Open problem, Arithmetic and number theory, Probability and statistics, posed 1936

Cramér's conjecture on prime gaps

Open

If $p_n$ denotes the $n$ -th prime number and $g_n = p_{n+1} - p_n$ , then $g_n = O\!\left((\log p_n)^2\right)$ as $n \to \infty$ ; in Cramér's stronger original form, $\limsup_{n \to \infty} \frac{p_{n+1} - p_n}{(\log p_n)^2} = 1$ .

Research frontier as of 2026

As of 2026, an enormous gulf separates the known upper and lower bounds for maximal prime gaps from Cramér's $(\log p_n)^2$ prediction. Unconditionally, the best upper bound on all prime gaps is $g_n \ll p_n^{0.525}$ (Baker–Harman–Pintz, 2001), while the 2024 Guth–Maynard zero-density estimate establishes the asymptotic prime number theorem in short intervals $[x, x+x^{17/30+\varepsilon}]$ for almost all scales. Even under the Riemann hypothesis, the best known bound is $g_n = O(\sqrt{p_n}\log p_n)$ , which is still a power of $p_n$ rather than polylogarithmic. In the opposite direction, the best unconditional lower bound for infinitely many gaps is $g_n \gg \frac{\log p_n \log\log p_n \log\log\log\log p_n}{\log\log\log p_n}$ (Ford–Green–Konyagin–Maynard–Tao, 2014), barely beyond $\log p_n$ .

Best known results

Unconditional upper bound: $p_{n+1} - p_n \ll p_n^{0.525}$ for all sufficiently large $n$ (Baker, Harman, and Pintz, 2001).
Conditional upper bound: $p_{n+1} - p_n = O(\sqrt{p_n}\log p_n)$ under the Riemann hypothesis (Cramér, 1920).
Unconditional lower bound: $p_{n+1} - p_n \gg \frac{\log p_n \log\log p_n \log\log\log\log p_n}{\log\log\log p_n}$ infinitely often (Ford, Green, Konyagin, Maynard, and Tao, 2014).

Tools and where they stop

Tool	Achieved	Where it stops
Zero-density estimates for $\zeta(s)$ and Harman's sieve	Control primes in short intervals $[x, x+x^{\theta}]$ , pushing the unconditional exponent down to $\theta = 0.525$ .	Even the full Riemann hypothesis only reaches $\theta = 1/2 + \varepsilon$ , because explicit formulas sum over zeros on the critical line $\mathrm{Re}(s)=1/2$ .
Rankin's sieve-out method with Maynard–Tao prime-tuple weights	Constructs long intervals of consecutive composite numbers by choosing residue classes modulo primes up to $z$ , breaking Erdős's longstanding Rankin-bound barrier.	By the prime number theorem, the product of primes below $z \approx \log x$ cannot exceed $x$ , limiting Jacobsthal-function constructions to $(\log x)(\log\log x)^{O(1)}$ and falling far short of $(\log x)^2$ .

Open questions

Can one prove $p_{n+1} - p_n \ll p_n^{\theta}$ for any exponent $\theta < 1/2$ , even conditionally on the Riemann hypothesis and Montgomery's pair correlation conjecture?
Is the true value of $\limsup_{n\to\infty} (p_{n+1}-p_n)/(\log p_n)^2$ equal to $1$ (Cramér), $2e^{-\gamma} \approx 1.1229$ (Granville), or larger?

References

Harald Cramér (1936). On the order of magnitude of the difference between consecutive prime numbers · DOI:10.4064/aa-2-1-23-46
Andrew Granville (1995). Harald Cramér and the distribution of prime numbers · DOI:10.1080/03461238.1995.10413946
Roger C. Baker, Glyn Harman, János Pintz (2001). The difference between consecutive primes, II · DOI:10.1112/plms/83.3.532
Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, Terence Tao (2018). Long gaps between primes · DOI:10.1090/jams/890 · arXiv:1412.5029