Off Topic  New Mersenne Conjecture.
Here is what they establish the New Mersenne Conjecture.
Let p be any odd natural number. If two of the following conditions hold, then so does the third:
1) p = 2^k +/ 1 or p = 4^k +/ 3
2) 2^p  1 is a prime
3) (2^p + 1)/3 is a prime
