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primenumbers@yahoogroups.com, "John" <mistermac39@...> wrote:

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> This is a copy of a post sent to me by Warren.

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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 diophantinea*X^4+b*Y^4=c*Z^2leading 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^2either 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 ifthe 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 willalways 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. Cohen notes that d=1513=17*89 is not there and notes d=2801 was a difficult case for his software.