WDS: Search II: a^4+b^4=d*c^2 with 0<a<b<=1024 and 0<d<=1024. Result: The following d are "special" (allow an apparent infinity of integer solutions, as opposed to none): 17, 68, 82, 97, 113, 153, 193, 257, 272, 274, 328, 337, 388, 425, 433, 452, 514, 577, 593, 612, 626, 641, 673, 706, 738, 772, 833, 873, 881, 914, 1017. [This sequence is not in OEIS.]
Phil Carmody: Whilst the existence of a solution for d = D doesn't guarantee on for d = k^2*d, I'd be tempted to fold all the non-quadratfrei numbers in that list into their core, moving the square part into c.
68 -> 17 153 -> 17 272 -> 17 (* a 4th power, so definitely worth dropping) 328 -> 82 388 -> 97 425 -> 17 452 -> 113 ... --WDS: Actually, the existence of an integer solution (a,b,c) for d=D DOES guarantee the existence of a RATIONAL solution for d=D*k^2, namely use c/k not c. If we then multiply a times k, b times k, and c/k times k^2 we get INTEGERS. So yes, Phil Carmody is entirely justified: we should only consider squarefree d. But the subsequence 17, 82, 97, 113,... of squarefree "special" d also is not in the OEIS. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)