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Old 2020-09-14, 13:16   #4
Dr Sardonicus
 
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Feb 2017
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If p is prime then 2p - 2 is divisible by 2p. So if q divides 2p - 1, gcd(q-1, 2p - 2) is divisible by 2p.

However, 2n - 2 is divisible by 2 but not by 4 if n > 1. If p is prime and the smallest prime factor q of 2p - 1 is congruent to 1 (mod 4), q-1 does not divide 2p - 2.

Consulting a table of factorizations of 2n - 1, I find that p = 29 (already noted) is the smallest p with this property. The next is p = 53 (q = 6361).
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