The Mobius Function and Congruent Numbers

TL;DR

This paper characterizes congruent numbers using Pythagorean triples and the Mobius function, providing a new criterion for identifying congruent numbers based on number-theoretic properties.

Contribution

It introduces a novel characterization of congruent numbers through Pythagorean triples and Mobius function conditions, linking geometric and number-theoretic concepts.

Findings

Every congruent number can be expressed via Pythagorean triples and specific divisibility conditions.

A new criterion involving the Mobius function and gcd conditions determines congruent numbers.

The characterization simplifies the process of identifying congruent numbers using algebraic and arithmetic properties.

Abstract

This work provides a complete characterization of congruent numbers in terms of Pythagorean triples. Specifically, we show that every congruent number can be written as $$\frac{nm\left(m-n\right)\left(m+n\right)}{\sigma^2}$$ were as $$\sigma \vert \rho\biggl(\left(m-n\right)\left(m+n\right)\biggr),\indent \text{or} \indent \sigma \vert \rho( nm )$$ were $\rho(\alpha)$ denotes the non-square free part of its argument $\alpha$. As a consequence, in order to find congruent numbers it suffices to devise a condition so that the equality $\mu(m-n)+1 = \gcd(m,n)$ or $\mu(m+n)+1 =\gcd(m,n)$ holds, were $\mu$ is the Mobius function.