3

This is similar to a question that has been asked but I am asking for something slightly different. It's been asked "are there prime gaps of every (even) size?" My question is, is there a simple proof that there are prime gaps of every (even) size? I'm not actually asking for the proof, and I'd prefer not to read such a proof right now*. So in your answer, if you do provide a proof, first please indicate whether there's a simple proof or not so I can stop there.

By "simple proof" I mean something like the simplicity of the proof that prime gaps can be arbitrarily large--just using math a smart middle schooler could understand, and just taking up a paragraph or two or so.

Thank you!

*Don't want to read it right now because I'd like to figure it out for myself if I can. I just figured out the proof that they can be arbitrarily large, and realized that I had not proven that every possible size of prime gap actually exists, so I'd like to do that as well.

user3752935
  • 141
  • Highly unlikely, considering that many basic results in number theory require complex analysis. – L. F. Jul 25 '22 at 06:33
  • 6
    Does this answer your question? Are there prime gaps of every size? - found in the "Related" section on the right. Note an answer links to Can every even integer be expressed as the difference of two primes?, where an answer there links to Prime Conjectures and Open Questions, where the $3$rd item states the more general "Every even number is the difference of two primes" is still open. – John Omielan Jul 25 '22 at 06:35
  • John, like I said, I'm not asking "are there prime gaps of every size," I'm asking "is there a simple proof" of that proposition, and I'm not asking what the proof is (and would prefer not to know for now). – user3752935 Jul 25 '22 at 06:38
  • 1
    @user3752935 Note that being an open question (i.e., a conjecture) means, by definition, that there are no proofs (at least, any generally accepted by the math community), simple or otherwise, of the statement being either true or false. – John Omielan Jul 25 '22 at 06:39
  • Is it an open question? – user3752935 Jul 25 '22 at 06:40
  • 1
    "It appears to be open if every even number is the difference of two primes" In other words, no one knows the answer to your question, let alone a simple proof. There isn't even a complicated proof. No one knows if there are prime gaps of every size or if there are not prime gaps of every size. – Adam Rubinson Jul 25 '22 at 06:41
  • That answers it! Thanks. – user3752935 Jul 25 '22 at 06:42
  • 1
    What I have found with number theory is that there are lots of open problems in number theory. It is not hard to come up with a question that is an open problem in number theory. I guess that's what makes number theory exciting to some people. – Adam Rubinson Jul 25 '22 at 06:44
  • Whether every even prime gap occurs is unknown. It is conjectured this is the case. The best known result is that infinite many prime gaps do not exceed $246$ , so one prime gap upto this limit must occur infinite many often. But this is a very deep result without an easy proof. – Peter Jul 25 '22 at 11:30
  • Even worse , it is unknown whether for every even positive integer $k$ , there are primes $p,q$ with $q-p=k$ – Peter Jul 25 '22 at 11:31

0 Answers0