[colbyh]

1. Essentially, yes there's a list. Since there doesn't exist a higher level formula to identify primes it's been verified by some variation of a brute force method.

2. That all smaller primes are known can't be verified. Primes of this size are usually found by a number of formulas that find "likely" primes and then verified by brute force (afaik). Because of that fuzzy match to start there might be smaller primes that haven't yet been identified.

[planteen]

Calling it brute force isn't quite right. There is an efficient algorithm for prime determination (Lucas-Lehmer primality test) on Mersenne Primes (numbers of the form 2^p-1).

https://en.wikipedia.org/wiki/Lucas–Lehmer_primality_test

[colbyh]

Thanks for the reminder! And this is my bad because I was paywalled on my phone and didn't see that it was, in fact, a mersenne prime that was found.

[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-...)