Abstract
Abstract We provide explicit formulae for primitive, integral solutions to the Diophantine equation x 2 + y 2 = M {x}^{2}+{y}^{2}=M , where M M is a product of powers of Pythagorean primes, i.e., of primes of the form 4 n + 1 4n+1 . It turns out that this is a nice application of the theory of Gaussian integers.
Highlights
The history of the Diophantine equation x2 + y2 = M has its roots in the study of Pythagorean triples
In 1625 Albert Girard, a French-born mathematician working in Leiden, the Netherlands, who coined the abbreviations sin, cos, and tan for the trigonometric functions and who was one of the first to use brackets in formulas, stated that every prime of the form 4n + 1 is the sum of two squares
It is easy to see that, if an odd prime is a sum of two squares, it must be of the form 4n + 1
Summary
For a Pythagorean prime p = 4n + 1, Gauss provided an explicit formula for the unique positive, primitive solution {x, y} of the Diophantine equation x2 + y2 = p. Hardy and Wright [17, Theorem 278] gave a formula which can be used to calculate the number of all integer solutions of equations of the form x2 + y2 = M for any given natural number M. The purpose of this paper is to provide explicit formulae for positive, primitive, integral solutions to the same Diophantine equation.
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