a/(b+c+d)+b/(a+c+d)+c/(a+b+d)+d/(a+b+c)=6的正整数解

wayne

求$\frac{a}{b+c+d}+\frac{d}{a+b+c}+\frac{c}{a+b+d}+\frac{b}{a+c+d}=6$的正整数解.

数论爱好者

知乎上有回答,a,b,c的值都很大,你自己验证一下,别人是否算对.
https://www.zhihu.com/question/276028810

数论爱好者

跟deepseek乱了一会,由于我知识有限,不能反驳它.我认为它是错的,它说我的maple不会化简.各位帮我看看deepseek错在哪里.
它说,一确解的形式满足:a=b=c=p,d=p*(8+sqrt(91))
我用a=8,d=d=8*(8+sqrt(91)),代入maple计算,输出24/(96+10*sqrt(91))+10/3+(5/12)*sqrt(91),近似等于6,
a=10,结果没有输出6,但是化成小数,好像是6
着重看一下它的化简过程有没有错.

数论爱好者

这里打开这个网页,等待10秒以后,等到变量d的解出现,会输出四次方程,我看不懂.可能还是没有用
https://www.wolframalpha.com/inp ... %29%3D6&lang=zh

wayne

我给出我的解答. 答案不完整,可能会漏掉. 这是20位数以内的解
https://www.zhihu.com/question/1 ... 1920832571228529819

更多的100位数的解在这里: https://nestwhile.com/res/abcn/ell-z-zuoye-100.txt

1. {3, 3, 4, 5, 245, 432, 3325, 23358}
   
2. {4, 4, 4, 5, 1181, 2077, 2252, 32211}
   
3. {4, 4, 4, 5, 2527, 3822, 6014, 72267}
   
4. {3, 5, 5, 6, 381, 19261, 30866, 295057}
   
5. {4, 5, 5, 6, 5372, 23023, 36735, 380588}
   
6. {4, 5, 5, 6, 5004, 13481, 59450, 455065}
   
7. {5, 5, 5, 6, 16430, 31341, 32629, 470005}
   
8. {5, 5, 5, 6, 20570, 23513, 40052, 491825}
   
9. {5, 5, 6, 7, 26553, 36057, 117130, 1050125}
   
10. {5, 5, 6, 7, 20350, 35875, 142687, 1161763}
    
11. {5, 6, 6, 7, 95775, 121295, 133098, 2047192}
    
12. {5, 6, 6, 7, 93583, 117300, 186645, 2323832}
    
13. {5, 5, 6, 7, 73851, 84849, 393460, 3225056}
    
14. {5, 6, 6, 7, 96051, 105281, 438643, 3738476}
    
15. {5, 6, 6, 7, 57433, 247180, 660835, 5639347}
    
16. {5, 6, 6, 7, 53634, 528979, 538122, 6548927}
    
17. {6, 6, 6, 7, 399379, 415492, 752004, 9159379}
    
18. {6, 6, 7, 8, 405783, 682993, 1210087, 13436500}
    
19. {6, 7, 7, 8, 649040, 1110029, 1236824, 17513771}
    
20. {6, 7, 7, 8, 662057, 1061682, 1283013, 17577202}
    
21. {6, 7, 7, 8, 621528, 2304605, 3729607, 38894100}
    
22. {6, 6, 7, 8, 244143, 653868, 7450069, 48701903}
    
23. {7, 7, 7, 8, 2544884, 3197139, 4752132, 61347439}
    
24. {6, 7, 7, 8, 419650, 4787125, 7826137, 76142713}
    
25. {5, 6, 8, 8, 90613, 917042, 14898837, 92762985}
    
26. {6, 7, 8, 9, 522364, 2441110, 14379841, 101224885}
    
27. {6, 7, 8, 9, 209902, 7817501, 11238217, 112552836}
    
28. {7, 7, 8, 9, 2425829, 4780615, 17071296, 141805716}
    
29. {7, 7, 8, 9, 1609049, 7028259, 26675992, 206199699}
    
30. {7, 8, 8, 9, 8678439, 13256530, 19146375, 240146471}
    
31. {7, 7, 8, 9, 2653775, 4157756, 60876089, 394862735}
    
32. {6, 8, 8, 9, 398376, 16542253, 58779071, 442064476}
    
33. {7, 8, 8, 9, 7739505, 11362865, 99771021, 693774634}
    
34. {8, 8, 9, 9, 10876668, 26108300, 105775615, 833689837}
    
35. {7, 8, 9, 10, 5765276, 71224679, 102051215, 1046079980}
    
36. {7, 8, 9, 10, 4286618, 51738093, 149854473, 1202274829}
    
37. {8, 8, 9, 10, 81978851, 87287767, 156493512, 1904272416}
    
38. {8, 9, 9, 10, 46277407, 147618163, 272702835, 2726542678}
    
39. {8, 9, 9, 10, 14781520, 160156609, 337929436, 2995846579}
    
40. {8, 9, 9, 10, 29085922, 104491681, 549169281, 3985543597}
    
41. {9, 9, 9, 10, 152292625, 194091599, 607405920, 5572846016}
    
42. {9, 9, 9, 10, 171685703, 519036945, 609976972, 7602430407}
    
43. {9, 9, 10, 11, 472116953, 757516277, 1135368630, 13824513790}
    
44. {9, 10, 10, 11, 152189013, 1115050377, 1624884650, 16898970997}
    
45. {9, 9, 10, 11, 548425549, 891815444, 2660104503, 23956228413}
    
46. {9, 10, 10, 11, 160953137, 2004013937, 2562768666, 27623844797}
    
47. {10, 10, 10, 11, 1102679165, 1134646274, 4479037381, 39237698580}
    
48. {7, 10, 10, 11, 2482825, 3509994125, 3676002848, 41997473802}
    
49. {9, 10, 10, 11, 446534957, 3152772400, 3707027463, 42695430960}
    
50. {9, 10, 10, 11, 666472025, 2379769387, 5369467428, 49167474120}
    
51. {10, 10, 10, 12, 4739974781, 6055117068, 8290914753, 111577045918}
    
52. {10, 10, 11, 12, 3395042301, 8282446806, 11396218949, 134863984486}
    
53. {9, 10, 11, 12, 835814545, 2219722657, 29033450908, 187185763975}
    
54. {10, 11, 11, 12, 6228099317, 15738611807, 16699088720, 226014756368}
    
55. {9, 10, 11, 12, 835148279, 6828329781, 45742838535, 311646698620}
    
56. {11, 11, 11, 12, 15625834979, 19800800096, 25406960649, 355641288640}
    
57. {10, 11, 11, 12, 7306358822, 11073627965, 50669231165, 403254691613}
    
58. {10, 11, 11, 12, 9731950880, 33789398240, 42869205949, 504908837331}
    
59. {11, 11, 11, 12, 15986601230, 33837645905, 36849020117, 506658412598}
    
60. {11, 11, 11, 12, 19616129079, 47341423604, 81530589847, 867800140214}
    
61. {10, 11, 12, 13, 9023798629, 42278790791, 139118581520, 1112038351815}
    
62. {10, 11, 12, 13, 6027956407, 15145760882, 201290859723, 1297710032570}
    
63. {12, 12, 12, 13, 155551243483, 192370429108, 215836732409, 3295910221608}
    
64. {11, 12, 12, 13, 87571288246, 178201035111, 308585620511, 3356864891308}
    
65. {11, 12, 12, 13, 51175091726, 129782929931, 415078012118, 3481495026443}
    
66. {12, 12, 12, 13, 170564240395, 304717255328, 456618660893, 5447202434157}
    
67. {12, 12, 12, 13, 192957345602, 252725519845, 561344192898, 5885543441337}
    
68. {12, 12, 12, 13, 129273380271, 383765320969, 744216136825, 7346606804416}
    
69. {12, 12, 13, 13, 129119184260, 129421622957, 1176736370135, 8377780826408}
    
70. {12, 12, 12, 13, 352321790435, 363596836724, 727870471796, 8439434008361}
    
71. {12, 12, 13, 14, 130276056344, 386729057267, 1383971104489, 11101915089192}
    
72. {12, 12, 13, 14, 125216177921, 395593654480, 1639627972110, 12615060243009}
    
73. {12, 12, 13, 14, 471077244911, 646323285400, 1184110269585, 13452619281729}
    
74. {12, 12, 13, 14, 466077161836, 617206156279, 1480206417580, 14981309860915}
    
75. {10, 13, 13, 14, 1201148175, 1787677204640, 1934188623769, 21751333189536}
    
76. {12, 12, 13, 14, 154884030725, 404775555028, 3437559996057, 23325130382580}
    
77. {12, 13, 13, 14, 787836362225, 1265687846125, 2469478500849, 26435037851426}
    
78. {12, 13, 13, 14, 415136475875, 2626301818035, 3653246629484, 39119691235661}
    
79. {13, 13, 13, 14, 1032659092003, 2149746030659, 4101099445863, 42565766145438}
    
80. {10, 13, 13, 14, 9278188529, 3697414438305, 5686329466819, 54871562191172}
    
81. {12, 13, 13, 14, 447575785825, 4542237526828, 5003800589395, 58395966172012}
    
82. {13, 13, 13, 14, 1857117409266, 3093618691711, 6444501640703, 66596384154991}
    
83. {11, 13, 14, 14, 94041874360, 3896578822920, 12410867908599, 95764316082461}
    
84. {13, 13, 14, 14, 2154703741641, 2374797813701, 12381795320544, 98765201872210}
    
85. {13, 13, 13, 15, 1127026782200, 8524067328749, 8543120198000, 106321762576901}
    
86. {12, 13, 14, 15, 285249598935, 7835748088443, 10221766084108, 107167059759914}
    
87. {13, 13, 14, 15, 4815523452572, 8901438805793, 23027250549133, 214694904608940}
    
88. {13, 14, 14, 15, 9061612773818, 15406834595343, 23759709713203, 281906756827404}
    
89. {14, 14, 14, 15, 11405700211304, 13196407192146, 26962372484575, 301397487182775}
    
90. {13, 14, 14, 15, 9919121392082, 21150149034525, 22630703029705, 313909759762813}
    
91. {14, 14, 14, 15, 18191717494011, 22846910515772, 38586607387657, 465452522108297}
    
92. {13, 13, 14, 15, 5586098286935, 7944477528231, 74923106122004, 516209329933745}
    
93. {13, 14, 14, 15, 4846085918971, 41779260574025, 46921840308744, 546639636763320}
    
94. {14, 14, 14, 15, 10237905353256, 11086976275427, 80161470502936, 592497375028377}
    
95. {13, 14, 14, 15, 3024501811266, 50933912375050, 52847828207485, 624067977496949}
    
96. {13, 14, 15, 16, 8748106043916, 14105216716919, 148651128318819, 1000741758805090}
    
97. {14, 15, 15, 16, 32417929742851, 103856097331511, 108036778837398, 1427984312344611}
    
98. {14, 14, 15, 16, 27381745766025, 57236038559812, 178639400108875, 1537853203730488}
    
99. {13, 14, 15, 16, 4619313743781, 15108842653736, 261536478802339, 1640369882213040}
    
100. {14, 14, 15, 16, 30985644485341, 66485799078689, 215040367143105, 1825497658781394}
     
101. {14, 15, 15, 16, 70015291682116, 103771350904051, 221448153116020, 2309901473834113}
     
102. {13, 15, 15, 16, 2829376411952, 105867256642895, 327478519699988, 2546797501938125}
     
103. {14, 15, 15, 16, 21417078931425, 138463841678697, 415237084178618, 3358707003219220}
     
104. {15, 15, 15, 17, 328731263716687, 447416339165237, 969373546245951, 10201693076682612}
     
105. {15, 15, 16, 17, 138994600697894, 866835811369326, 1017875363368395, 11826073758921235}
     
106. {14, 15, 16, 17, 81889571219456, 736734549460992, 1524377532162817, 13686881913365247}
     
107. {15, 16, 16, 17, 212518745375717, 1030770536277600, 1158866203811040, 14038634568163363}
     
108. {15, 15, 16, 17, 879232153096440, 910344125232113, 1581834476403040, 19708433264049957}
     
109. {15, 16, 16, 17, 421396584326809, 1141830813884452, 3408413326547047, 29041083581619847}
     
110. {16, 16, 16, 17, 1225129663629648, 1777505051165545, 2316902595869104, 31097639347361503}
     
111. {15, 16, 16, 17, 934139990316701, 1516777428173186, 5624791077880121, 47171999502360594}
     
112. {15, 16, 16, 17, 109969433945368, 3415063477061808, 9144658758612809, 73987544060823393}
     
113. {15, 16, 17, 17, 853589184088991, 1113822001921514, 11720030235841836, 79873606739554835}
     
114. {15, 16, 16, 17, 523118409058987, 6789821791506972, 8052588443633537, 89781271576656891}
     
115. {16, 16, 16, 18, 2870369647532617, 5661566161908144, 9301502207173039, 104232808046878544}
     
116. {15, 16, 17, 18, 152362952874561, 8826169411521084, 10505457140620975, 113833594260224255}
     
117. {15, 16, 17, 18, 399701796472348, 8160530802319772, 11739210850119045, 118596930825229765}
     
118. {15, 16, 17, 18, 349483115617436, 1120632337821191, 20705296070579036, 129325519096140843}
     
119. {15, 16, 17, 18, 441594882553011, 5488291089174536, 22463552226465763, 165754859081355901}
     
120. {16, 17, 17, 18, 2387660199575816, 10178905885565120, 17875948022277659, 177880838305571351}
     
121. {16, 16, 17, 18, 9081219449328904, 9920131901370145, 11538275289110161, 178546230358300000}
     
122. {16, 17, 17, 18, 1252915478036754, 14150150179408881, 18308611783323290, 196976831029104529}
     
123. {17, 17, 17, 18, 11100616029467633, 16482187531502213, 18640995992859282, 270229299086853445}
     
124. {17, 17, 17, 18, 10731344508769139, 12550229370068387, 31797714292127924, 321890982581761649}
     
125. {16, 17, 17, 18, 6224020498475255, 12001379705511735, 44553492941525184, 366678521249280421}
     
126. {16, 17, 17, 18, 1479408167610505, 26720443022281162, 42071296870582525, 410536506412939933}
     
127. {16, 17, 17, 18, 5809456295968478, 42564169199175679, 60674129725482868, 637183636696835943}
     
128. {17, 17, 18, 18, 22140734967121637, 34181799729037989, 107002192064441104, 954199738158854527}
     
129. {17, 17, 18, 19, 21224759044246076, 67612207523035500, 116435948644904375, 1199558507766440549}
     
130. {17, 17, 18, 19, 53722404237037143, 55568923377147980, 100543812735395623, 1226621528949694537}
     
131. {18, 18, 18, 19, 102536017245923014, 178881794524951271, 621290921606013159, 5273114459204475630}
     
132. {18, 18, 18, 19, 229417512308309426, 252358455510499787, 444240671325001047, 5413151464112108090}
     
133. {18, 18, 18, 19, 164070608965325241, 250519686080568811, 880682529183004736, 7566595125675086368}
     
134. {18, 18, 19, 20, 352922405155385540, 604758998957186788, 1713830935036919055, 15608702856601836527}
     
135. {18, 19, 19, 20, 194235029603703824, 1205497998407204752, 2603194159895977929, 23384211986215352679}
     
136. {18, 19, 19, 20, 650715528787401376, 1169285287860219355, 2486499782135164664, 25167085116881865165}
     
137. {18, 19, 19, 20, 289432496952157141, 4582204638794248620, 5809120723365052539, 62405406129656807189}
     
138. {19, 19, 19, 20, 1561459405687489523, 2053746924185344640, 9681223191242639592, 77655219978521217445}
     
139. {19, 19, 19, 20, 1897229584946669383, 5830348805279917237, 6553283401997574640, 83470518177149770288}
     
140. {19, 19, 19, 20, 1467396800972677315, 5000733626605364908, 8326482600045050707, 86455766239491641805}
     
141. {19, 19, 19, 20, 3488676775661611848, 4599332914440149123, 7816853683085419561, 92970924333918955891}

复制代码

hujunhua

有些丢番图方程，比如 $a^4+b^4+c^4=d^4$, 人们从费尔马大定理外推一度怀疑无解，结果是因为它的解比较大，人们搜索范围未达到。
丢番图方程       $a/(b+c)+b/(c+a)+c/(a+b)=6$ 是一个更极端的例子，右边的数字 6 应该是精心选择的。
这些例子告诉我们，不要因为暴力搜索不得就轻易得出无解的结论。对于哥猜、黎曼猜想、3X+1猜想等，都应作如是观。

从构造极端例子的意义上说，本主帖的方程失败了，因为最小解真的小。
我相信，可以把 6 改成一个相似的不起眼小数字，使得方程有巨大的最小解。

wayne

hujunhua 发表于 2025-6-26 22:33
有些丢番图方程，比如 $a^4+b^4+c^4=d^4$, 人们从费尔马大定理外推一度怀疑无解，结果是因为它的解比较大， ...

是的.关于 $a/(b+c)+b/(c+a)+c/(a+b)=k$, k取这些数的时候,方程有正整数解. 我计算了所有的k<1000的情况,提交了, https://oeis.org/A369896
具体的值,我已经计算出来,放在这里了. https://nestwhile.com/res/abcn/38.txt, https://nestwhile.com/res/abcn/