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Updated Oct 7

Problem 680

Statement. Is it true that, for all sufficiently large $n$ , there exists some $k$ such that

p(n+k)>k^2+1,

where $p(m)$ denotes the least prime factor of $m$ ?

Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$ , for all $\epsilon>0$ , where $C_\epsilon>0$ is some constant?

Status. Open.

Source. erdosproblems.com/680, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #680, https://www.erdosproblems.com/680.

References.

[Gr95] Granville, Andrew, Harald Cramér and the distribution of prime numbers. Scand. Actuar. J. (1995), 12-28.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (5)

Primes primes/gafni_2025_rough_numbers_between_consecutive_primes primes/granville_1995_harald_cramer_distribution_prime_numbers Equation (14), p. 21: Cramér's conjecture that the largest prime gap up to x is asymptotic to log^2 x Heuristic (pp. 23-24): a sieve-corrected Cramér model suggests max prime gap up to x is at least about 2e^{-gamma} log^2 x

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