Prove that $(2^n-1)(3^n-1)$ is not a perfect square
Solution 1:
That there are no solutions was proved by Szalay in 1997; a generalization to the equation $$ (2^n-1)(3^m-1) = z^2 $$ was given by Walsh in 2000 or so :
http://mysite.science.uottawa.ca/gwalsh/slov1.pdf
The proof follows from elementary arguments about (binary) recurrence sequences and local considerations at the primes $2$ and $3$.