1.Introduction


I show the collections of parametric solution for x1^k + x2^k + x3^k = y1^k + y2^k + y3^k where k = 2,6.




2.Collections
         
     
@@ 
    1. Degree 4

      (Brudno[2],Bremner[1],Delorme[3])

      x1 = 2a^4 + 4a^3 - 5a^2 - 12a - 9
      x2 = 3a^4 + 9a^3 + 18a^2 + 21a + 9
      x3 = -a^4 - 10a^3 - 17a^2 - 12a

      y1 = a^4 - 3a^3 - 14a^2 - 15a - 9
      y2 = 3a^4 + 8a^3 + 9a^2
      y3 = 2a^4 + 12a^3 + 19a^2 + 18a + 9


    2. Degree 5

      (Bremner,Delorme)

      x1 = 3a^5 + 8a^4 + 9a^3 - 4a^2 - 9a - 2
      x2 = a^5 + 9a^4 + 13a^3 + 7a^2 + 7a + 3
      x3 = 2a^5 + a^4 - 12a^3 - 13a^2 - 4a + 1

      y1 = 2a^5 + 9a^4 + 4a^3 - 9a^2 - 8a - 3
      y2 = a^5 - 4a^4 - 13a^3 - 12a^2 + a + 2
      y3 = 3a^5 + 7a^4 + 7a^3 + 13a^2 + 9a + 1

      (Bremner)

      x1 = 27a^5 - 54a^4 + 54a^3 + 4a^2 - 33a + 18
      x2 = 9a^5 - 27a^4 + 46a^3 - 74a^2 + 57a - 27
      x3 = -18a^5 + 57a^4 - 124a^3 + 126a^2 - 66a + 9

      y1 = 18a^5 - 33a^4 + 4a^3 + 54a^2 - 54a + 27
      y2 = 9a^5 - 66a^4 + 126a^3 - 124a^2 + 57a - 18
      y3 = -27a^5 + 57a^4 - 74a^3 + 46a^2 - 27a + 9

      (Delorme)

      x1 = 16a^5 + 64a^4 + 104a^3 + 116a^2 + 78a + 27
      x2 = 16a^5 + 24a^4 + 48a^3 + 52a^2 + 57a + 18
      x3 = 16a^5 + 32a^4 + 52a^3 + 48a^2 + 21a - 9

      y1 = 16a^5 + 48a^4 + 84a^3 + 76a^2 + 33a + 18
      y2 = 16a^5 + 56a^4 + 112a^3 + 108a^2 + 81a + 27
      y3 = 16a^5 + 16a^4 + 8a^3 - 28a^2 - 18a - 9

      (Tomita)

      x1 = -48a^5 - 224a^4 - 688a^3 - 956a^2 - 912a - 603
      x2 = 80a^5 + 192a^4 + 204a^3 - 232a^2 - 189a - 333
      x3 = 16a^5 - 232a^4 - 464a^3 - 828a^2 - 351a - 324

      y1 = -80a^5 - 208a^4 - 236a^3 - 492a^2 - 519a - 468
      y2 = -48a^5 - 16a^4 - 272a^3 - 244a^2 - 408a + 135
      y3 = 16a^5 + 312a^4 + 624a^3 + 988a^2 + 921a + 585




    3. Degree 6

      (Bremner)

      x1 = a^6 + 20a^5 + 25a^4 + 16a^3 + 11a^2 - 6a - 2
      x2 = -3a^6 + a^5 - 4a^4 + 13a^3 + 4a^2 + a + 3
      x3 = -5a^6 - 10a^5 + 2a^4 - 16a^3 - 14a^2 - 10a + 1

      y1 = -a^6 + 14a^5 + 18a^4 + 16a^3 - 6a^2 - 2a - 3
      y2 = -5a^6 - 9a^5 + 4a^4 + 27a^3 + 16a^2 + 3a + 1
      y3 = -3a^6 - 12a^5 - 35a^4 - 16a^3 - a^2 + 2a - 2



    4. Degree 7

      (Delorme)

      x1 = 2a^7 + 7a^6 + 15a^5 + 18a^4 + 9a^3 + 2a^2 + a + 1
      x2 = a^7 + a^6 - 4a^5 - 17a^4 - 22a^3 - 15a^2 - 7a - 2
      x3 = -a^7 - 6a^6 - 17a^5 - 21a^4 - 17a^3 - 11a^2 - 6a - 1

      y1 = -a^7 - a^6 - 2a^5 - 9a^4 - 18a^3 - 15a^2 - 7a - 2
      y2 = a^7 + 6a^6 + 11a^5 + 17a^4 + 21a^3 + 17a^2 + 6a + 1
      y3 = 2a^7 + 7a^6 + 15a^5 + 22a^4 + 17a^3 + 4a^2 - a - 1

     
    5. Degree 8

      (Delorme)

      x1 = 13a^8 - 28a^7 + 31a^6 + 68a^5 - 160a^4 + 336a^3 - 256a^2 + 192a - 64
      x2 = -21a^8 + 55a^7 - 151a^6 + 240a^5 - 380a^4 + 368a^3 - 240a^2 + 128a
      x3 = a^8 - 58a^7 + 166a^6 - 308a^5 + 460a^4 - 368a^3 + 304a^2 - 128a + 64

      y1 = -21a^8 + 54a^7 - 154a^6 + 268a^5 - 300a^4 + 368a^3 - 272a^2 + 192a - 64
      y2 = 13a^8 - 39a^7 - a^6 + 48a^5 - 200a^4 + 256a^3 - 288a^2 + 128a - 64
      y3 = -a^8 - 64a^7 + 161a^6 - 320a^5 + 420a^4 - 432a^3 + 272a^2 - 128a

      (Tomita)

      x1 = 20a^8 + 88a^7 - 251a^6 + 596a^5 - 140a^4 + 496a^3 + 176a^2 - 128a - 128
      x2 = 35a^8 - 72a^7 + 161a^6 - 598a^5 + 96a^4 - 88a^3 - 832a^2 - 96a - 64
      x3 = -15a^8 + 129a^7 + 21a^6 - 144a^5 + 576a^4 - 720a^3 - 192a^2 - 192a - 192
      y1 = 15a^8 + 14a^7 - 83a^6 + 120a^5 + 608a^4 + 208a^3 + 512a + 64
      y2 = -20a^8 + 28a^7 - 195a^6 + 610a^5 - 488a^4 - 216a^3 - 32a^2 - 224a - 192
      y3 = 35a^8 - 67a^7 + 277a^6 - 64a^5 + 308a^4 - 320a^3 + 560a^2 + 128a - 128



    6. Degree 9

      (Delorme)

      x1 = 16a^9 + 64a^8 + 160a^7 + 308a^6 + 416a^5 + 397a^4 + 262a^3 + 139a^2 + 54a + 9
      x2 = -16a^9 - 56a^8 - 80a^7 - 12a^6 + 119a^5 + 234a^4 + 233a^3 + 126a^2 + 27a
      x3 = 16a^9 + 96a^8 + 236a^7 + 368a^6 + 387a^5 + 261a^4 + 75a^3 - 41a^2 - 39a - 9

      y1 = -16a^9 - 48a^8 - 44a^7 + 60a^6 + 225a^5 + 290a^4 + 231a^3 + 106a^2 + 21a
      y2 = 16a^9 + 88a^8 + 208a^7 + 324a^6 + 369a^5 + 355a^4 + 281a^3 + 149a^2 + 51a + 9
      y3 = -16a^9 - 80a^8 - 224a^7 - 364a^6 - 360a^5 - 199a^4 - 34a^3 + 51a^2 + 42a + 9

    7. Degree 11

      (Delorme)

      x1 = 8a^11+43a^10+167a^9+547a^8+1296a^7+1863a^6+1631a^5+927a^4+352a^3+63a^2 - 17a - 5
      x2 = -a^11+39a^10+213a^9+610a^8+1005a^7+939a^6+587a^5+412a^4+341a^3+177a^2+45a+8
      x3 = 5a^11+38a^10+42a^9-155a^8-769a^7-1554a^6-1744a^5-1217a^4-613a^3-232a^2-50a-1

      y1 = -5a^11-17a^10+63a^9+352a^8+927a^7+1631a^6+1863a^5+1296a^4+547a^3+167a^2+43a+8
      y2 = a^11+50a^10+232a^9+613a^8+1217a^7+1744a^6+1554a^5+769a^4+155a^3-42a^2-38a-5
      y3 = -8a^11-45a^10-177a^9-341a^8-412a^7-587a^6-939a^5-1005a^4-610a^3-213a^2-39a+1

    8. Degree 12

      (Tomita)

      x1 = 5a^12-122a^11+382a^10-306a^9-990a^8+2476a^7-4221a^6+4014a^5-3680a^4+2104a^3-1408a^2+480a-192
      x2 = 35a^12+101a^11+70a^10-623a^9+2150a^8-3507a^7+4935a^6-4672a^5+3600a^4-1808a^3+576a^2-64a-64
      x3 = 40a^12+32a^11-181a^10+1318a^9-1911a^8+2630a^7-1149a^6+324a^5+1436a^4-1264a^3+1168a^2-384a+128

      y1 = -40a^12-92a^11-a^10+396a^9-1723a^8+3768a^7-4753a^6+4838a^5-3192a^4+2040a^3-800a^2+352a-64
      y2 = 5a^12+217a^11-180a^10+57a^9+944a^8-719a^7+1043a^6+104a^5+316a^4+400a^2-128a+128
      y3 = -35a^12+16a^11+482a^10-882a^9+1874a^8-1980a^7+2967a^6-3064a^5+3216a^4-2224a^3+1408a^2-512a+192

    9. Degree 13

      (Tomita)

      x1 = 13a^13+86a^12-11a^11-590a^10-310a^9+2266a^8+3589a^7+326a^6-3478a^5-3178a^4-875a^3-114a^2-147a-52
      x2 = -8a^13-35a^12-342a^11-1719a^10-4238a^9-5450a^8-2694a^7+649a^6-434a^5-2898a^4-2266a^3-703a^2-122a-15
      x3 = a^13+123a^12+711a^11+1539a^10+1964a^9+3132a^8+6159a^7+7647a^6+4968a^5+2264a^4+1571a^3+1023a^2+383a+65

      y1 = -8a^13-69a^12-546a^11-2021a^10-3878a^9-4738a^8-4250a^7-1301a^6+2918a^5+3950a^4+2186a^3+907a^2+358a+67
      y2 = 13a^13+83a^12-29a^11-1489a^10-4640a^9-6160a^8-4333a^7-3029a^6-3804a^5-4172a^4-3001a^3-1069a^2-57a+37
      y3 = a^13-110a^12-687a^11-1550a^10-666a^9+3006a^8+6057a^7+6102a^6+5022a^5+2930a^4+265a^3-762a^2-347a-36

   10. Degree 14

      (Tomita)

      x1 = 311a^14+2366a^13+6498a^12+6860a^11+5349a^10+11908a^9+24569a^8+32672a^7+28453a^6+14988a^5+3857a^4-236a^3-218a^2+26a+15
      x2 = -99a^14+756a^13+6901a^12+20028a^11+30230a^10+35446a^9+33657a^8+21752a^7+3542a^6-6830a^5-6047a^4-2120a^3-255a^2+2a-20
      x3 = 173a^14+1885a^13+5538a^12+7271a^11-2017a^10-17762a^9-30411a^8-30453a^7-20037a^6-7474a^5-991a^4+343a^3+222a^2+77a+35

      y1 = 199a^14+972a^13+4727a^12+18068a^11+35938a^10+43202a^9+27803a^8+7912a^7+210a^6+1526a^5+2643a^4+1768a^3+691a^2+118a-20
      y2 = 37a^14+1098a^13+4830a^12+11460a^11+13855a^10+16860a^9+23171a^8+21792a^7+10559a^6+180a^5-2053a^4-740a^3-46a^2-82a-35
      y3 = 309a^14+2437a^13+6450a^12+9071a^11+339a^10-24510a^9-41799a^8-37653a^7-21681a^6-9942a^5-4291a^4-1321a^3+34a^2+169a+15

   11. Degree 15

      (Tomita)

      x1 = -a^14-3a^13+2a^12+57a^11+241a^10+662a^9+1309a^8+2025a^7+2453a^6+2346a^5+1789a^4+1093a^3+482a^2+125a+15
      x2 = a^15+8a^14+37a^13+121a^12+304a^11+609a^10+1009a^9+1416a^8+1752a^7+1861a^6+1669a^5+1168a^4+563a^3+137a^2-5
      x3 = a^15+7a^14+32a^13+105a^12+271a^11+548a^10+867a^9+1045a^8+887a^7+396a^6-189a^5-595a^4-634a^3-371a^2-125a-20

      y1 = -a^15-7a^14-30a^13-87a^12-191a^11-326a^10-463a^9-555a^8-627a^7-738a^6-887a^5-943a^4-726a^3-379a^2-125a-20
      y2 = -a^14-11a^13-50a^12-163a^11-397a^10-804a^9-1361a^8-1975a^7-2409a^6-2420a^5-1869a^4-1067a^3-448a^2-125a-15
      y3 = -a^15-8a^14-39a^13-129a^12-324a^11-651a^10-1063a^9-1436a^8-1572a^7-1379a^6-903a^5-460a^4-203a^3-67a^2+5





3.References

      1. A. Bremner, A geometric approach to equal sums of sixth powers,Proc.London Math.Soc.(3) 43 (1981).

      2. S. Brudno, Triples of sixth powers with equal sums,Math.Comp. 30 (1976).

      3. J. Delorme, On the diophantine equation x1^6 +x2^6 +x3^6 = y1^6 +y2^6 +y3^6,Math. Comp. 59 (1992).



     
  
       





 














HOME