# Prime of the form $4p+1$

I have googled this but found nothing. Do we know if there are infinitely many (or finitely many) primes of the form $4p+1$ where $p$ itself prime?

I proved something only for this case of prime and want to know if it covers only finitely many cases or not (at least I know I didn't prove an empty case :) )

Thank you.

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I believe this result is true if you consider if it was false for all p large enough has to be $4n+3$ which is a formula for the primes number in sense. – checkmath Mar 7 '12 at 0:07
See OEIS A090866, A023212 and a longer list of the latter. These are quite common. – Henry Mar 7 '12 at 0:55
– David Speyer Mar 7 '12 at 1:39

## 1 Answer

We "know" that there are infinitely many primes $p$ such that $4p+1$ is prime, in the sense that we are absolutely certain that it is true. However, no one has been able to prove that it is true.

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May I ask from where this confidence comes from? – user9077 Mar 6 '12 at 23:13
Yeah, is that a conjecture? – checkmath Mar 6 '12 at 23:18
@user9077 Yes, it is conjectural. But this is of a large collection of possible statements about primes where probabilistic arguments suggest firmly there ought to be infinitely many. And, as you are finding, some of the most useful conjectures, in the sense of coming up fairly naturally, are the farthest from proof. – Will Jagy Mar 6 '12 at 23:27
We not only "know" that there are infinitely many such primes, we even have a simple formula that tells us, as a function of $x$, approximately how many such primes there are with $p\lt x$. The formula is very close to the actual counts for even the largest values of $x$ for which calculations have been done. For more information, search for Dickson's conjecture. – Gerry Myerson Mar 6 '12 at 23:34