# Prime numbers stretch to infinity, but what about the distance between them?

That is, let $p_n$ be the nth positive prime number. Does $$L = \lim\limits_{n \to \infty} \left( p_{n+1} - p_n \right)$$ equal infinity?

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I doubt whether the behavior of that sequence is completely known, since it is still unsolved whether there are infinitely many twin primes. –  Tobias Kildetoft May 6 '11 at 7:47

No, the limit (probably) does not exist.

The sequence $p_{n+1} - p_n$ has a name: it is called the sequence of prime gaps. Define $g_n = p_{n+1} - p_n$, then you are interested in the sequence $g_1, g_2, \dots$. The following facts are known:

• For any $k$, it is easy to see that the $(k-1)$ numbers $k!+2, k!+3, \dots, k!+k$ are all non-prime. Thus there exist arbitrarily long sequences of composite numbers; in other words, $g_n$ can get arbitrarily large: for any $k$ there exists $n$ such that $g_n \ge k$. Equivalently, $$\limsup_{n \to \infty} (p_{n+1} - p_n) = \infty.$$

• There is the (believed) twin-prime conjecture which states that there exist infinitely many primes that differ by 2; this means that $g_n$ takes the value $2$ for infinitely many $n$, or $$\liminf_{n \to \infty} (p_{n+1} - p_n) = 2.$$ Even if it turns out to be false for 2, there is Polignac's conjecture that for any even integer $k$, there exist infinitely many primes that differ by $k$; this has not been proved or disproved for any $k$. Even better, it has been proved in 2005, assuming a certain conjecture (which I think is weaker than the Riemann hypothesis) that there are infinitely many $n$ for which $g_n$ is at most $16$, thus $$\liminf_{n \to \infty} (p_{n+1} - p_n) \le 16.$$

• The "average" gap between the $n$th prime and the next increases logarithmically: $g_n \approx \ln p_n$. So you can say that the distance between primes does become infinite "on average"; though infinitely often it touches very small numbers.

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I accepted your answer essentially because of the last topic. I should have thought in terms of average too, quite interesting. –  Luke May 7 '11 at 21:37
I just found this survey article that has a lot more details: ams.org/journals/bull/2007-44-01/S0273-0979-06-01142-6/… ("Small gaps between prime numbers: The work of Goldston-Pintz-Yildirim") –  ShreevatsaR May 15 '13 at 6:06
Polignac turned out to be right: golem.ph.utexas.edu/category/2013/05/…. –  Luke May 20 '13 at 23:48
@Luke: AFAICT, Zhang's result does not yet prove Polignac's conjecture. It says that there are infinitely many prime gaps less than $C \approx 7 \times 10^7$, so technically all we can say is that at least one gap less than $C$ occurs infinitely often (not all gaps). But, it's a step in the relevant direction. –  ShreevatsaR May 22 '13 at 6:40

Assuming some fairly reasonable conjectures, then $(p_{n+1}-p_n)$ is a divergent sequence (i.e. a limit does not exist). For example:

Conjecture: There exists an infinite number of twin primes. (where $p_{n+1}-p_n=2$ an infinite number of times)

Conjecture: There exists an infinite number of prime pairs that differ by 4. (where $p_{n+1}-p_n=4$ an infinite number of times)

However, we can observe that $\lim\sup (p_{n+1}-p_n)=\infty$, since $n!+k$ is composite (properly divisible by k) for all $n \geq 2$ and $2 \leq k \leq n$. Therefore there are arbitrarily large prime gaps.

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Though, maybe it is of interest to you that there are arbitrarily large gaps between primes, which is a related question. The proof is pretty simple: consider the numbers following $n!$...
The numbers from $n!+2$ through to $n!+n$ are clearly all composite, though these sized sequences of $n-1$ consecutive composite numbers in fact appear a lot earlier for $n>3$. –  Henry May 11 '11 at 1:17