Various people & I made some progress, but then it turned out the full solution of this problem was already known. Theorem: a^4+b^4=d*c^2 for d fixed either has an infinity of nonzero solutions (a,b,c) or none (A.Desboves & WDS). Theorem: When d=1, none (P.de Fermat 1640 or so, see T.Nagell: Intro to number theory Wiley 1951, pp.227-229; L.J.Mordell: Diophantine equations, Academic Press 1969 chapter 4) Theorem: when d=2 the only solutions (a,b,c) are the trivial ones with a=b (Euler 1738, see Mordell: Diophantine Equations, Academic Press 1969 chapter 4 page 18.) Theorem: Wlog we can take d to be squarefree (Phil Carmody). Theorem: Then the prime factors of d all must be of the form 8*k+1 or 2 only (Jaroslaw Wroblewski). Actually both Carmody & Wronblewki's theorems were given on p110 of H.C.Pocklington: Some diophantine impossibilities, Proc Cambridge Philos Soc 17 (1913) 108-121. Corollary of JW's theorem: every allowed squarefree d is a sum of two squares [WDS using Euler 1738]. 16-Theorem: Also d must equal 0,1,2,4,8,9, or 12 (mod 16) [WDS]. 81-theorem: Also (d mod 81) must be in {0,1,2,4,5,8,9,11,13,16,17,18,19,26,27,32,36,38,40,41, 43,45,46,54,59,61,63,65,71,72,76,77,80} [WDS]. Similar mod-p^4 theorems [WDS]: (density of allowed d mod 2^4)=7/16=0.437500 (density of allowed d mod 3^4)=33/81=0.407407 (density of allowed d mod 5^4)=233/625=0.372800 (density of allowed d mod 7^4)=878/2401=0.365681 (density of allowed d mod 11^4)=5406/14641=0.369237 (density of allowed d mod 13^4)=10298/28561=0.360562 (density of allowed d mod 17^4)=35453/83521=0.424480 (density of allowed d mod 19^4)=48901/130321=0.375235 Theorem [WDS proved using an idea by E.B.Escott 1900] If d squarefree then 2*d must be a a "congruent number" i.e. member of this sequence http://oeis.org/A003273 . But most of the above theorems are subsumed by the full solution theorem...: Upon looking in Henri Cohen: Number Theory volume I, Springer 2007 GTM 239, pages 392-395, lo and behold, he has a discussion of the diophantine a*X^4+b*Y^4=c*Z^2 leading to a complete solution of our problem! This 2-volume book by Cohen is packed with very powerful modern stuff and dispenses with a lot of old gunk. Specialized to our case a^4+b^4=d*c^2, here is the result. THEOREM [H.Cohen]. For integer d>=3 fixed, a^4+b^4=d*c^2 either has an infinite number of inequivalent nonzero solutions(a,b,c), or no nonzero solutions. For infinite solutions d must have squarefree part divisible only by 2 and primes 8*k+1. Then the question is related to this elliptic curve EC: y^2 = x*(x^2 + d^2) and we have infinite solutions if and only if the class of d modulo squares belongs to the image of EC's 2-descent map (described in section 8.3 of Cohen). Here is how the problem maps to EC: x=d*a^(-2)*b^2, y=d*d*b*c*a^(-3). Under the Birch/Swinnerton-Dyer conjecture, the rank of EC will always be even. Cohen p395 gives the following table, computed with MAGMA and MWRANK using above theorem, listing every d with 3<=d<=10001 such that infinite solutions exist. 17, 82, 97, 113, 193, 257, 274, 337, 433, 514, 577, 593, 626, 641, 673, 706, 881, 914, 929, 1153, 1217, 1297, 1409, 1522, 1777, 1873, 1889, 1921, 2129, 2402, 2417, 2434, 2482, 2498, 2642, 2657, 2753, 2801, 2833, 2897, 3026, 3121, 3137, 3298, 3329, 3457, 3649, 3697, 4001, 4097, 4129, 4177, 4226, 4289, 4481, 4546, 4561, 4721, 4817, 4993, 5281, 5554, 5617, 5666, 5729, 5906, 6002, 6353, 6449, 6481, 6497, 6562, 6577, 6673, 6817, 6866, 7057, 7186, 7489, 7522, 7537, 7633, 7762, 8017, 8081, 8737, 8753, 8882, 8962, 9281, 9298, 9553, 9586, 9649, 9778, 9857, 10001.