Two Theorems on the structure of Pythagorean triples and some diophantine consequences

TL;DR

This paper proves two main theorems about the structure of Pythagorean triples and their implications for certain Diophantine equations, revealing restrictions on the existence of specific primitive triples and solutions.

Contribution

It introduces new theorems that characterize conditions under which certain Pythagorean triples and Diophantine equations have no solutions, expanding understanding of their algebraic structure.

Findings

No primitive Pythagorean triples with legs of specific quadratic residue forms exist under certain conditions.

Certain Diophantine equations involving prime squares and fourth powers have no solutions when primes satisfy particular residue properties.

The theorems connect quadratic reciprocity with the nonexistence of solutions in specific number-theoretic contexts.

Abstract

Even though four theorems are actually proved in this paper, two are the main ones,Teorems 1 and 3. In Theorem 1 we show that if a and be are odd squarefree positive integers satisfying certain quadratic residue conditions; then there exists no primitive Pythagorean triangle one of whose leglengths is equal to a times an integer square, while the other leglength is equal to b times a perfect square. The family of all such pairs (a,b) is slightly complicated in its description. A subfamily of the said family consists of pairs (a,b), with a being congruent to 1, while b being congruent to 5 modulo8; and also with both a and b being primes, and with a being a quadratic nonresidue ofb(and so by the quadratic reciprocity law, b also being a nonresidue of a). Theorem 3 is similar in nature, but less complicated in its hypothesis. It states that if p and q are primes, both congruent to 1 modulo4, and one of them being a quadratic nonresidue of the other.Then the diophantine equation, p^2x^4 + q^2y^4 = z^2, Has no solutions in positive integers x, y, and z, satisfying (px, qy)=1.