# A family of diophantine equations of the form x^4 2nx^2y^2 my^4=z^2 with no solutions in Z ^3 - Mathematics > Number Theory

A family of diophantine equations of the form x^4 2nx^2y^2 my^4=z^2 with no solutions in Z ^3 - Mathematics > Number Theory - Download this document for free, or read online. Document in PDF available to download.

Abstract: In this work, we prove the following resultTheorem 1: Suppose that n is apositive integer, p an odd prime, and such that either n is congruent to 0modulo4 and p congruent to 3 modulo8; or alternatively, n is congruent to 2modulo4 and p congruent to 7 modulo 8. In addition to the above, assume thatone of the following holds: Either i n^2-p>0 and the positive integer is aprime, Or ii n^2-p<0 and the positive integer N=-m=-n^2-p is a prime. Thenthe diophantine equation x^4 +2nx^2y^2+my^4=z^2 has no positive integersolutions. The method of proof is elementary in that it only uses congruencearguments, the method of infinite descent as originally applied by P.Fermat,and the general solution in positive inteagers to the 3-variable diophantineequation x^2+ly^2=z^2, l a positive integer. We offer 53 numerical examples inthe form of two tableson pages 9 and 10; and historical commentary on pages10-12.

Author: ** Konstantine Zelator**

Source: https://arxiv.org/