The Mobius Function and Congruent Numbers

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.