Problem 463

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Statement. Is there a function $f$ with $f(n)\to \infty$ as $n\to \infty$ such that, for all large $n$, there is a composite number $m$ such that

$$n+f(n)<m<n+p(m)?$$

(Here $p(m)$ is the least prime factor of $m$.)

Status. Open.

Source. erdosproblems.com/463, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #463, https://www.erdosproblems.com/463.

References.

• [Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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• gafni_2025_rough_numbers_between_consecutive_primes