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Abstract: Even though four theorems are actually proved in this paper, two are the mainones,Teorems 1 and 3. In Theorem 1 we show that if a and be are odd squarefreepositive integers satisfying certain quadratic residue conditions; then thereexists no primitive Pythagorean triangle one of whose leglengths is equal to atimes an integer square, while the other leglength is equal to b times aperfect square. The family of all such pairs a,b is slightly complicated inits description. A subfamily of the said family consists of pairs a,b, with abeing congruent to 1, while b being congruent to 5 modulo8; and also with botha and b being primes, and with a being a quadratic nonresidue ofband so by thequadratic reciprocity law, b also being a nonresidue of a. Theorem 3 issimilar in nature, but less complicated in its hypothesis. It states that if pand q are primes, both congruent to 1 modulo4, and one of them being aquadratic 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, satisfyingpx, qy=1.



Autor: Konstantine Hermes Zelator

Fuente: https://arxiv.org/







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