3
$\begingroup$

A proof I'm writing rests on something I can't prove, probably beyond my knowledge, but it seems right:

For any two primes $p_k, p_l$ (not necessarily consecutive) such that the distance between them $|p_l - p_k| = n$, there exist infinitely many other primes such that the distance between them is also $n$.

I can't figure out a way to show this; I'm guessing it's probably a known result and referring to it would be enough.

  • 0
    Maybe not false, but not yet proved? It is the twin-prime-conjecture.2015-01-26

1 Answers 1

12

Nobody knows. The twin primes conjecture is still a conjecture. Same for your $n=4,$ or $n=6,$ and so on. Nobody knows. As pointed out, you do need to take your $n$ even.

http://en.wikipedia.org/wiki/Twin_prime

There seem to be doubts. Let me point out that the Prime Number Theorem says that the next larger prime above some prime $p$ is approximately $ p + \log p,$ where the logarithm is base $e = 2.718281828459...$ At the same time, conjectures of, for example, Shanks, are consistent with the suggestion that the next larger prime is never larger than $p \; + \; 3 \; (\log p)^2.$ What is missing is small prime gaps, maybe there is some slowly increasing function ( monotone increasing and unbounded) $f(p)$ such that the next prime is larger than $p + f(p).$ If so, you are out of luck. Nothing is known for certain except the Prime Number Theorem.

  • 0
    Well, these days there might be hope for proving that there are primes fairly close together. See https://www.simonsfoundation.org/quanta/20131119-together-and-alone-closing-the-prime-gap/ for a broad-strokes overview of recent development in the area.2013-11-22