  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.
