Let be $p$ and $q$ two arbitrary Mersenne numbers.

Is there a simple proof that $p\cdot q-1$ can never been a square?

$p\cdot q-1$ can instead be a power of 3 for:

$p=3,q=3$

$p=7,q=31$

$p=63,q=127$

In these cases it is interesting to see that $p\cdot q+1$ is an even semi-prime, as in the case $3\cdot 3 +1=10$ or $7\cdot 31+1=218$ or $63\cdot 127 +1=4001\cdot 2$