[Someone]

We know for sure that we don’t know all smaller primes.

For example, there is at least one prime between 2^77,232,915 and 2^77,232,916 that we don’t know (https://en.wikipedia.org/wiki/Bertrand%27s_postulate)

(That’s a very loose bound. I think that, for any N > 2, there are more primes than squares of integers less than it)

[throwacide]

What’s the significance of primes being one less than large powers of 2? Does this mean base-2 is the most natural base?

[Someone]

2 is the only integer n such that n-1 = 1.

That’s significant because (n^m) - 1 is divisible by n - 1.

So, for example, 125320078^35785332 - 1 is divisible by 125320077 and, hence, not a prime.

So, this problem has been solved for all n>2.

n^m + 1 is a more difficult problem. For example, for n=10, we get 2: prime, 11: prime, 101: prime, 1001: composite, 10001: composite, etc (at least the 8,000 following are composite. See https://math.stackexchange.com/questions/2108085/is-there-a-...)