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

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.