Problem 1184

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Statement. Let $f(n,k)$ count the number of $1\leq i\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\alpha>1$ is such that $n=k^{\alpha+o(1)}$, then

$$f(n,k)=(1-\rho(\alpha)+o(1))k,$$

where $\rho$ is the Dickman function?

Status. Open.

Source. erdosproblems.com/1184, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1184, https://www.erdosproblems.com/1184.

References.

• [Er76e] Erdős, P., Problems and results on consecutive integers. Publ. Math. Debrecen (1976), 271-282.
• [RST75b] Ramachandra, K. and Shorey, T. N. and Tijdeman, R., On Grimm's problem relating to factorisation of a block of consecutive integers. II. J. Reine Angew. Math. (1976), 192-201.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

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.

• erdos_1976_problems_results_consecutive_integers
• ramachandra_1976_grimm_s_problem_relating_factorisation_block
• ramachandra_1976_grimm_s_problem_relating_factorisation_block / theorem_2