# Does there exist a prime $p > 5$ such that $3p+1 = 2^n, n \in \mathbb{N}$?

Looking at prime factorization, and I was wondering if, in general, there were any primes larger than 5 so that 3 times the prime plus one is a power of two.

If there is, is there a method of determining that without multiplication - i.e. is there a pattern in which they appear?

I have tested all the low primes and have not found one that does this like $5$.

If $3p+1=2^n$ then $2^n\equiv 1 \pmod{3}$ and hence $n$ is even. Let$n=2m$.
Then $$3p=2^{2m}-1=(2^m-1)(2^m+1)$$
This means that either $$2^m-1=1 \\ 2^m+1=3p$$ which is not possible, or $$2^m-1=3 \\ 2^m+1=p$$ which gives $p=5$.
Therefore, $p=5$ is the only possibility.