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

TL;DR

This paper proves that certain families of diophantine equations of the form x^4 + 2nx^2y^2 + my^4 = z^2 have no positive integer solutions under specific conditions, using elementary methods like congruences and descent.

Contribution

It introduces new non-existence results for solutions of a class of quartic diophantine equations with specific parameter constraints, employing elementary proof techniques.

Findings

No positive solutions for the specified equations under given conditions.

53 numerical examples illustrating the results.

Historical commentary on related diophantine problems.

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.