# (In)finitely many natural numbers are not the sum or difference of two perfect powers

Are there infinitely many positive integers which are neither a sum nor a difference of two perfect powers?

This question was proposed some years ago at KoMaL.

It's easy to see that the odd numbers can be written as the difference of two consecutive perfect squares.

A similiar problem was posted at MSE and the same problem was posted at MSE. According to OEIS there are many numbers that are hard (and probably open) to prove they aren't a difference of powers.

I'm looking for a proof or any reference of this result.
Any help would be appreciated, thanks.

• Given that the difference question seems to be known to be hard, and sums of two perfect powers are sparse for obvious reasons, this new question, although logically a little easier, might still be hard. Is there any reason to believe that mathematics is ready for this problem? – znt Sep 3 '16 at 15:55
• @znt Yes, there is. The question with difference only is an open problem. The question with difference or sum is not an open problem (open problems are not proposed at KoMaL). – jack Sep 3 '16 at 16:22
• If you're saying that this is a puzzle, and someone somewhere knows the answer, then some people would say that this question isn't appropriate for this site and you should take it to MSE. – znt Sep 3 '16 at 18:06
• @znt This is a research level question, that is why I posted it here. Please, read the 'comments' and 'references' from the OEIS site. They say: 'Conjectured list of positive numbers which are not of the form r^i-s^j'. – jack Sep 3 '16 at 18:27
• I think what is conjectured at OEIS, jack, is not that only the numbers listed satisfy the property, but that all the numbers listed do satisfy the property. For any given number on the list, it is hard (and probably open) to prove that it isn't a difference of powers. Think of how hard it was to prove Catalan's Conjecture. – Gerry Myerson Sep 3 '16 at 22:21