Problem A3 in Richard Guy's `Unsolved Problems in Number Theory'
includes this question, by D.H. Lehmer:
Let Mp = 2^p - 1 be a Mersenne prime, where p > 2.
Denote S[1] = 4 and S[k+1] = S[k]^2 - 2 for k >= 1.
Then S[p-2] == +- 2^((p+1)/2) mod Mp.
Predict which congruence occurs.
For example, when p = 3, S[1] = 4 == 2^2 (mod 7).
When p = 5, S[3] = 194 == 2^3 (mod 31).
When p = 7, S[5] = 1416317954 == -2^4 (mod 127).
The sign is + for p = 3 and p = 5 but - for p = 7.
Do we have the pattern through M38?
Peter Montgomery
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