# Cramér's conjecture

In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936,[1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that

$p_{n+1}-p_n=O((\log p_n)^2),\$

where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement

$\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1,$

and this formulation is often called Cramér's conjecture in the literature.

Neither form of Cramér's conjecture has yet been proven or disproven.

## Conditional proven results on prime gaps

Cramér also gave a conditional proof of the much weaker statement that

$p_{n+1}-p_n = O(\sqrt{p_n}\,\log p_n)$

on the assumption of the Riemann hypothesis.[1]

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,[2]

$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.$

## Heuristic justification

Cramér's conjecture is based on a probabilistic model (essentially a heuristic) of the primes, in which one assumes that the probability of a natural number of size x being prime is 1/log x. This is known as the Cramér model of the primes. Cramér proved that in this model, the above conjecture holds true with probability one.[1]

## Heuristics of Shanks and Cramér–Granville conjectures

Prime gap function

Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture.[3]

In the Cramér random model,

$\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = c,$ with $c = 1.$

But this constant, $c$, may not apply to all the primes, by Maier's theorem. As pointed out by Andrew Granville,[4] a refinement of Cramér's model taking into account divisibility by small primes suggests that $c \ge 2e^{-\gamma}\approx1.1229\ldots$, where $\gamma$ is the Euler–Mascheroni constant.

Thomas Nicely has calculated many large prime gaps.[5] He measures the quality of fit to Cramér's conjecture by measuring the ratio

$R = \frac{\log(p_n)}{\sqrt{p_{n+1}-p_n}}.$

He writes, “For the largest known maximal gaps, $R$ has remained near 1.13.” However, $1/R^2$ is still less than 1, and it does not provide support to Granville's refinement that c should be greater than 1.