  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.
