[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-...)