A family of diophantine equations of the form x⁴ +2nx²y²+my⁴=z² with no solutions in (Z+)³

2009/05/20 by Konstantine Zelator, Zelator, Konstantine
Computer Science · Mathematics · #11 D #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Polynomial and algebraic computation #math.NT #msc:11

paper · pdf · doi:10.48550/arxiv.0905.3346

13 pages, no figures

arxiv created 2009/05/20 · openalex publication_date 2009/05/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) n²-p>0 and the positive integer is a prime, Or (ii) n²-p<0 and the positive integer N=-m=-(n²-p) is a prime. Then the diophantine equation x⁴ +2nx²y²+my⁴=z² 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 x²+ly²=z², 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.

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