Erdős Problem #218

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.

Natural density in this entry is density of prime indices $n$, not density of the prime values among all integers. Both weak inequalities count equal neighbouring gaps; if both densities are one half, equality must have index density zero even if it occurs infinitely often.

$p_j$ denotes the $j$th positive prime in increasing order, starting with $p_1=2$. The prime-counting function is $\pi(x)=\#\{p\leq x:p\text{ is prime}\}$. Consecutive primes are consecutive in this full sequence. [S1] [S2]

Statement recorded in the supplied source.

Let $d_n=p_{n+1}-p_n$. The set of $n$ such that $d_{n+1}\geq d_n$ has density $1/2$, and similarly for $d_{n+1}\leq d_n$. Furthermore, there are infinitely many $n$ such that $d_{n+1}=d_n$. [S1]

Short English statement

Do increases and decreases of consecutive prime gaps each have density one half, and do equal neighbouring gaps occur infinitely often?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 1997 (EP-218), 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 T. F. Bloom, Erdős Problems, Problem 218, https://www.erdosproblems.com/218. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Erdős, P., On some of my problems in number theory I would most like to see solved. Number theory (Ootacamund, 1984) (1985), 74-84. Bibliographic information from the supplied record.