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Old 2018-02-27, 15:36   #16
Dr Sardonicus
 
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Feb 2017
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Even with your conditions in post #8 and post #11 to this thread [M(p) divides p-1, and if p = 2^e - 1 then M(p) = e], your "function" is only defined unambiguously if either

(1) p is a Mersenne prime, or

(2) 2 is a primitive root (mod p).

Up to the limit 200, the following primes do not satisfy either condition:

For p = 17, the possible values of M(p) are 8 times k for k in [1, 2].
For p = 23, the possible values of M(p) are 11 times k for k in [1, 2].
For p = 41, the possible values of M(p) are 20 times k for k in [1, 2].
For p = 43, the possible values of M(p) are 14 times k for k in [1, 3].
For p = 47, the possible values of M(p) are 23 times k for k in [1, 2].
For p = 71, the possible values of M(p) are 35 times k for k in [1, 2].
For p = 73, the possible values of M(p) are 9 times k for k in [1, 2, 4, 8].
For p = 79, the possible values of M(p) are 39 times k for k in [1, 2].
For p = 89, the possible values of M(p) are 11 times k for k in [1, 2, 4, 8].
For p = 97, the possible values of M(p) are 48 times k for k in [1, 2].
For p = 103, the possible values of M(p) are 51 times k for k in [1, 2].
For p = 109, the possible values of M(p) are 36 times k for k in [1, 3].
For p = 113, the possible values of M(p) are 28 times k for k in [1, 2, 4].
For p = 137, the possible values of M(p) are 68 times k for k in [1, 2].
For p = 151, the possible values of M(p) are 15 times k for k in [1, 2, 5, 10].
For p = 157, the possible values of M(p) are 52 times k for k in [1, 3].
For p = 167, the possible values of M(p) are 83 times k for k in [1, 2].
For p = 191, the possible values of M(p) are 95 times k for k in [1, 2].
For p = 193, the possible values of M(p) are 96 times k for k in [1, 2].
For p = 199, the possible values of M(p) are 99 times k for k in [1, 2].

If you want M(p) to be the multiplicative order of 2 (mod p), please just say so.
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