Focus on the right-hand side, we can distribute the first factor into the second one:

The first term gives

and the second term gives

When adding up these two terms, it's easy to see that all terms but the first and the last cancel out proving the identity.

So this basically proves that one Mersenne number (which isn't prime) can be factored out into another Mersenne number. And it also proves that if the exponent is of the form it cannot be a Mersenne prime.

Happy to see other people post puzzles as well :)

Anyone with time and coding skills, check out this issue to add spoiler formatting in SN. In addition to the usual SN contribution reward, I am giving out a 30k sat bounty.