<s>[INST] Product of arbitrary Mersenne numbers 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 3rd power 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 $63\cdot 127 +1=4001\cdot 2$
 [/INST] Write $p=2^m-1$ and $q=2^n-1$ where $m>n$ without loss of generality. If $pq-1=(2^m-2^{m-n}-1)2^n$ is a square, then so is $2^m-2^{m-n}-1$. But then $m=n+1$, as $m\ge n+2$ implies $2^m-2^{m-n}-1\equiv 3\pmod 4$. Now, $2^m-2^{m-n}-1=2^m-3$, which clearly is not a square if $m$ is even, and which is not a square if $m$ is odd either as in this case $2^m-3\equiv (-1)^m\equiv -1\pmod 3$.
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