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

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

where p_n 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 heuristic actually supports the stronger statement

${\displaystyle \limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{(\log p_{n})^{2}}}=1,}$

and sometimes this formulation is called Cramér's conjecture. However, this stronger version is not supported by more accurate heuristic models, which nevertheless support the first version of Cramér's conjecture.

The strongest form of all, which was never claimed by Cramér but is the one used in experimental verification computations and the plot in this article, is simply

${\displaystyle p_{n+1}-p_{n}<(\log p_{n})^{2}.}$

None of the three forms has yet been proven or disproven.

Cramér gave a conditional proof of the much weaker statement that

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

on the assumption of the Riemann hypothesis.^[1] The best known unconditional bound is

${\displaystyle p_{n+1}-p_{n}=O(p_{n}^{0.525})}$

due to Baker, Harman, and Pintz.^[2]

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,^[3]

${\displaystyle \limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=\infty .}$

His result was improved by R. A. Rankin,^[4] who proved that

${\displaystyle \limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}\cdot {\frac {\left(\log \log \log p_{n}\right)^{2}}{\log \log p_{n}\log \log \log \log p_{n}}}>0.}$

Paul Erdős conjectured that the left-hand side of the above formula is infinite, and this was proven in 2014 by Kevin Ford, Ben Green, Sergei Konyagin, and Terence Tao,^[5] and independently by James Maynard.^[6] The two sets of authors eliminated one of the factors of ${\displaystyle \log \log \log p_{n}}$ later that year,^[7] showing that, infinitely often,

${\displaystyle \ {p_{n+1}-p_{n}}{>}{\frac {c\cdot \log p_{n}\cdot \log \log p_{n}\cdot \log \log \log \log p_{n}}{\log \log \log p_{n}}}}$

where ${\displaystyle c>0}$ is some constant.

Cramér's conjecture is based on a probabilistic model—essentially a heuristic—in which the probability that a number of size x is prime is 1/log x. This is known as the Cramér random model or Cramér model of the primes.^[8]

In the Cramér random model,

${\displaystyle \limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{\log ^{2}p_{n}}}=1}$

with probability one.^[1] However, as pointed out by Andrew Granville,^[9] Maier's theorem shows that the Cramér random model does not adequately describe the distribution of primes on short intervals, and a refinement of Cramér's model taking into account divisibility by small primes suggests that the limit should not be 1, but a constant ${\displaystyle c\geq 2e^{-\gamma }\approx 1.1229\ldots }$ (OEIS: A125313), where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. János Pintz has suggested that the limit sup may be infinite,^[10] and similarly Leonard Adleman and Kevin McCurley write

As a result of the work of H. Maier on gaps between consecutive primes, the exact formulation of Cramér's conjecture has been called into question [...] It is still probably true that for every constant ${\displaystyle c>2}$, there is a constant ${\displaystyle d>0}$ such that there is a prime between ${\displaystyle x}$ and ${\displaystyle x+d(\log x)^{c}}$.^[11]

Similarly, Robin Visser writes

In fact, due to the work done by Granville, it is now widely believed that Cramér's conjecture is false. Indeed, there [are] some theorems concerning short intervals between primes, such as Maier's theorem, which contradict Cramér's model.^[12]

(internal references removed).

Daniel Shanks conjectured the following asymptotic equality, stronger than Cramér's conjecture,^[13] for record gaps: ${\displaystyle G(x)\sim \log ^{2}x.}$

J.H. Cadwell^[14] has proposed the formula for the maximal gaps: ${\displaystyle G(x)\sim \log ^{2}x-\log x\log \log x,}$ which is formally identical to the Shanks conjecture but suggests a lower-order term.

Marek Wolf^[15] has proposed the formula for the maximal gaps ${\displaystyle G(x)}$ expressed in terms of the prime-counting function ${\displaystyle \pi (x)}$:

${\displaystyle G(x)\sim {\frac {x}{\pi (x)}}(2\log \pi (x)-\log x+c),}$

where ${\displaystyle c=\log(2C_{2})=0.2778769...}$ and ${\displaystyle C_{2}=0.6601618...}$ is the twin primes constant; see OEIS: A005597, A114907. This is again formally equivalent to the Shanks conjecture but suggests lower-order terms

${\displaystyle G(x)\sim \log ^{2}x-2\log x\log \log x-(1-c)\log x.}$.

Thomas Nicely has calculated many large prime gaps.^[16] He measures the quality of fit to Cramér's conjecture by measuring the ratio

${\displaystyle R={\frac {\log p_{n}}{\sqrt {p_{n+1}-p_{n}}}}.}$

He writes, "For the largest known maximal gaps, ${\displaystyle R}$ has remained near 1.13."

• Prime number theorem
• Legendre's conjecture and Andrica's conjecture, much weaker but still unproven upper bounds on prime gaps
• Firoozbakht's conjecture
• Maier's theorem on the numbers of primes in short intervals for which the model predicts an incorrect answer

• Guy, Richard K. (2004). Unsolved problems in number theory (3rd ed.). Springer-Verlag. A8. ISBN 978-0-387-20860-2. Zbl 1058.11001.
• Pintz, János (2007). "Cramér vs. Cramér. On Cramér's probabilistic model for primes". Functiones et Approximatio Commentarii Mathematici. 37 (2): 361–376. doi:10.7169/facm/1229619660. ISSN 0208-6573. MR 2363833. Zbl 1226.11096.
• Soundararajan, K. (2007). "The distribution of prime numbers". In Granville, Andrew; Rudnick, Zeév (eds.). Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11--22, 2005. NATO Science Series II: Mathematics, Physics and Chemistry. Vol. 237. Dordrecht: Springer-Verlag. pp. 59–83. ISBN 978-1-4020-5403-7. Zbl 1141.11043.

• Weisstein, Eric W. "Cramér Conjecture". MathWorld.
• Weisstein, Eric W. "Cramér-Granville Conjecture". MathWorld.