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Old 2021-03-23, 12:39   #14
Dr Sardonicus
 
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Feb 2017
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The fact that S(P-2) mod p doesn't tell you anything about whether 2^P - 1 may be prime when P = 2^p - 1 is prime, disproves the existence of a general "S(p-2) mod p" test for Mersenne primes.

More important than this specific case IMO is the fact that S(p-2) == trace(u^e) (mod p), where u = Mod(x+2,x^2 - 3) and e = 2^(p-2)%m, where

m = p - 1 if p == 1 or 11 (mod 12)

m = p + 1 if p == 5 or 7 (mod 12).

We know that u^m == 1 (mod p) so the exact multiplicative order of u (mod p) is a divisor of m. I have not investigated when it might be known to be a proper divisor.

If there were some pattern in S(p-2) (mod p) that indicated whether 2^p - 1 was prime, presumably it would somehow be manifest in the "reduced" exponent e = 2^(p-2)%m. I am not aware of any convenient formula for this remainder, but I don't know that there isn't one.

So far the "trivial" pattern when p is a Mersenne prime is the only consistent one I am aware of, and in that case S(p-2) mod p utterly fails to indicate whether 2^p - 1 is prime.
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