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Does the diophantine equation $a^4+b^4=tc^4$ have any integer solutions when $\pm t\neq 1$ and $a,b,c>1$ are pairwise coprime integers? We all know that Fermat proved the case where $t=\pm 1$. I was wondering about this more generalized case.

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    $\begingroup$ Well, $t=17$ will certainly work $\endgroup$ – Hagen von Eitzen Oct 3 '16 at 19:01
  • $\begingroup$ $t=0$ would be a special case with many solutions, too. $\endgroup$ – JB King Oct 3 '16 at 19:03
  • $\begingroup$ First of all I think you should discard the obvious solution $(0,0,0)$ as then we have a solution for every $t$. Then we can discard the negative values for $t$ too. $\endgroup$ – Stefan4024 Oct 3 '16 at 19:08
  • $\begingroup$ Thanks. It's clear we can find infinitely many trivial solutions. $\endgroup$ – user97615 Oct 3 '16 at 19:10
  • $\begingroup$ @HagenvonEitzen, do you have an example that does not involve one of those integers to be equal to 0 or 1? $\endgroup$ – user97615 Oct 3 '16 at 19:14
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 a 149 b 25 a^4 + b^4 493275026 =  2 17^4 2953
 a 486 b 479 a^4 + b^4 108431722897 =  17^4 113 11489

Any odd prime $p$ dividing the result has $p \equiv 1 \pmod 8,$ as $-1$ is a quartic residue. Took a while to find anything with $41^4.$

Ran it over night, here are the first half dozen with each prime:

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 149 b 25 a^4 + b^4 493275026 =  = 2 17^4  2953
 a 486 b 479 a^4 + b^4 108431722897 =  = 17^4 113  11489
 a 504 b 337 a^4 + b^4 77422046017 =  = 17^4  926977
 a 529 b 188 a^4 + b^4 79560183617 =  = 17^4 73  13049
 a 554 b 39 a^4 + b^4 94199744497 =  = 17^4  1127857
 a 579 b 110 a^4 + b^4 112532938081 =  = 17^4 73  18457
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 1002 b 751 a^4 + b^4 1326121160017 =  = 41^4 89  5273
 a 1875 b 1261 a^4 + b^4 14888103935266 =  = 2 41^4  2634353
 a 2263 b 1124 a^4 + b^4 27822490843937 =  = 41^4  9846017
 a 2626 b 259 a^4 + b^4 47557605667937 =  = 17 41^4  990001
 a 3265 b 373 a^4 + b^4 113659753929266 =  = 2 41^4 401  50153
 a 3377 b 743 a^4 + b^4 130358916347042 =  = 2 17 41^4 457  2969
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 4233 b 1324 a^4 + b^4 324138005544097 =  = 73^4 113  101009
 a 5985 b 2314 a^4 + b^4 1311760217373841 =  = 17 73^4 313  8681
 a 6547 b 4661 a^4 + b^4 2309226878233922 =  = 2 73^4  40657921
 a 7309 b 1919 a^4 + b^4 2867415887752082 =  = 2 17 41 73^4 113  641
 a 8633 b 6152 a^4 + b^4 6986928765245537 =  = 73^4 1993  123449
 a 10385 b 9957 a^4 + b^4 21460345858789426 =  = 2 73^4 521 761  953
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 4813 b 2128 a^4 + b^4 557122050422417 =  = 89^4  8879537
 a 9099 b 9013 a^4 + b^4 13453472525944162 =  = 2 89^4  107212241
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 7961 b 6052 a^4 + b^4 5358225614009057 =  = 41 97^4  1476217
 a 9305 b 2388 a^4 + b^4 7529139226222561 =  = 97^4 2441  34841
 a 10349 b 3253 a^4 + b^4 11582774691919682 =  = 2 97^4 1153  56737
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
 a 6069 b 2570 a^4 + b^4 1400276986893121 =  = 113^4  8588161
=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=

original longer output from yesterday:

 a 149 b 25 a^4 + b^4 493275026 =  2 17^4 2953
 a 486 b 479 a^4 + b^4 108431722897 =  17^4 113 11489
 a 504 b 337 a^4 + b^4 77422046017 =  17^4 926977
 a 529 b 188 a^4 + b^4 79560183617 =  17^4 73 13049
 a 554 b 39 a^4 + b^4 94199744497 =  17^4 1127857
 a 579 b 110 a^4 + b^4 112532938081 =  17^4 73 18457
 a 604 b 259 a^4 + b^4 137590574417 =  17^4 1647377
 a 635 b 454 a^4 + b^4 205074206081 =  17^5 97 1489
 a 654 b 557 a^4 + b^4 279195418657 =  17^4 1009 3313
 a 706 b 679 a^4 + b^4 460997249777 =  17^4 233 23689
 a 784 b 429 a^4 + b^4 411673088017 =  17^4 4928977
 a 855 b 704 a^4 + b^4 780032770081 =  17^4 9339361
 a 933 b 404 a^4 + b^4 784390561777 =  17^4 9391537
 a 983 b 823 a^4 + b^4 1392489005762 =  2 17^4 41 203321
 a 1002 b 751 a^4 + b^4 1326121160017 =  41^4 89 5273      ******
 a 1004 b 729 a^4 + b^4 1298525792737 =  17^4 15547297
 a 1033 b 525 a^4 + b^4 1214648074546 =  2 17^4 7271513
 a 1082 b 379 a^4 + b^4 1391227421057 =  17^4 113 147409
 a 1083 b 227 a^4 + b^4 1378323844162 =  2 17^4 8251361
 a 1121 b 933 a^4 + b^4 2336897702002 =  2 17^4 337 41513
 a 1133 b 71 a^4 + b^4 1647882860402 =  2 17^4 9865081
 a 1153 b 754 a^4 + b^4 2090538731537 =  17^4 25030097
 a 1183 b 369 a^4 + b^4 1977109279042 =  2 17^4 73 281 577
 a 1231 b 354 a^4 + b^4 2312023060177 =  17^4 89 311033
 a 1233 b 667 a^4 + b^4 2509204865842 =  2 17^4 3617 4153
 a 1283 b 965 a^4 + b^4 3576788996546 =  2 17^4 21412513
 a 1302 b 779 a^4 + b^4 3241972600897 =  17^4 38816257
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  • $\begingroup$ thanks. you are such a resourceful person in the mathematical community. Very much appreciated. $\endgroup$ – user97615 Oct 6 '16 at 12:13

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