Infinite number of primes of the form $2^x \cdot 3^y + 1$?

Question

Are there an infinite number of primes of the form $2^x \cdot 3^y + 1$?

I really have no idea where to start with this. I thought of it because it would imply an affirmative answer to this recent question: Does Euler's $\phi$ function have the same value in arbitrarily large subsets of $\mathbb{N}$?.

It seems likely to be unknown - such questions are rarely known. — Thomas Andrews, Jul 09 '15 at 23:03
Seems to be an open problem https://en.wikipedia.org/wiki/Pierpont_prime — Dan Brumleve, Jul 09 '15 at 23:07
Such questions are rarely known, but almost always it's obviously "yes". — Trevor, Feb 12 '22 at 05:18

score 2 · Accepted Answer · answered Jul 09 '15 at 23:09

2

This is Guy's Unsolved Problems number A18. This doesn't necessarily mean that it's not doable, but I do think it's not currently been done. [I think it is not currently doable, however].

answered Jul 09 '15 at 23:09

davidlowryduda

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Infinite number of primes of the form $2^x \cdot 3^y + 1$?

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