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3

This problem has been studied already; see this Wikipedia article. Does there a exist a prime $q$ such that $p = 2^kq + 1$ is also prime? A number $q$ such that $p$ is never prime is called a SierpiƄski number; such numbers do exist, and some of them are prime. The smallest known such prime is $q = 271129$; the next known ones are $322523, 327739, ...


5

There is numerical evidence that for the vast majority of primes $p$, there exists $k$ making $q=2^k\,p+1$ prime. See first exceptions in A137715. The only practical way I see to find which $(p,k)$ make $q=2^k\,p+1$ prime in the question's context is testing $q$ for primality using a fast test. As noted in the other answer, sieving for small primes can be ...


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