a^4 + 2n(ab)^2 + b^4 = c^4 + 2n(cd)^2 + d^4 …. Part 2

 
 
a^4 \; + \; 2 \, n \, (a \, b)^2 \; + \; b^4 \; = \; c^4 \; + \; 2 \, n \, (c \, d)^2 \; + \; d^4

for   n = 1, 2, 3, …, 10
 

17^4 + 4(17\times 8)^2 + 8^4 \; = \; 1^4 + 4(1\times 20)^2 + 20^4

1279^4 + 6(1279\times 1127)^2 + 1127^4 \; = \; 523^4 + 6(523\times 1829)^2 + 1829^4

1893^4 + 8(1893\times 2714)^2 + 2714^4 \; = \; 1381^4 + 8(1381\times 3258)^2 + 3258^4

287^4 + 10(287\times 601)^2 + 601^4 \; = \; 299^4 + 10(299\times 587)^2 + 587^4

133^4 + 12(133\times 380)^2 + 380^4 \; = \; 181^4 + 12(181\times 320)^2 + 320^4

63^4 + 14(63\times 235)^2 + 235^4 \; = \; 107^4 + 14(107\times 177)^2 + 177^4

541^4 + 16(541\times 2558)^2 + 2558^4 \; = \; 1117^4 + 16(1117\times 1766)^2 + 1766^4

5623^4 + 18(5623\times 33029)^2 + 33029^4 \; = \; 13891^4 + 18(13891\times 21263)^2 + 21263^4

6333^4 + 20(6333\times 45584)^2 + 45584^4 \; = \; 18541^4 + 20(18541\times 27708)^2 + 27708^4

 

For   n = 11, 12, 13, …, 100

31^4 + 22(31\times 271)^2 + 271^4 =  107^4 + 22(107\times 157)^2 + 157^4

1505^4 + 24(1505\times 15898)^2 + 15898^4 = 6113^4 + 24(6113\times 8842)^2 + 8842^4

7959^4 + 26(7959\times 101417)^2 + 101417^4 = 38083^4 + 26(38083\times 54459)^2 + 54459^4

917^4 + 28(917\times 14116)^2 + 14116^4 =  5189^4 + 28(5189\times 7352)^2 + 7352^4

8383^4 + 30(8383\times 156659)^2 + 156659^4 = 56491^4 + 30(56491\times 79433)^2 + 79433^4

9^4 + 32(9\times 206)^2 + 206^4 =  73^4 + 32(73\times 102)^2 + 102^4

179^4 + 34(179\times 5089)^2 + 5089^4 = 1775^4 + 34(1775\times 2467)^2 + 2467^4

7549^4 + 36(7549\times 272312)^2 + (272312)^4
    = 93613^4 + 36(93613\times 129524)^2 + 129524^4

6783^4 + 38(6783\times 320759)^2 + (320759)^4
    = 108811^4 + 38(108811\times 149973)^2 + 149973^4

49^4 + 40(49\times 3202)^2 + 3202^4 = 1073^4 + 40(1073\times 1474)^2 + 1474^4

175^4 + 42(175\times 17369)^2 + 17369^4 = 5755^4 + 42(5755\times 7883)^2 + 7883^4

537^4 + 44(537\times 99964)^2 + 99964^4 = 32777^4 + 44(32777\times 44784)^2 + 44784^4

71^4 + 46(71\times 63523)^2 + 63523^4 = 20627^4 + 46(20627\times 28121)^2 + 28121^4

1787^4 + 48(1787\times 650174)^2 + (650174)^4
    = \; 209221^4 + 48(209221\times 284678)^2 + 284678^4

4617^4 + 50(4617\times 735509)^2 + (735509)^4
    = \; 234691^4 + 50(234691\times 318783)^2 + 318783^4

7^4 + 52(7\times 736)^2 + 736^4 = 233^4 + 52(233\times 316)^2 + 316^4

2317^4 + 54(2317\times 185587)^2 + 185587^4
    = \; 58319^4 + 54(58319\times 78985)^2 + 78985^4

15771^4 + 56(15771\times 1035602)^2 + 1035602^4
    = \; 323173^4 + 56(323173\times 437154)^2 + 437154^4

2273^4 + 58(2273\times 127921)^2 + 127921^4
    = \; 39659^4 + 58(39659\times 53587)^2 + 53587^4

25667^4 + 60(25667\times 1275284)^2 + (1275284)^4
    = \; 392941^4 + 60(392941\times 530408)^2 + 530408^4

1257^4 + 62(1257\times 56315)^2 + 56315^4
    = \; 17251^4 + 62(17251\times 23265)^2 + 23265^4

839^4 + 64(839\times 34430)^2 + 34430^4
    = \; 10489^4 + 64(10489\times 14134)^2 + 14134^4

3437^4 + 66(3437\times 130769)^2 + 130769^4
    = \; 39631^4 + 66(39631\times 53363)^2 + 53363^4

52227^4 + 68(52227\times 1860104)^2 + 1860104^4
    = \; 560941^4 + 68(560941\times 754788)^2 + 754788^4

6713^4 + 70(6713\times 225551)^2 + 225551^4
    = \; 67699^4 + 70(67699\times 91037)^2 + 91037^4

2771^4 + 72(2771\times 88394)^2 + 88394^4 = 26413^4 + 72(26413\times 35498)^2 + 35498^4

15765^4 + 74(15765\times 480013)^2 + 480013^4
    = \; 142823^4 + 74(142823\times 191847)^2 + 191847^4

9899^4 + 76(9899\times 288988)^2 + 288988^4
    = \; 85637^4 + 76(85637\times 114976)^2 + 114976^4

100097^4 + 78(100097\times 2812619)^2 + 2812619^4
    = \; 830251^4 + 78(830251\times 1114193)^2 + 1114193^4

111867^4 + 80(111867\times 3035534)^2 + 3035534^4
    = \; 892741^4 + 80(892741\times 1197558)^2 + 1197558^4

553^4 + 82(553\times 14533)^2 + 14533^4 = 4259^4 + 82(4259\times 5711)^2 + 5711^4

27559^4 + 84(27559\times 703216)^2 + 703216^4
    = \; 205385^4 + 84(205385\times 275308)^2 + 275308^4

152001^4 + 86(152001\times 3774287)^2 + 3774287^4
    = \; 1098763^4 + 86(1098763\times 1472349)^2 + 1472349^4

18563^4 + 88(18563\times 449426)^2 + 449426^4
    = \; 130429^4 + 88(130429\times 174722)^2 + 174722^4

183017^4 + 90(183017\times 4328009)^2 + 4328009^4
    = \; 1252291^4 + 90(1252291\times 1677083)^2 + 1677083^4

615^4 + 92(615\times 14228)^2 + 14228^4 = 4105^4 + 92(4105\times 5496)^2 + 5496^4

4837^4 + 94(4837\times 109631)^2 + 109631^4
    = 31543^4 + 94(31543\times 42221)^2 + 42221^4

236411^4 + 96(236411\times 5256182)^2 + 5256182^4
    = 1508293^4 + 96(1508293\times 2018414)^2 + 2018414^4

256137^4 + 98(256137\times 5592749)^2 + 5592749^4
    = 1600771^4 + 98(1600771\times 2141703)^2 + 2141703^4

30763^4 + 100(30763\times 660376)^2 + 660376^4
    = 188549^4 + 100(188549\times 252212)^2 + 252212^4

 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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