New Mersenne conjecture

for any odd natural number p, if any two of the three conditions p = 2^k \pm 1 or p = 4^k \pm 3, 2^p - 1 is prime, and (2^{p} + 1)/3 is prime are true, then the third condition is also true.

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Source: List of unsolved problems in mathematics (CC BY-SA 4.0)