Peter-Lawrence.Montgomery wrote:
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] == +-
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.
Dear Peter and All,
This is as far as I can go in Ubasic:
p Result
3 +
5 +
7 -
13 +
17 -
19 -
31 +
61 +
89 -
107 -
127 +
Dear All,
Following up my own msg here.
First, there is an obvious linear relationship between my two
conjectures, so they are equivalent.
Second, predictions where possible (U=Unknown):
p (p+1)/2 mod 31 Conj 1 (p-2) mod 31 Conj 2
4423 11 U 19 U
9689 9 - 15 -
9941 11 U 19 U
11213 27 U
