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 $$σ\vert ρ\biggl(\left(m-n\right)\left(m+n\right)\biggr),\indent \text{or} \indent σ\vert ρ( nm )$$ 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.