Abstract

We show that if  a square-free and odd (respectively, even) positive integer is a congruent number, then $$\#\{(x,y,z) \in \mathbb{Z}^3|n=x^2+2y^2+32z^2\}=\#\{(x,y,z) \in \mathbb{Z}^3|n=2x^2+4y^2+9z^2-4yz\},$$\#\{(x,y,z) \in \mathbb{Z}^3|\frac{n}{2}=x^2+4y^2+32z^2\}=\#\{(x,y,z) \in \mathbb{Z}^3|\frac{n}{2}=4x^2+4y^2+9z^2-4yz\}.$$

We shall also discuss some applications of the proposed method. In particular, for a prime , we show that  if   is a congruent number, then the -rank of equals one  and if with then is not a congruent number.