Improved Bounds for Distinct Multiples in Intervals

For the Erdős–Pomerance functions $F(n)$ and $h_{\mathbb{P}}(n)$ counting how many consecutive integers are needed to contain a distinct multiple of each integer, respectively prime, up to $n$, the paper proves $F(n) \ge h_{\mathbb{P}}(n) \ge n\exp\left(\left(\frac{\log 2}{2} - o(1)\right)\frac{\log n}{\log\log n}\right)$, disproving Kominers' conjecture that $F(n) \ll n\log n$. The paper also significantly improves known upper bounds (which were on the order of $n^{3/2}$) to $F(n) \le n^{4/3 + o(1)}$ and $h_{\mathbb{P}}(n) \le n^{4/3 - o(1)}$.

Result

Disproved

Status

Resolved

AI contribution

AI co-developed

Method

Argument

Field

Number Theory, Erdős–Pomerance Functions

Posed by

Scott Duke Kominers; functions introduced by Erdős and Pomerance

Year posed

2026

Years open

0y

Solved

2026-07-29

Model

ChatGPT 5.x

Vendor

OpenAI

Collaborators

Kaizhe Chen, Samuel Korsky

Verification

Unreviewed

Publication

Preprint

Significance

13 / 100*

Disclosed cost

—

Wikipedia

No dedicated article