A family of diophantine equations of the form x^4 +2nx^2y^2+my^4=z^2 with no solutions in (Z+)^3

Abstract

In this work, we prove the following result(Theorem 1): Suppose that n is a positive integer, p an odd prime, and such that either n is congruent to 0 modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2 modulo4 and p congruent to 7 modulo 8. In addition to the above, assume that one of the following holds: Either (i) n^2-p>0 and the positive integer is a prime, Or (ii) n^2-p<0 and the positive integer N=-m=-(n^2-p) is a prime. Then the diophantine equation x^4 +2nx^2y^2+my^4=z^2 has no positive integer solutions. The method of proof is elementary in that it only uses congruence arguments, the method of (infinite) descent as originally applied by P.Fermat, and the general solution in positive inteagers to the 3-variable diophantine equation x^2+ly^2=z^2, l a positive integer. We offer 53 numerical examples in the form of two tableson pages 9 and 10; and historical commentary on pages 10-12.