The following problem came up in something I was doing recently. I have no idea how difficult it is – it looks hard to me – but I would be glad to hear from anyone who knows more than I do.
As is well known, a Mersenne prime is one of the form 2n−1, and a Fermat prime is one of the form 2n+1. In each case, only finitely many examples are known, but it is thought that there may be infinitely many Mersenne primes but only finitely many Fermat primes.
If we relax “prime” to “prime power”, we get Catalan’s equation, which only gives us one new solution: 23+1 = 32.
But what I want is the following. For which positive integers n is it the case that each of 2n−1 and 2n+1 is the product of at most two distinct primes?
It happens that, in many small cases, if one of these numbers is prime, the other is a product of two primes; but this may be just the law of small numbers. But there are other cases. For example,
211−1 = 23×89, 211+1 = 3×683.
As usual, thoughts welcome.