Least Primitive Root of Mersenne Primes

Least Primitive Root of Mersenne Primes: All the more reason the C217 composite cofactor of 2^1101+1 ought to be factored. The final terms of the sequence were calculated using the Mersenne factorizations mentioned earlier. The earlier terms were calculated using znprimroot in Pari/GP. Proving a primitive root X of a prime p requires modular exponentiation of X to all powers (p-1)/q where q is a prime factor of (p-1). At first I thought it had to be to all divisors of (p-1), which would have required something like 2^50 work for 2^2281-1; fortunately, it was not.

The least primitive root of M2203 is at least 3, and it probably is 3. 3 is probably the least primitive root for M4253, M9689, M21701, as well. This comes from the observation that 2 never seems to be a primitive root. When 3 is not a primitive root, then 3^((p-1)/3)==1 (mod p) empirically. It's interesting that all mersenne primes except the first seem to have remainder 1 when divided by 3 (true for all odd exponents).

for(i=2, length(p), if((Mod(3,2^p[i]-1)^((2^p[i]-2)/3))!=Mod(1,2^p[i]-1), print(p[i]))) where p is a list of mersenne exponents.

P-1 for some of the higher Mersenne prime exponents are surprisingly smooth: 216090 = 2*3*3*5*7*7*7*7, 20996010 = 2*3*3*3*3*5*7*7*23*23. These would be more interesting targets for the ElevenSmooth project than their number 2*2*2*2*2*2*3*3*3*5*5*7*11 = 3326400

In an unrelated topic, in June I submitted a superseeker query for [7,11,32,40,75,87,136,152,215,235,312,336,427], but I no longer know the significance of the sequence. I think it might have something to do with the Pell Equation. The generating function guesser generated the continuation [455, 560, 592, 711, 747, 880], which I recall the first few terms being correct.