040, 7592431981391 (Seiji Tomita, 03/13/2006)
26] 573646321871961; 514818101299289, 440804942580160, 130064300991400 (Seiji Tomita, 09/15/2008)
27] 20249506709579721; 18565945114216720, 14890026433468471, 3579087147375440 (Seiji Tomita, 08/13/2008)
28] 62940516903410601; 56827813308111785, 47886740272114976, 8813425670440240 (Seiji Tomita, 08/13/2008)
29] 2778996090487120353; 2556827383749699103, 2024155336530384440, 585715960903147640 (Andrew Bremner, 2015)
30] 31293260543726494476580617; 27386104940472276169105720, 25024939958628554701755145, 8089277164034877786318544 (Andrew Bremner, 2015)
31] 1677479490238223823661446513; 1507524066882038472584786800, 1288056982586427591062203384, 169218021322170204480680305 (Seiji Tomita, 03/13/2006)

Tito Piezas III
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This is a 2024 update of the 2015 table which just had 30 old entries of primitive $a^4+b^4+c^4 = d^4$ with upper bound $\color{red}{d<10^{27}}$. Within the same bound, there are new primitive solutions: Tomita found 34 more, Piezas found 20, Fulea found 6, and Bremner found 3, for a total of $T = 30+34+20+6+3 = 93$ solutions. Statistics are,

\begin{array}{|c|c|c|} \hline \text{Range #} & \text{Range of d} & \text{# of sol} \\ \hline 1 & 10^3-10^9 & 15 \\ \hline 2 & 10^9-10^{15} & 31 \\ \hline 3 & 10^{15}-10^{21} & 15 \\ \hline 4 & 10^{21}-10^{27} & 32 \\ \hline - & \text{Total} & 93 \\ \hline\end{array}

$\hskip1.6in$enter image description here

The bar chart looks strange. Range 1 is covered by the brute-search of team GDRZ (Robert Gerbicz, Leonid Durman, Yuri Radaev, and Alexey Zubkov) for $d<2\times10^9$ which found the smallest 19. But higher ranges will have more solutions, so a lot are missing. And if we plot $d/10^6$ from #10 to #20 we get,

$\hskip1.6in$enter image description here

Since the brute-force stopped at #19, the steep jump from #19 to #20 implies that #20, #21, etc are not the next smallest. The table below of 93 primitive solutions is of form $(d;a,b,c)$ and arranged from the smallest $d$. The C_k is the elliptic curve family, while $u$ is the elliptic curve parameter described in this post. (And since #20, #21, are not the next smallest, there are probably other $u$ of similar height to the list below.)

  1. 422481; 414560, 217519, 95800 (Frye, 1988); C1, u = -9/20.
  2. 2813001; 2767624, 1390400, 673865 (MacLeod 1997); C1, u = -9/20.
  3. 8707481; 8332208, 5507880, 1705575 (Bernstein, 2001); u = -29/12.
  4. 12197457; 11289040, 8282543, 5870000 (Bernstein, 2001); u = -93/80, -400/37.
  5. 16003017; 14173720, 12552200, 4479031 (Bernstein, 2001); u = -136/133, 201/4.
  6. 16430513; 16281009, 7028600, 3642840 (Bernstein, 2001); u = 12185/432.
  7. 20615673; 18796760, 15365639, 2682440 (Elkies, 1986); C0, u = -5/8.
  8. 44310257; 41084175, 31669120, 2164632 (Gerbicz, 2006); u = -817/660.
  9. 68711097; 65932985, 42878560, 10409096 (Gerbicz, 2006); u = -21021/9788.
  10. 117112081; 106161120, 87865617, 34918520 (Gerbicz, 2006); u = (big).
  11. 145087793; 122055375, 121952168, 1841160 (Rathmann, 2007); u = -361/540.
  12. 156646737; 146627384, 108644015, 27450160 (Rathmann, 2007); u = -136/133.
  13. 589845921; 582665296, 260052385, 186668000 (Tomita, 2006); C0, u = -5/8.
  14. 638523249; 630662624, 275156240, 219076465 (MacLeod, 1998); C0, u = -5/8.
  15. 873822121; 769321280, 606710871, 558424440 (GDRZ, 2007); u = -12285/4112.
  16. 1259768473; 1166705840, 859396455, 588903336 (GDRZ, 2008); C5, u = -41/36.
  17. 1679142729; 1670617271, 632671960, 50237800 (Tomita, 2006); C1, u = -9/20.
  18. 1787882337; 1662997663, 1237796960, 686398000 (GDRZ, 2007); u = -93/80.
  19. 1871713857; 1593513080, 1553556440, 92622401 (GDRZ, 2007); u = -865/592.
  20. 3393603777; 3134081336, 2448718655, 664793200 (Tomita, 2007); C0, u = -5/8.
  21. 5179020201; 24743080, 3971389576, 4657804375 (Fulea, Feb 2024); u = 553/80.
  22. 12558554489; 11988496761, 7813353720, 4707813440 (Bremner, 2015); u = 233/60.
  23. 15434547801; 15355831360, 5821981400, 140976551 (Tomita, 2007); C1
  24. 39871595729; 36295982895, 29676864960, 11262039896 (Tomita, Feb 2024)
  25. 46055390617; 18125123544, 41714673255, 34169217200 (Tomita, 2024); C5
  26. 64244765937; 52289667920, 17111129720, 55479193841 (Piezas, 2024); C9, u = -125/92
  27. 76973733409; 39110088360, 49796687200, 71826977313 (Tomita, Feb 2024)
  28. 521084370137; 372623278887, 435210480720, 369168502640 (Tomita, Feb 2024)
  29. 597385645737; 443873167360, 142485966505, 544848079888 (Tomita, Feb 2024)
  30. 820234293081; 78558599440, 814295112544, 337210257575 (Tomita, Feb 2024)
  31. 1059621884297; 535914713672, 1041572957760, 187577183625 (Piezas, 2024)
  32. 1367141947873; 1226022682752, 1047978087905, 408600530760 (Bremner, 2015)
  33. 1682315502153; 468405247415, 1657554153472, 801719896720 (Tomita, Feb 2024)
  34. 2051764828361; 125777308440, 894416022327, 2032977944240 (Tomita, 2024)
  35. 5062297699257; 4987588419655, 2480452675600, 502038853976 (Tomita, 2008); C0
  36. 6014017311081; 66822832760, 1313903832425, 6010589044544 (Fulea, Feb 2024)
  37. 6382441853233; 2927198165920, 613935345969, 6310500741600 (Tomita, Feb 2024)
  38. 7082388012473; 4408757988560, 5819035124295, 5611660306848 (Tomita, Feb 2024)
  39. 25866132798297; 23449050222680, 18776929334105, 12035933588696 (Piezas, 2024); C9
  40. 26969608212297; 487814048600, 8528631804200, 26901926181047 (Fulea, Feb 2024)
  41. 27497822498977, 19031674138785, 25762744660064, 2054845288320 (Tomita, Feb 2024)
  42. 29999857938609; 27239791692640, 22495595284040, 7592431981391 (Tomita, 2006); C1
  43. 45556888578449; 27546142170735, 7908038161032, 43940127884360 (Tomita, 2024)
  44. 58844817090201; 34511786481280, 56329979520665, 26636493544576 (Tomita, Feb 2024)
  45. 230791363907489, 148739531603136, 32467583677535, 220093974949320 (Tomita, Feb 2024)
  46. 573646321871961; 514818101299289, 440804942580160, 130064300991400 (Tomita, 2008); C1
  47. 5380742305932201; 1554532675059625, 1841841620201576, 5352683902805120 (Fulea, Feb 2024)
  48. 20249506709579721; 18565945114216720, 14890026433468471, 3579087147375440 (Tomita, 2008); C0
  49. 62940516903410601; 56827813308111785, 47886740272114976, 8813425670440240 (Tomita, 2008); C0
  50. 87486470529871881; 16306696482461560, 21794572772239369, 87375622888246360 (Fulea, Feb 2024)
  51. 103117303193818953; 4092004076331400, 24975412054750025, 103028409596553328; (Fulea, Feb 2024)
  52. 481334894209428521; 343651286746207896, 438980913824794665, 225712385669145920 (Piezas, 2024)
  53. 1592672455342770513; 208032601069058735, 851144034922098880, 1559028675188874616 (Piezas, 2024)
  54. 2778996090487120353; 2556827383749699103, 2024155336530384440, 585715960903147640 (Bremner, 2015); C7
  55. 10816708329115215113; 9585769407872803575, 8510180374729994520, 985735303963754488 (Piezas, 2024)
  56. 20234461127553384633; 19399184483902029008, 2329747842666412840, 12696186158476139705 (Tomita, 2024); C7
  57. 77107030404994920297; 69320669852667799672, 38320435200564613600, 56375727168307546985 (Tomita, 2024); C9
  58. 101783028910511968041; 99569174129827461335, 21710111037730547416, 54488888702794271560 (Piezas, 2024); C1
  59. 108593344076382641697; -46196947347028916440, -107238802094189542120, 38751631463616255521 (Bremner, 2024); C0
  60. 202540855134365138633; 201236910265023650505, 705558147137161920, 80940380256877627544 (Piezas, 2024)
  61. 228746036963039501833; 163180699054891578792, 84616109521023161865, 210878774189729581880 (Piezas, 2024)
  62. 375075545025537358721; 335981923744570504065, 188195571677171463096, 275897431444390465240 (Piezas, 2024)
  63. 1671674986261410994097; 1199828498161126807800, 1534990269771364822095, 655960628418767673472 (Piezas, 2024)
  64. 2711100675240842912689; 1977857900813232827064, 1617105485720597938520, 2376217986337223238735 (Piezas, 2024)
  65. 3037467718844497770129; 2877363855098380947880, 444897078221606141840, 2016612085130087009647 (Tomita, 2024); C7
  66. 4069249774250960557713; 3819055879832290430609, 842141328509923524200, 2794258267049888616280 (Piezas, 2024)
  67. 9649219915259253551497; 7599957410902753037705, 8407785501400674212160, 4280294741983707700872 (Tomita, Feb 2024)
  68. 10739931407728904606857; 772654695228940017240, 10320518856970101984393, 6653143628547990852040 (Piezas, 2024)
  69. 11305555143522867817873; 10539980352556633840239, 7799922278924748599160, 4141571237269338150920 (Tomita, 2024); C5
  70. 12214291847502204701241; 7745659501403353894384, 2120589250533219579335, 11684173258429439467360 (Tomita, Feb 2024)
  71. 17503689286309573964097; 15876595946759369395903, 7188470920864810763360, 12896301483090810351440 (Tomita, 2024)
  72. 18276027741543869996617; 13226266181198583365544, 16841033682021117865520, 4780632380106855105975 (Tomita, 2024); C5
  73. 24504057146788194291849; 1519814187310380835480, 23896480714429616100215, 13623248018235893097232 (Tomita, 2024); C7
  74. 29998124444432653523113; 12036780855644297767488, 23415987016826083521705, 26432693245716083746520 (Tomita, Feb 2024)
  75. 120175486227071990769561; 30248376090268690676600, 118508989446504950664160, 56915898438422390129561 (Tomita, 2024); C1
  76. 409840652625395469143913; 381461080909525552802665, 168213921178037201816584, 281048473715879152495040 (Tomita, 2024)
  77. 587020625514136613276553; 179164925544119666072000, 222787467202130880567415, 582653975191641098286104 (Piezas, 2024)
  78. 864745895259187110399737; 57810716855047169409080, 591519768111748983750888, 812937165464036006213895 (Piezas, 2024)
  79. 1088768337585323892067521; 76399836994875695614145, 974071910293355929264000, 842960029363955380661896 (Piezas, Feb 2024)
  80. 1171867103503245199920081; 1165970778032514255823760, 440517744543240750721000, 59421842165791512201169 (Tomita, 2024); C1
  81. 1779979592349189232414713; 1724845107301282006322000, 574585584668340612894713, 1018986340666195845813760 (Tomita, 2024)
  82. 1796867575393608033006561; 21904850878998429166561, 1771894249938641198780200, 867970652747799735398360 (Piezas, 2024)
  83. 3122849928997768901912409; 1891988836723177605880960, 985329200220584284726784, 3003225858017812695181145 (Tomita, Feb 2024)
  84. 6714012701109174954871521; 1758067984180618846616200, 6632467268281371571709360, 3057432874236989781768479 (Tomita, 2024); C1
  85. 8997319881974346759473697; 8281143989708209432415360, 1749772249172099623115896, 6550300128305909879699935 (Tomita, 2024)
  86. 19874054816411213708481009; -5967420362778572362681840, -19270755733101284410120384, 11389900458885552539102735 (Bremner, 2024); C0
  87. 21291952935426564624339201; 5328636655728999148343576, 20991236668646283695879935, 10137374115207940432133560 (Tomita, 2024); C1
  88. 31293260543726494476580617; 27386104940472276169105720, 25024939958628554701755145, 8089277164034877786318544 (Bremner, 2015); C9
  89. 34497456764264994703368889; 12209879806944320496330055, 26621272474250391413865480, 30730370351168229154149048 (Tomita, Feb 2024)
  90. 96242977191578497031965033; 47172089378698523207965335, 26409847035187091768472744, 94680315476024517009462320 (Tomita, 2024); C5
  91. 133140691304639620846181457; 129410861225043592041256520, 41328162329293632574512440, 74522041242387759937530799 (Bremner, 2024); C0
  92. 227529118288906398066378489, 85818832944459457142858489, 226369052354324181334408840, 2650718685573298353948640 (Piezas, 2024); C1
  93. 452835938257547709708389177, 451622371501239854889723705, 10453265194894185904695360, 145561707582919191804801464 (Piezas, 2024)

Note 1: The largest within the bound is $d\approx 4.52\times10^{26}.$

Note 2: Just five elliptic curves are responsible for $35$, namely $(10+12+5+4+4=35)$, or more than third of known solutions, namely, $$C_0: X^3-X^2+2815805388X-94443526967868 = Y^2$$ $$C_1: X^3+ 2265722465761X -3154189403034549278 = Y^2$$ $$C_5: X^3 + 2639323244332897X -20156152630838819347102 = Y^2$$ $$C_7: X^3 - X^2 + 349942184229228X -11167797929528591502588 = Y^2$$ $$C_9: X^3 - X^2+9243195710310751148X -761969307339454319105751548 =Y^2$$ all with rank 3 and also discussed by Tomita here, here, here, and here.

  1. $C_0$ ($u=-\frac{5}{8}$) is the one studied by Elkies.
  2. $C_1$ ($u=-\frac{9}{20}$) yields the smallest $d = 422481$, as well as the most $d$ < bound.
  3. $C_5$ ($u=-\frac{41}{36}$) was found by the GDRZ team.
  4. $C_7$ ($u=-\frac{5}{44}$) is by Bremner.
  5. $C_9$ ($u=-\frac{125}{92}$) is also by Bremner, found in 2015.

The parameters $(u,v)$ with small height discussed in this post tend to yield more $d$ within a bound.

Tito Piezas III
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1

In response to (1), this behaviour of the function is indeed asymptotic. This was proved in D N Lehmer (1900) Asymptotic Evaluation of Certain Totient Sums American Journal of Mathematics Vol 22, freely available via JSTOR here. This result (although the term 'Pythagorean triple' is not used) is on pp 327-8.

Adam Bailey
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