Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

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.

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.