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

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, 6, 6, 7}, {95775, 121295, 133098, 2047192}}
    
11. {{5, 6, 6, 7}, {93583, 117300, 186645, 2323832}}
    
12. {{5, 5, 6, 7}, {73851, 84849, 393460, 3225056}}
    
13. {{5, 6, 6, 7}, {96051, 105281, 438643, 3738476}}
    
14. {{5, 6, 6, 7}, {57433, 247180, 660835, 5639347}}
    
15. {{5, 6, 6, 7}, {53634, 528979, 538122, 6548927}}
    
16. {{6, 6, 7, 8}, {405783, 682993, 1210087, 13436500}}
    
17. {{6, 7, 7, 8}, {649040, 1110029, 1236824, 17513771}}
    
18. {{6, 7, 7, 8}, {662057, 1061682, 1283013, 17577202}}
    
19. {{6, 7, 7, 8}, {621528, 2304605, 3729607, 38894100}}
    
20. {{6, 6, 7, 8}, {244143, 653868, 7450069, 48701903}}
    
21. {{7, 7, 7, 8}, {2544884, 3197139, 4752132, 61347439}}
    
22. {{6, 7, 7, 8}, {419650, 4787125, 7826137, 76142713}}
    
23. {{5, 6, 8, 8}, {90613, 917042, 14898837, 92762985}}
    
24. {{6, 7, 8, 9}, {522364, 2441110, 14379841, 101224885}}
    
25. {{6, 7, 8, 9}, {209902, 7817501, 11238217, 112552836}}
    
26. {{7, 8, 8, 9}, {8678439, 13256530, 19146375, 240146471}}
    
27. {{7, 7, 8, 9}, {2653775, 4157756, 60876089, 394862735}}
    
28. {{7, 8, 8, 9}, {7739505, 11362865, 99771021, 693774634}}
    
29. {{8, 8, 9, 9}, {10876668, 26108300, 105775615, 833689837}}
    
30. {{7, 8, 9, 10}, {5765276, 71224679, 102051215, 1046079980}}
    
31. {{7, 8, 9, 10}, {4286618, 51738093, 149854473, 1202274829}}
    
32. {{8, 8, 9, 10}, {81978851, 87287767, 156493512, 1904272416}}
    
33. {{8, 9, 9, 10}, {46277407, 147618163, 272702835, 2726542678}}
    
34. {{8, 9, 9, 10}, {29085922, 104491681, 549169281, 3985543597}}
    
35. {{9, 9, 9, 10}, {171685703, 519036945, 609976972, 7602430407}}
    
36. {{9, 9, 10, 11}, {472116953, 757516277, 1135368630, 13824513790}}
    
37. {{9, 10, 10, 11}, {160953137, 2004013937, 2562768666, 27623844797}}
    
38. {{10, 10, 10, 11}, {1102679165, 1134646274, 4479037381, 39237698580}}
    
39. {{9, 10, 10, 11}, {666472025, 2379769387, 5369467428, 49167474120}}
    
40. {{10, 10, 10, 12}, {4739974781, 6055117068, 8290914753, 111577045918}}
    
41. {{10, 10, 11, 12}, {3395042301, 8282446806, 11396218949, 134863984486}}
    
42. {{9, 10, 11, 12}, {835814545, 2219722657, 29033450908, 187185763975}}
    
43. {{10, 11, 11, 12}, {6228099317, 15738611807, 16699088720, 226014756368}}
    
44. {{9, 10, 11, 12}, {835148279, 6828329781, 45742838535, 311646698620}}
    
45. {{10, 11, 11, 12}, {7306358822, 11073627965, 50669231165, 403254691613}}
    
46. {{10, 11, 11, 12}, {9731950880, 33789398240, 42869205949, 504908837331}}
    
47. {{10, 11, 12, 13}, {9023798629, 42278790791, 139118581520, 1112038351815}}
    
48. {{10, 11, 12, 13}, {6027956407, 15145760882, 201290859723, 1297710032570}}
    
49. {{12, 12, 12, 13}, {155551243483, 192370429108, 215836732409, 3295910221608}}
    
50. {{11, 12, 12, 13}, {87571288246, 178201035111, 308585620511, 3356864891308}}
    
51. {{11, 12, 12, 13}, {51175091726, 129782929931, 415078012118, 3481495026443}}
    
52. {{12, 12, 12, 13}, {170564240395, 304717255328, 456618660893, 5447202434157}}
    
53. {{12, 12, 12, 13}, {129273380271, 383765320969, 744216136825, 7346606804416}}
    
54. {{12, 12, 13, 13}, {129119184260, 129421622957, 1176736370135, 8377780826408}}
    
55. {{12, 12, 12, 13}, {352321790435, 363596836724, 727870471796, 8439434008361}}
    
56. {{12, 12, 13, 14}, {471077244911, 646323285400, 1184110269585, 13452619281729}}
    
57. {{12, 12, 13, 14}, {466077161836, 617206156279, 1480206417580, 14981309860915}}
    
58. {{10, 13, 13, 14}, {1201148175, 1787677204640, 1934188623769, 21751333189536}}
    
59. {{12, 12, 13, 14}, {154884030725, 404775555028, 3437559996057, 23325130382580}}
    
60. {{12, 13, 13, 14}, {415136475875, 2626301818035, 3653246629484, 39119691235661}}
    
61. {{13, 13, 13, 14}, {1032659092003, 2149746030659, 4101099445863, 42565766145438}}
    
62. {{10, 13, 13, 14}, {9278188529, 3697414438305, 5686329466819, 54871562191172}}
    
63. {{12, 13, 13, 14}, {447575785825, 4542237526828, 5003800589395, 58395966172012}}
    
64. {{11, 13, 14, 14}, {94041874360, 3896578822920, 12410867908599, 95764316082461}}
    
65. {{12, 13, 14, 15}, {285249598935, 7835748088443, 10221766084108, 107167059759914}}
    
66. {{13, 13, 14, 15}, {4815523452572, 8901438805793, 23027250549133, 214694904608940}}
    
67. {{13, 14, 14, 15}, {9061612773818, 15406834595343, 23759709713203, 281906756827404}}
    
68. {{14, 14, 14, 15}, {11405700211304, 13196407192146, 26962372484575, 301397487182775}}
    
69. {{13, 14, 14, 15}, {9919121392082, 21150149034525, 22630703029705, 313909759762813}}
    
70. {{14, 14, 14, 15}, {18191717494011, 22846910515772, 38586607387657, 465452522108297}}
    
71. {{13, 13, 14, 15}, {5586098286935, 7944477528231, 74923106122004, 516209329933745}}
    
72. {{14, 14, 14, 15}, {10237905353256, 11086976275427, 80161470502936, 592497375028377}}
    
73. {{13, 14, 15, 16}, {8748106043916, 14105216716919, 148651128318819, 1000741758805090}}
    
74. {{14, 14, 15, 16}, {27381745766025, 57236038559812, 178639400108875, 1537853203730488}}
    
75. {{13, 14, 15, 16}, {4619313743781, 15108842653736, 261536478802339, 1640369882213040}}
    
76. {{14, 14, 15, 16}, {30985644485341, 66485799078689, 215040367143105, 1825497658781394}}
    
77. {{14, 15, 15, 16}, {70015291682116, 103771350904051, 221448153116020, 2309901473834113}}
    
78. {{13, 15, 15, 16}, {2829376411952, 105867256642895, 327478519699988, 2546797501938125}}
    
79. {{14, 15, 15, 16}, {21417078931425, 138463841678697, 415237084178618, 3358707003219220}}
    
80. {{14, 15, 16, 17}, {81889571219456, 736734549460992, 1524377532162817, 13686881913365247}}
    
81. {{15, 16, 16, 17}, {212518745375717, 1030770536277600, 1158866203811040, 14038634568163363}}
    
82. {{15, 15, 16, 17}, {879232153096440, 910344125232113, 1581834476403040, 19708433264049957}}
    
83. {{15, 16, 16, 17}, {421396584326809, 1141830813884452, 3408413326547047, 29041083581619847}}
    
84. {{15, 16, 16, 17}, {934139990316701, 1516777428173186, 5624791077880121, 47171999502360594}}
    
85. {{15, 16, 16, 17}, {109969433945368, 3415063477061808, 9144658758612809, 73987544060823393}}
    
86. {{15, 16, 17, 17}, {853589184088991, 1113822001921514, 11720030235841836, 79873606739554835}}
    
87. {{15, 16, 16, 17}, {523118409058987, 6789821791506972, 8052588443633537, 89781271576656891}}
    
88. {{16, 16, 16, 18}, {2870369647532617, 5661566161908144, 9301502207173039, 104232808046878544}}
    
89. {{15, 16, 17, 18}, {152362952874561, 8826169411521084, 10505457140620975, 113833594260224255}}
    
90. {{15, 16, 17, 18}, {399701796472348, 8160530802319772, 11739210850119045, 118596930825229765}}
    
91. {{15, 16, 17, 18}, {349483115617436, 1120632337821191, 20705296070579036, 129325519096140843}}
    
92. {{15, 16, 17, 18}, {441594882553011, 5488291089174536, 22463552226465763, 165754859081355901}}
    
93. {{16, 17, 17, 18}, {2387660199575816, 10178905885565120, 17875948022277659, 177880838305571351}}
    
94. {{16, 16, 17, 18}, {9081219449328904, 9920131901370145, 11538275289110161, 178546230358300000}}
    
95. {{16, 17, 17, 18}, {1252915478036754, 14150150179408881, 18308611783323290, 196976831029104529}}
    
96. {{16, 17, 17, 18}, {6224020498475255, 12001379705511735, 44553492941525184, 366678521249280421}}
    
97. {{16, 17, 17, 18}, {1479408167610505, 26720443022281162, 42071296870582525, 410536506412939933}}
    
98. {{17, 17, 18, 18}, {22140734967121637, 34181799729037989, 107002192064441104, 954199738158854527}}
    
99. {{17, 17, 18, 19}, {21224759044246076, 67612207523035500, 116435948644904375, 1199558507766440549}}
    
100. {{17, 17, 18, 19}, {53722404237037143, 55568923377147980, 100543812735395623, 1226621528949694537}}
     
101. {{18, 18, 18, 19}, {102536017245923014, 178881794524951271, 621290921606013159, 5273114459204475630}}
     
102. {{18, 18, 18, 19}, {229417512308309426, 252358455510499787, 444240671325001047, 5413151464112108090}}
     
103. {{18, 18, 18, 19}, {164070608965325241, 250519686080568811, 880682529183004736, 7566595125675086368}}
     
104. {{18, 18, 19, 20}, {352922405155385540, 604758998957186788, 1713830935036919055, 15608702856601836527}}
     
105. {{18, 19, 19, 20}, {194235029603703824, 1205497998407204752, 2603194159895977929, 23384211986215352679}}
     
106. {{18, 19, 19, 20}, {650715528787401376, 1169285287860219355, 2486499782135164664, 25167085116881865165}}
     
107. {{18, 19, 19, 20}, {289432496952157141, 4582204638794248620, 5809120723365052539, 62405406129656807189}}
     
108. {{19, 19, 19, 20}, {1561459405687489523, 2053746924185344640, 9681223191242639592, 77655219978521217445}}
     
109. {{19, 19, 19, 20}, {1467396800972677315, 5000733626605364908, 8326482600045050707, 86455766239491641805}}
     
110. {{19, 19, 19, 20}, {3488676775661611848, 4599332914440149123, 7816853683085419561, 92970924333918955891}}

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