Hardy-Littlewood Conjecture B (Second Conjecture)

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

Let $\pi(t)=\#\{p\leq t:p\text{ is prime}\}$. For integers $x,y\geq2$, the inequality is equivalent to $\#\{p:x<p\leq x+y\}\leq\pi(y)$. The lower endpoint is excluded and the upper endpoint included. This is an assertion for every indicated pair, not an averaged interval estimate. [S1] [S2]

Statement recorded in the supplied source.

For all integers $x, y \geq 2$, we have $\pi(x+y) \leq \pi(x) + \pi(y)$, where $\pi(n)$ denotes the prime counting function (the number of primes less than or equal to $n$). [S1]

Short English statement

Can an interval of a given integer length ever contain more primes than the initial interval of that same length?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1518 (HL-B), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 Douglas Hensley and Ian Richards, Primes in intervals, Acta Arithmetica 25 (1974), 375-391. https://doi.org/10.4064/aa-25-4-375-391. Bibliographic information from the supplied record.

• S3 Bittu Chahal, Ertan Elma, Nic Fellini, Akshaa Vatwani, and Do Nhat Tan Vo, On the Second Hardy-Littlewood Conjecture, arXiv:2503.02766 (2025). https://arxiv.org/abs/2503.02766. Bibliographic information from the supplied record.

• S4 Matt Visser, The second Hardy-Littlewood conjecture is true, arXiv:2101.03283 (2021). https://arxiv.org/abs/2101.03283. Bibliographic information from the supplied record.