## A^3 + B^3 = C^2, {(A+B), (A^2-A*B+B^2)} are 3 times squares

Find positive integers   $A$   and   $B$   such that

$A^3 \; + \; B^3 \; = \; C^2$

$A \; + \; B \; = \; 3 \, x^2$

$A^2 \; - \; A \, B \; + \; B^2 \; = \; 3 \, y^2$

Here the solutions I found for   $A \; \leq \; 1000$

Check whether I miss any other solutions.

Paul found :

math grad - Interest: Number theory
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### 2 Responses to A^3 + B^3 = C^2, {(A+B), (A^2-A*B+B^2)} are 3 times squares

1. Paul says:

Here is what I found

(1 , 2), 3^2 , 3 x 1^2 , 3 x 1^2
(2 , 1), 3^2 , 3 x 1^2 , 3 x 1^2
(2 , 46), 312^2 , 3 x 4^2 , 3 x 26^2
(4 , 8), 24^2 , 3 x 2^2 , 3 x 4^2
(8 , 4), 24^2 , 3 x 2^2 , 3 x 4^2
(8 , 184), 2496^2 , 3 x 8^2 , 3 x 104^2
(9 , 18), 81^2 , 3 x 3^2 , 3 x 9^2
(10 , 65), 525^2 , 3 x 5^2 , 3 x 35^2
(11 , 37), 228^2 , 3 x 4^2 , 3 x 19^2
(16 , 32), 192^2 , 3 x 4^2 , 3 x 16^2
(18 , 9), 81^2 , 3 x 3^2 , 3 x 9^2
(18 , 414), 8424^2 , 3 x 12^2 , 3 x 234^2
(22 , 26), 168^2 , 3 x 4^2 , 3 x 14^2
(25 , 50), 375^2 , 3 x 5^2 , 3 x 25^2
(26 , 22), 168^2 , 3 x 4^2 , 3 x 14^2
(32 , 16), 192^2 , 3 x 4^2 , 3 x 16^2
(32 , 736), 19968^2 , 3 x 16^2 , 3 x 416^2
(36 , 72), 648^2 , 3 x 6^2 , 3 x 36^2
(37 , 11), 228^2 , 3 x 4^2 , 3 x 19^2
(40 , 260), 4200^2 , 3 x 10^2 , 3 x 140^2
(44 , 148), 1824^2 , 3 x 8^2 , 3 x 76^2
(46 , 2), 312^2 , 3 x 4^2 , 3 x 26^2
(49 , 98), 1029^2 , 3 x 7^2 , 3 x 49^2
(50 , 25), 375^2 , 3 x 5^2 , 3 x 25^2
(64 , 128), 1536^2 , 3 x 8^2 , 3 x 64^2
(65 , 10), 525^2 , 3 x 5^2 , 3 x 35^2
(65 , 610), 15075^2 , 3 x 15^2 , 3 x 335^2
(72 , 36), 648^2 , 3 x 6^2 , 3 x 36^2
(78 , 354), 6696^2 , 3 x 12^2 , 3 x 186^2
(81 , 162), 2187^2 , 3 x 9^2 , 3 x 81^2
(88 , 104), 1344^2 , 3 x 8^2 , 3 x 56^2
(90 , 585), 14175^2 , 3 x 15^2 , 3 x 315^2
(98 , 49), 1029^2 , 3 x 7^2 , 3 x 49^2
(99 , 333), 6156^2 , 3 x 12^2 , 3 x 171^2
(100 , 200), 3000^2 , 3 x 10^2 , 3 x 100^2
(104 , 88), 1344^2 , 3 x 8^2 , 3 x 56^2
(121 , 242), 3993^2 , 3 x 11^2 , 3 x 121^2
(128 , 64), 1536^2 , 3 x 8^2 , 3 x 64^2
(144 , 288), 5184^2 , 3 x 12^2 , 3 x 144^2
(148 , 44), 1824^2 , 3 x 8^2 , 3 x 76^2
(162 , 81), 2187^2 , 3 x 9^2 , 3 x 81^2
(169 , 338), 6591^2 , 3 x 13^2 , 3 x 169^2
(176 , 592), 14592^2 , 3 x 16^2 , 3 x 304^2
(183 , 249), 4644^2 , 3 x 12^2 , 3 x 129^2
(184 , 8), 2496^2 , 3 x 8^2 , 3 x 104^2
(196 , 392), 8232^2 , 3 x 14^2 , 3 x 196^2
(198 , 234), 4536^2 , 3 x 12^2 , 3 x 126^2
(200 , 100), 3000^2 , 3 x 10^2 , 3 x 100^2
(225 , 450), 10125^2 , 3 x 15^2 , 3 x 225^2
(234 , 198), 4536^2 , 3 x 12^2 , 3 x 126^2
(242 , 121), 3993^2 , 3 x 11^2 , 3 x 121^2
(242 , 433), 9765^2 , 3 x 15^2 , 3 x 217^2
(249 , 183), 4644^2 , 3 x 12^2 , 3 x 129^2
(256 , 512), 12288^2 , 3 x 16^2 , 3 x 256^2
(260 , 40), 4200^2 , 3 x 10^2 , 3 x 140^2
(275 , 925), 28500^2 , 3 x 20^2 , 3 x 475^2
(288 , 144), 5184^2 , 3 x 12^2 , 3 x 144^2
(289 , 578), 14739^2 , 3 x 17^2 , 3 x 289^2
(305 , 895), 27300^2 , 3 x 20^2 , 3 x 455^2
(324 , 648), 17496^2 , 3 x 18^2 , 3 x 324^2
(330 , 345), 8775^2 , 3 x 15^2 , 3 x 195^2
(333 , 99), 6156^2 , 3 x 12^2 , 3 x 171^2
(338 , 169), 6591^2 , 3 x 13^2 , 3 x 169^2
(345 , 330), 8775^2 , 3 x 15^2 , 3 x 195^2
(352 , 416), 10752^2 , 3 x 16^2 , 3 x 224^2
(354 , 78), 6696^2 , 3 x 12^2 , 3 x 186^2
(361 , 722), 20577^2 , 3 x 19^2 , 3 x 361^2
(392 , 196), 8232^2 , 3 x 14^2 , 3 x 196^2
(400 , 800), 24000^2 , 3 x 20^2 , 3 x 400^2
(414 , 18), 8424^2 , 3 x 12^2 , 3 x 234^2
(416 , 352), 10752^2 , 3 x 16^2 , 3 x 224^2
(433 , 242), 9765^2 , 3 x 15^2 , 3 x 217^2
(441 , 882), 27783^2 , 3 x 21^2 , 3 x 441^2
(450 , 225), 10125^2 , 3 x 15^2 , 3 x 225^2
(470 , 730), 22200^2 , 3 x 20^2 , 3 x 370^2
(484 , 968), 31944^2 , 3 x 22^2 , 3 x 484^2
(485 , 715), 21900^2 , 3 x 20^2 , 3 x 365^2
(512 , 256), 12288^2 , 3 x 16^2 , 3 x 256^2
(546 , 777), 25137^2 , 3 x 21^2 , 3 x 399^2
(550 , 650), 21000^2 , 3 x 20^2 , 3 x 350^2
(578 , 289), 14739^2 , 3 x 17^2 , 3 x 289^2
(585 , 90), 14175^2 , 3 x 15^2 , 3 x 315^2
(592 , 176), 14592^2 , 3 x 16^2 , 3 x 304^2
(610 , 65), 15075^2 , 3 x 15^2 , 3 x 335^2
(648 , 324), 17496^2 , 3 x 18^2 , 3 x 324^2
(650 , 550), 21000^2 , 3 x 20^2 , 3 x 350^2
(715 , 485), 21900^2 , 3 x 20^2 , 3 x 365^2
(722 , 361), 20577^2 , 3 x 19^2 , 3 x 361^2
(730 , 470), 22200^2 , 3 x 20^2 , 3 x 370^2
(732 , 996), 37152^2 , 3 x 24^2 , 3 x 516^2
(736 , 32), 19968^2 , 3 x 16^2 , 3 x 416^2
(777 , 546), 25137^2 , 3 x 21^2 , 3 x 399^2
(792 , 936), 36288^2 , 3 x 24^2 , 3 x 504^2
(800 , 400), 24000^2 , 3 x 20^2 , 3 x 400^2
(851 , 877), 35928^2 , 3 x 24^2 , 3 x 499^2
(877 , 851), 35928^2 , 3 x 24^2 , 3 x 499^2
(882 , 441), 27783^2 , 3 x 21^2 , 3 x 441^2
(895 , 305), 27300^2 , 3 x 20^2 , 3 x 455^2
(925 , 275), 28500^2 , 3 x 20^2 , 3 x 475^2
(936 , 792), 36288^2 , 3 x 24^2 , 3 x 504^2
(968 , 484), 31944^2 , 3 x 22^2 , 3 x 484^2
(996 , 732), 37152^2 , 3 x 24^2 , 3 x 516^2

Paul.

• benvitalis says:

Wow! I missed lots of solutions. I used one family of solutions.