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Are there an infinite number of sizes of gaps between primes? let $p_n$ be the nth prime number. Let $g_n = p_{n+1} - p_n$ (i.e. size of gaps between consecutive primes). As $p_n$ goes to infinity, does $g_n$ go to infinity also?

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  • $\begingroup$ It is not known whether any difference occurs infinite many often, but it is known that differences $\le 246$ occur infinit many often (See also the comment below) and that arbitary large gaps must exist (but I think it is not known whether every large difference occurs, but here I might be wrong). $\endgroup$ – Peter Oct 17 '16 at 15:08
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    $\begingroup$ It is conjectured that every even difference occurs infinite many often, part of this conjecture is the twin-prime-conjecture. $\endgroup$ – Peter Oct 17 '16 at 15:19
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    $\begingroup$ @Peter: More precisely (with regards to your first comment), there exists some integer $N\leq246$, for which there are infinitely many pairs of primes that differ by $N$. $\endgroup$ – barak manos Oct 17 '16 at 16:02
  • $\begingroup$ @barakmanos Indeed, though your statement is a consequence of Peter's. $\endgroup$ – Tavian Barnes Oct 17 '16 at 16:14
  • $\begingroup$ @T.C. Your headline and your first sentence pose a different question to your second and third sentences. As the question currently stands, the question the second and final sentence ask is: "Is there a gap of infinite size between two consecutive primes?" $\endgroup$ – user24000 Oct 17 '16 at 18:29
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You can easily find as long a string of composites as you wish, so the gaps between primes can be arbitrarily large, so must have infinitely many different values.

Consecutive composite numbers

But that does not mean the size of the gap goes to infinity. In fact it's less than 70 million infinitely often.

https://en.wikipedia.org/wiki/Yitang_Zhang

As @DunstanLevenstein comments. 70 million was the bound in Zhang's revolutionary paper. It's since been reduced to 246.

It's thought that in fact there are infinitely many twin primes, so the conjecture is that the bound is actually 2.

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    $\begingroup$ The latest definitive news on the prime gaps problem, as of April 14, 2014, is that it has been reduced to 246 - significantly lower than 70,000,000. en.wikipedia.org/wiki/Twin_prime#History $\endgroup$ – Dustan Levenstein Oct 17 '16 at 14:50
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    $\begingroup$ To drive the point home, it follows from Zhang's result that $g_n\not\to\infty $ as $n\to\infty $. From the results mentioned in this answer, we get $\liminf_n g_n\le 246$ and $\limsup_n g_n=+\infty $. And the twin prime conjecture is equivalent to the claim that $\liminf_n g_n=2$. $\endgroup$ – Andrés E. Caicedo Jan 11 '18 at 0:50

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