A^4 + B^4 + C^4 = D^4的正整数解

wayne · 发表于 2025-5-24 19:31:55

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关于这个方程$A^4 + B^4 + C^4 = D^4$的正整数解,
可否在计算上打破他们在2008年的记录 https://euler413.narod.ru/
https://oeis.org/A003828

N A B C D Autor
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 15434547801 15355831360 5821981400 140976551 (Seiji Tomita, 10/24/2007)
22 5062297699257 4987588419655 2480452675600 502038853976 (Seiji Tomita, 05/15/2008)
23 29999857938609 27239791692640 22495595284040 7592431981391 (Seiji Tomita, 03/13/2006)
24 573646321871961 514818101299289 440804942580160 130064300991400 (Seiji Tomita, 09/15/2008)
25 20249506709579721 18565945114216720 14890026433468471 3579087147375440 (Seiji Tomita, 08/13/2008)
26 62940516903410601 56827813308111785 47886740272114976 8813425670440240 (Seiji Tomita, 08/13/2008)
27 1677479490238223823661446513 1507524066882038472584786800 1288056982586427591062203384 169218021322170204480680305 (Seiji Tomita, 03/13/2006)

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northwolves · 发表于 2025-5-24 21:17:53

NEW!


      9033964577482532388059482429398457291004947925005743028147465732645880^4 +
      4417264698994538496943597489754952845854672497179047898864124209346920^4 +
      1439965710648954492268506771833175267850201426615300442218292336336633^4 =
      9161781830035436847832452398267266038227002962257243662070370888722169^4 (M.A.FAJJAL, 5/31/2009)

northwolves · 发表于 2025-5-24 21:20:26

这些记录应该都是大型机或者分布式运算得到的吧？打破纪录难度比较大，就看你的了

northwolves · 发表于 2025-5-24 21:23:03

http://www.maroon.dti.ne.jp/fermat/dioph46e.html
http://www.maroon.dti.ne.jp/fermat/grouplaw1e.html

northwolves · 发表于 2025-5-24 22:00:12

A274015
Least number of the form x^2 + y^4 in exactly n ways where x and y are nonzero integers.

2, 17, 3026, 141457, 4740625, 113260225, 205117028945, 3234825286225

northwolves · 发表于 2025-5-24 22:00:39

3234825286225 = 1235^4+ 953160^2
            =  780^4+ 1692535^2
            =  637^4+ 1752192^2
            =  416^4+ 1790217^2
            =  195^4+ 1798160^2
            = 1092^4+ 1346423^2
            = 1341^4+ 31792^2
            =  979^4+ 1521912^2

nyy · 发表于 2025-5-24 22:57:26

northwolves 发表于 2025-5-24 21:17
NEW!

怎么得到的？

mathe · 发表于 2025-5-25 17:02:19

这个除了硬搜两数四次方和和四次方差有其它技巧吗？

nyy · 发表于 7 天前

N A B C D Autor
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 15434547801 15355831360 5821981400 140976551 (Seiji Tomita, 10/24/2007)
22 5062297699257 4987588419655 2480452675600 502038853976 (Seiji Tomita, 05/15/2008)
23 29999857938609 27239791692640 22495595284040 7592431981391 (Seiji Tomita, 03/13/2006)
24 573646321871961 514818101299289 440804942580160 130064300991400 (Seiji Tomita, 09/15/2008)
25 20249506709579721 18565945114216720 14890026433468471 3579087147375440 (Seiji Tomita, 08/13/2008)
26 62940516903410601 56827813308111785 47886740272114976 8813425670440240 (Seiji Tomita, 08/13/2008)
27 1677479490238223823661446513 1507524066882038472584786800 1288056982586427591062203384 169218021322170204480680305 (Seiji Tomita, 03/13/2006)

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wayne · 发表于 6 天前

无意间搜到了这个帖子, 内有乾坤, 记录已经被2024年stackexchange的一些人给打破了.
https://math.stackexchange.com/q ... 4b4/4857107#4857107

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[讨论] A^4 + B^4 + C^4 = D^4的正整数解

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