Why primes are either of the form 8n+3 or 8n+7
 

2, 3, 7, 19, 67, 79, 359, 2131, 3371, 331259 are the k's (k is prime) such that pg(k) is also prime

First I note that the primes have either the form 8n+3 or 8n+7 or 8n+2 (why?)

The primes that are not of the form 8n+3, that is 2, 7 and 359 are of the form s^2-2 where s is a prime... Infact 2=2^2-2, 7=3^2-2 and 359=19^2-2

One of the best ways to answer this question (without posting) is to try and find the answer in a text book or peer reviewed paper. If you haven't found it after a quick search then you haven't looked hard enough.

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if a prime p has the form 8s+1 or 8s+5


then 10^d*(2^p-1)+2^(p-1)-1 is divisible by 5