# Cramér's conjecture

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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 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 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. Neither form has yet been proven or disproven.

## Conditional proven results on prime gaps

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

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

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]

## Heuristic justification

Cramér's conjecture is based on a probabilistic model—essentially a heuristic—in which the probability 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.[6]

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,[7] 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 ${\displaystyle c\geq 2e^{-\gamma }\approx 1.1229\ldots }$ (), where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. János Pintz has suggested that the limit sup may be infinite,[8] 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 ${\displaystyle x+d(\log x)^{c}}$. [9]

## Related conjectures and heuristics

Prime gap function

Daniel Shanks conjectured the following asymptotic equality, stronger than Cramér's conjecture,[10] for record gaps: ${\displaystyle G(x)\sim \log ^{2}x.}$

In the paper [11] J.H. Cadwell has proposed the formula for the maximal gaps: ${\displaystyle G(x)\sim \log x(\log x-\log \log x),}$ which is formally identical to the Shanks conjecture but suggests a lower-order term.

In the paper [12] Marek Wolf has proposed the formula for the maximal gaps ${\displaystyle G(x)}$ expressed in terms of the prime counting function ${\displaystyle \pi (x)}$:

${\displaystyle G(x)\sim {\frac {x}{\pi (x)}}(2\log \pi (x)-\log x+c_{0}),}$

where ${\displaystyle c_{0}=\log(C_{2})=0.2778769...}$ and ${\displaystyle C_{2}=1.3203236...}$ is the twin primes constant; see . Using Gauss's approximation ${\displaystyle \pi (x)\sim x/\log(x)}$ this gives

${\displaystyle G(x)\sim \log(x)(\log x-2\log \log x),}$

which for large ${\displaystyle x}$ is also asymptotically equivalent to the Cramér and Shanks conjectures: ${\displaystyle G(x)\sim \log ^{2}(x)}$.

Thomas Nicely has calculated many large prime gaps.[13] 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.” However, ${\displaystyle 1/R^{2}}$ is still less than 1.

## References

1. ^ a b c Cramér, Harald (1936), "On the order of magnitude of the difference between consecutive prime numbers" (PDF), Acta Arithmetica, 2: 23–46
2. ^ R. C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes. II. Proc. London Math. Soc. (3), 83 (2001), no. 3, 532-562
3. ^ Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM 57.0186.02, Zbl 0003.24601.
4. ^ R. A. Rankin, The difference between consecutive prime numbers, J. London Math. Soc. 13 (1938), 242-247
5. ^ K. Ford, B. Green, S. Konyagin, and T. Tao, Large gaps between consecutive prime numbers. Ann. of Math. (2) 183 (2016), no. 3, 935–974
6. ^ Terry Tao, 254A, Supplement 4: Probabilistic models and heuristics for the primes (optional), section on The Cramér random model, January 2015.
7. ^ Granville, A. (1995), "Harald Cramér and the distribution of prime numbers" (PDF), Scandinavian Actuarial Journal, 1: 12–28, doi:10.1080/03461238.1995.10413946.
8. ^ János Pintz, Very large gaps between consecutive primes, Journal of Number Theory 63:2 (April 1997), pp. 286–301.
9. ^ Leonard Adleman and Kevin McCurley, Open Problems in Number Theoretic Complexity, II. Algorithmic number theory (Ithaca, NY, 1994), 291–322, Lecture Notes in Comput. Sci., 877, Springer, Berlin, 1994.
10. ^ Shanks, Daniel (1964), "On Maximal Gaps between Successive Primes", Mathematics of Computation, American Mathematical Society, 18 (88): 646–651, doi:10.2307/2002951, JSTOR 2002951, Zbl 0128.04203.
11. ^ Cadwell, J. H. (1971), "Large Intervals Between Consecutive Primes", Mathematics of Computation, 25 (116): 909–913, doi:10.2307/2004355, JSTOR 2004355
12. ^ Wolf, Marek (2014), "Nearest-neighbor-spacing distribution of prime numbers and quantum chaos", Phys. Rev. E, 89: 022922, arXiv:1212.3841, Bibcode:2014PhRvE..89b2922W, doi:10.1103/physreve.89.022922
13. ^ Nicely, Thomas R. (1999), "New maximal prime gaps and first occurrences", Mathematics of Computation, 68 (227): 1311–1315, doi:10.1090/S0025-5718-99-01065-0, MR 1627813.