## a^4 + 14 a^2 b^2 + b^4 = c^4 + 14 c^2 d^2 + d^4

Find distinct positive integers   $a, b, c, d$   such that

$a^4 \; + \; 14 \, a^2 \, b^2 \; + \; b^4 \; = \; c^4 \; + \; 14 \, c^2 \, d^2 \; + \; d^4$

Note that

$(a+b)^8 \; + \; (a-b)^8 \; + \; (4 \, a \, b)^4 \; = \; 2 \, (a^4 + 14 \, a^2 \, b^2 + b^4)^2$

$a^4 \; + \; 14 \, a^2 \, b^2 \; + \; b^4 \; = \; c^4 \; + \; 14 \, c^2 \, d^2 \; + \; d^4$

implies

$(a+b)^8 \; + \; (a-b)^8 \; + \; (4 \, a \, b)^4 \; = \; (c+d)^8 \; + \; (c-d)^8 \; + \; (4 \, c \, d)^4$

Using Paul’s first few solutions:

$(a+b)^8 \; + \; (a-b)^8 \; + \; (4 \, a \, b)^4 \; = \; (c+d)^8 \; + \; (c-d)^8 \; + \; (4 \, c \, d)^4 \; = \; N$

math grad - Interest: Number theory
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### 2 Responses to a^4 + 14 a^2 b^2 + b^4 = c^4 + 14 c^2 d^2 + d^4

1. Paul says:

Here are 50 solutions out of 503 where a < b < 1000 and c < d < 1000.

{1,9,2,8}
1^4 + 14*1^2*9^2 + 9^4 = 2^4 + 14*2^2*8^2 + 8^4 = 7696

{1,25,8,18}
1^4 + 14*1^2*25^2 + 25^4 = 8^4 + 14*8^2*18^2 + 18^4 = 399376

{1,49,18,32}
1^4 + 14*1^2*49^2 + 49^4 = 18^4 + 14*18^2*32^2 + 32^4 = 5798416

{1,81,32,50}
1^4 + 14*1^2*81^2 + 81^4 = 32^4 + 14*32^2*50^2 + 50^4 = 43138576

{1,121,50,72}
1^4 + 14*1^2*121^2 + 121^4 = 50^4 + 14*50^2*72^2 + 72^4 = 214563856

{1,169,72,98}
1^4 + 14*1^2*169^2 + 169^4 = 72^4 + 14*72^2*98^2 + 98^4 = 816130576

{1,225,98,128}
1^4 + 14*1^2*225^2 + 225^4 = 98^4 + 14*98^2*128^2 + 128^4 = 2563599376

{1,289,128,162}
1^4 + 14*1^2*289^2 + 289^4 = 128^4 + 14*128^2*162^2 + 162^4 = 6976926736

{1,361,162,200}
1^4 + 14*1^2*361^2 + 361^4 = 162^4 + 14*162^2*200^2 + 200^4 = 16985387536

{1,441,200,242}
1^4 + 14*1^2*441^2 + 441^4 = 200^4 + 14*200^2*242^2 + 242^4 = 37825582096

{1,529,242,288}
1^4 + 14*1^2*529^2 + 529^4 = 242^4 + 14*242^2*288^2 + 288^4 = 78314903056

{1,625,288,338}
1^4 + 14*1^2*625^2 + 625^4 = 288^4 + 14*288^2*338^2 + 338^4 = 152593359376

{1,729,338,392}
1^4 + 14*1^2*729^2 + 729^4 = 338^4 + 14*338^2*392^2 + 392^4 = 282436976656

{1,841,392,450}
1^4 + 14*1^2*841^2 + 841^4 = 392^4 + 14*392^2*450^2 + 450^4 = 500256314896

{1,961,450,512}
1^4 + 14*1^2*961^2 + 961^4 = 450^4 + 14*450^2*512^2 + 512^4 = 852903966736

{2,18,4,16}
2^4 + 14*2^2*18^2 + 18^4 = 4^4 + 14*4^2*16^2 + 16^4 = 123136

{2,32,9,25}
2^4 + 14*2^2*32^2 + 32^4 = 9^4 + 14*9^2*25^2 + 25^4 = 1105936

{2,50,16,36}
2^4 + 14*2^2*50^2 + 50^4 = 16^4 + 14*16^2*36^2 + 36^4 = 6390016

{2,72,25,49}
2^4 + 14*2^2*72^2 + 72^4 = 25^4 + 14*25^2*49^2 + 49^4 = 27164176

{2,96,23,79}
2^4 + 14*2^2*96^2 + 96^4 = 23^4 + 14*23^2*79^2 + 79^4 = 85450768

{2,98,36,64}
2^4 + 14*2^2*98^2 + 98^4 = 36^4 + 14*36^2*64^2 + 64^4 = 92774656

{2,128,49,81}
2^4 + 14*2^2*128^2 + 128^4 = 49^4 + 14*49^2*81^2 + 81^4 = 269352976

{2,162,64,100}
2^4 + 14*2^2*162^2 + 162^4 = 64^4 + 14*64^2*100^2 + 100^4 = 690217216

{2,200,81,121}
2^4 + 14*2^2*200^2 + 200^4 = 81^4 + 14*81^2*121^2 + 121^4 = 1602240016

{2,242,100,144}
2^4 + 14*2^2*242^2 + 242^4 = 100^4 + 14*100^2*144^2 + 144^4 = 3433021696

{2,288,121,169}
2^4 + 14*2^2*288^2 + 288^4 = 121^4 + 14*121^2*169^2 + 169^4 = 6884352016

{2,338,144,196}
2^4 + 14*2^2*338^2 + 338^4 = 144^4 + 14*144^2*196^2 + 196^4 = 13058089216

{2,392,169,225}
2^4 + 14*2^2*392^2 + 392^4 = 169^4 + 14*169^2*225^2 + 225^4 = 23621230096

{2,450,196,256}
2^4 + 14*2^2*450^2 + 450^4 = 196^4 + 14*196^2*256^2 + 256^4 = 41017590016

{2,512,225,289}
2^4 + 14*2^2*512^2 + 512^4 = 225^4 + 14*225^2*289^2 + 289^4 = 68734156816

{2,578,256,324}
2^4 + 14*2^2*578^2 + 578^4 = 256^4 + 14*256^2*324^2 + 324^4 = 111630827776

{2,648,289,361}
2^4 + 14*2^2*648^2 + 648^4 = 289^4 + 14*289^2*361^2 + 361^4 = 176342883856

{2,722,324,400}
2^4 + 14*2^2*722^2 + 722^4 = 324^4 + 14*324^2*400^2 + 400^4 = 271766200576

{2,800,361,441}
2^4 + 14*2^2*800^2 + 800^4 = 361^4 + 14*361^2*441^2 + 441^4 = 409635840016

{2,882,400,484}
2^4 + 14*2^2*882^2 + 882^4 = 400^4 + 14*400^2*484^2 + 484^4 = 605209313536

{2,968,441,529}
2^4 + 14*2^2*968^2 + 968^4 = 441^4 + 14*441^2*529^2 + 529^4 = 878066449936

{3,27,6,24}
3^4 + 14*3^2*27^2 + 27^4 = 6^4 + 14*6^2*24^2 + 24^4 = 623376

{3,75,24,54}
3^4 + 14*3^2*75^2 + 75^4 = 24^4 + 14*24^2*54^2 + 54^4 = 32349456

{3,147,54,96}
3^4 + 14*3^2*147^2 + 147^4 = 54^4 + 14*54^2*96^2 + 96^4 = 469671696

{3,243,96,150}
3^4 + 14*3^2*243^2 + 243^4 = 96^4 + 14*96^2*150^2 + 150^4 = 3494224656

{3,363,150,216}
3^4 + 14*3^2*363^2 + 363^4 = 150^4 + 14*150^2*216^2 + 216^4 = 17379672336

{3,507,216,294}
3^4 + 14*3^2*507^2 + 507^4 = 216^4 + 14*216^2*294^2 + 294^4 = 66106576656

{3,675,294,384}
3^4 + 14*3^2*675^2 + 675^4 = 294^4 + 14*294^2*384^2 + 384^4 = 207651549456

{3,867,384,486}
3^4 + 14*3^2*867^2 + 867^4 = 384^4 + 14*384^2*486^2 + 486^4 = 565131065616

{4,36,8,32}
4^4 + 14*4^2*36^2 + 36^4 = 8^4 + 14*8^2*32^2 + 32^4 = 1970176

{4,64,18,50}
4^4 + 14*4^2*64^2 + 64^4 = 18^4 + 14*18^2*50^2 + 50^4 = 17694976

{4,100,32,72}
4^4 + 14*4^2*100^2 + 100^4 = 32^4 + 14*32^2*72^2 + 72^4 = 102240256

{4,144,50,98}
4^4 + 14*4^2*144^2 + 144^4 = 50^4 + 14*50^2*98^2 + 98^4 = 434626816

{4,192,46,158}
4^4 + 14*4^2*192^2 + 192^4 = 46^4 + 14*46^2*158^2 + 158^4 = 1367212288

{4,196,72,128}
4^4 + 14*4^2*196^2 + 196^4 = 72^4 + 14*72^2*128^2 + 128^4 = 1484394496

Here are the last 3 of the 503

{316,604,384,520}
316^4 + 14*316^2*604^2 + 604^4 = 384^4 + 14*384^2*520^2 + 520^4 = 653068865536

{395,755,480,650}
395^4 + 14*395^2*755^2 + 755^4 = 480^4 + 14*480^2*650^2 + 650^4 = 1594406410000

{474,906,576,780}
474^4 + 14*474^2*906^2 + 906^4 = 576^4 + 14*576^2*780^2 + 780^4 = 3306161131776

{534,790,646,666}
534^4 + 14*534^2*790^2 + 790^4 = 646^4 + 14*646^2*666^2 + 666^4 = 2962336788736

Paul.

• benvitalis says:

I used some of your solutions, where a = 1