On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor

摘要

The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x~2+y~2=z~2 It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and nm are mutually Prime numbers of different parity, and this happens in cases where, for example, n=4, and m is a Prime number ending in 3 or 7. In such cases, the hypotenuse z=m~2+n~2 is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.