From: Warren Smith <warren.wds@gmail.com>
Search II: a^4+b^4=d*c^2 with 0<a<b<=1024 and 0<d<=1024. Result: The following d are "special": 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.]
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 -> 113 612 -> 17 738 -> 82 772 -> 193 833 -> 17 873 -> 97 1017 -> 113 I noticed this because hte GP script I wrote split a^4+b^4 into its core and square part, and sorted things by core. Consequently I didn't see any of the above NQF d's. Alas I don't have time to investigate, but I hope a number-theorist wakes up and stokes this rather intersting looking fire. Phil