求\(a^2+ab+b^2=c^k\)的解

cn8888

其中\[0 \lt a \lt b,\ c \gt 0,\ k \gt 2\]\[a^2+ab+b^2=c^k\]

cn8888

k=3的解答:

1. Do[out={ToRules@Reduce[a^2+a*b+b^2==c^3&&0<a&&a<b&&c==n&&GCD[a,b]==1,{a,b,c},Integers]};(*调用reduce函数求解*)
   
2.   If[Not[out==={}],(*如果解答存在的话*)
   
3.      outabc={a,b,c}/.out;(*去掉a->这类符号,只保留结果*)
   
4.      Map[Print,outabc](*通过映射打印出所有解答*)
   
5.   ],
   
6.    {n,1,400}
   
7. ]
   

复制代码

1. {1,18,7}
   
2. {17,36,13}
   
3. {17,73,19}
   
4. {90,109,31}
   
5. {71,181,37}
   
6. {126,197,43}
   
7. {37,323,49}
   
8. {179,361,61}
   
9. {251,378,67}
   
10. {216,487,73}
    
11. {127,630,79}
    
12. {270,701,91}
    
13. {359,631,91}
    
14. {269,792,97}
    
15. {594,613,103}
    
16. {307,953,109}
    
17. {629,1009,127}
    
18. {233,1404,133}
    
19. {396,1297,133}
    
20. {703,1170,139}
    
21. {71,1819,151}
    
22. {468,1691,157}
    
23. {1007,1386,163}
    
24. {1007,1513,169}
    
25. {739,1980,181}
    
26. {937,2087,193}
    
27. {1170,2033,199}
    
28. {630,2701,211}
    
29. {1511,2161,217}
    
30. {1819,1872,217}
    
31. {73,3293,223}
    
32. {703,3060,229}
    
33. {720,3329,241}
    
34. {629,3529,247}
    
35. {2142,2339,247}
    
36. {1097,3510,259}
    
37. {1530,3187,259}
    
38. {2159,2971,271}
    
39. {379,4409,277}
    
40. {577,4446,283}
    
41. {2071,3869,301}
    
42. {2591,3420,301}
    
43. {918,4861,307}
    
44. {2736,3637,313}
    
45. {2969,3961,331}
    
46. {811,5741,337}
    
47. {1026,5777,343}
    
48. {3060,4427,349}
    
49. {2771,5040,361}
    
50. {1691,6031,367}
    
51. {4033,4284,373}
    
52. {827,6930,379}
    
53. {3959,5149,397}
    

复制代码

cn8888

k=4的解答

1. {16,39,7}
   
2. {15,161,13}
   
3. {185,231,19}
   
4. {455,649,31}
   
5. {511,1040,37}
   
6. {1056,1079,43}
   
7. {1265,1504,49}
   
8. {1089,3055,61}
   
9. {1001,3904,67}
   
10. {2431,3680,73}
    
11. {1001,5680,79}
    
12. {1991,7104,91}
    
13. {3401,6039,91}
    
14. {224,9295,97}
    
15. {4329,7760,103}
    
16. {5136,8449,109}
    
17. {3289,14231,127}
    
18. {3519,15665,133}
    
19. {6049,13871,133}
    
20. {3520,17319,139}
    
21. {10089,16016,151}
    
22. {7775,19824,157}
    
23. {6919,22425,163}
    
24. {5055,25696,169}
    
25. {209,32656,181}
    
26. {12224,29601,193}
    
27. {21616,24089,199}
    
28. {12151,37184,211}
    
29. {7361,42975,217}
    
30. {18031,35409,217}
    
31. {20999,35785,223}
    
32. {6864,48671,229}
    
33. {14849,49215,241}
    
34. {26320,43431,247}
    
35. {34416,36025,247}
    
36. {3289,65376,259}
    
37. {32384,44745,259}
    
38. {10279,67760,271}
    
39. {38064,50255,277}
    
40. {19175,68761,283}
    
41. {23920,76241,301}
    
42. {25185,75344,301}
    
43. {21385,81719,307}
    
44. {49984,62895,313}
    
45. {13881,101959,331}
    
46. {62959,68145,337}
    
47. {25271,102960,343}
    
48. {66079,74481,349}
    
49. {19136,119695,361}
    
50. {63184,91481,367}
    
51. {51425,106096,373}
    
52. {36105,122144,379}
    
53. {18239,147696,397}
    

复制代码

cn8888

k=5的解答

1. {62,87,7}
   
2. {116,543,13}
   
3. {93,1525,19}
   
4. {1626,4349,31}
   
5. {1076,7737,37}
   
6. {5257,8609,43}
   
7. {3725,14632,49}
   
8. {6775,25076,61}
   
9. {801,36337,67}
   
10. {25559,27009,73}
    
11. {27050,36757,79}
    
12. {23574,64525,91}
    
13. {40049,50950,91}
    
14. {30349,73688,97}
    
15. {23441,94018,103}
    
16. {6293,120775,109}
    
17. {62363,142374,127}
    
18. {86459,146532,133}
    
19. {108407,126909,133}
    
20. {16757,218950,139}
    
21. {93699,221326,151}
    
22. {120899,230113,157}
    
23. {8834,334707,163}
    
24. {139432,281393,169}
    
25. {128325,362351,181}
    
26. {97191,461993,193}
    
27. {269925,372418,199}
    
28. {219449,508425,211}
    
29. {128176,620637,217}
    
30. {277551,511912,217}
    
31. {267641,571693,223}
    
32. {360043,549732,229}
    
33. {286799,723376,241}
    
34. {126401,889362,247}
    
35. {238187,817287,247}
    
36. {386082,833443,259}
    
37. {606407,640018,259}
    
38. {507050,872901,271}
    
39. {336612,1075001,277}
    
40. {776891,778843,283}
    
41. {76399,1532276,301}
    
42. {423049,1317051,301}
    
43. {466649,1367838,307}
    
44. {611059,1344957,313}
    
45. {866899,1413050,331}
    
46. {382213,1867299,337}
    
47. {583109,1828009,343}
    
48. {899900,1687857,349}
    
49. {292299,2316976,361}
    
50. {1937,2579301,367}
    
51. {449841,2433716,373}
    
52. {1579425,1649318,379}
    
53. {1405788,2191849,397}
    

复制代码

cn8888

k=6的解答

1. {37,323,7}
   
2. {1007,1513,13}
   
3. {2771,5040,19}
   
4. {3781,27720,31}
   
5. {27720,30743,37}
   
6. {22933,65520,43}
   
7. {25271,102960,49}
   
8. {98280,161279,61}
   
9. {79883,252757,67}
   
10. {190513,257040,73}
    
11. {176149,380771,79}
    
12. {269280,581939,91}
    
13. {418501,451440,91}
    
14. {498457,554903,97}
    
15. {22933,1081080,103}
    
16. {679391,813960,109}
    
17. {622440,1664963,127}
    
18. {708553,1916927,133}
    
19. {1184040,1525393,133}
    
20. {874691,2139229,139}
    
21. {263339,3303720,151}
    
22. {1801800,2640457,157}
    
23. {906947,3805453,163}
    
24. {1275120,4061231,169}
    
25. {3374279,3472561,181}
    
26. {3477600,4789007,193}
    
27. {2764189,6126120,199}
    
28. {3800160,6898501,211}
    
29. {195623,10119097,217}
    
30. {2386800,8813663,217}
    
31. {486107,10838520,223}
    
32. {4796569,8869391,229}
    
33. {5312160,10563841,241}
    
34. {882757,14608440,247}
    
35. {4835123,12058200,247}
    
36. {7816069,12093120,259}
    
37. {8904349,11116691,259}
    
38. {4165560,17490059,271}
    
39. {3485663,19295640,277}
    
40. {5463613,19433987,283}
    
41. {4983119,24435721,301}
    
42. {10680120,20314439,301}
    
43. {9767520,22786597,307}
    
44. {5742073,27387360,313}
    
45. {6874560,32335379,331}
    
46. {9969623,32301360,337}
    
47. {12907080,32321053,343}
    
48. {10234729,36456840,349}
    
49. {17723159,35610121,361}
    
50. {23256323,33513480,367}
    
51. {2087567,50819833,373}
    
52. {12146149,47340971,379}
    
53. {10838520,56443463,397}
    

复制代码

cn8888

奇怪,最后一列似乎是一样的!

cn8888

k=7的解答:

1. {249,757,7}
   
2. {3532,5541,13}
   
3. {9578,23935,19}
   
4. {8815,161286,31}
   
5. {18651,298361,37}
   
6. {72078,481573,43}
   
7. {438987,511048,49}
   
8. {153716,1690915,61}
   
9. {1224461,1609533,67}
   
10. {1267064,2503873,73}
    
11. {90730,4336157,79}
    
12. {1294015,6453714,91}
    
13. {3315069,4932901,91}
    
14. {2322947,7599304,97}
    
15. {1955763,9981983,103}
    
16. {2300765,12222492,109}
    
17. {5632698,19746461,127}
    
18. {1290731,26463532,133}
    
19. {15595283,15733873,133}
    
20. {17788670,18768217,139}
    
21. {20205346,28416785,151}
    
22. {26063857,29883684,157}
    
23. {24113617,39139142,163}
    
24. {18227657,51616648,169}
    
25. {24701055,64504429,181}
    
26. {2224649,98742663,193}
    
27. {23682217,97420178,199}
    
28. {63900901,92778854,211}
    
29. {33487259,130961392,217}
    
30. {49028104,119898567,217}
    
31. {73294939,116307078,223}
    
32. {13211873,174762492,229}
    
33. {95558401,153145455,241}
    
34. {31466722,219525029,247}
    
35. {98794339,171439614,247}
    
36. {60173082,244622183,259}
    
37. {93054795,221215178,259}
    
38. {115754931,254046230,271}
    
39. {201297077,207148039,277}
    
40. {45576953,356450478,283}
    
41. {65348631,437061229,301}
    
42. {192422145,346603339,301}
    
43. {198601957,377604629,307}
    
44. {9711088,537586059,313}
    
45. {247733630,500054929,331}
    
46. {340160696,467772963,337}
    
47. {277555573,568871874,343}
    
48. {64619873,759840987,349}
    
49. {481146141,550236944,361}
    
50. {226368333,813260426,367}
    
51. {247118239,855586893,373}
    
52. {149194562,977329843,379}
    
53. {490815853,926607516,397}
    

复制代码

cn8888

谁能说明一下为什么最后一列的c值都是一样的呢?

cn8888

\[57983399659992357633594^2 + 57983399659992357633594*91621807513192521639905 + 91621807513192521639905^2 = 31^{31} \]

cn8888

\[235871^2 + 235871*17284173 + 17284173^2 = 13^{13}\]