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Erdős #1181 · number theory

Let q(n,k)q(n,k) denote the least prime which does not divide ∏1≤i≤k(n+i)\prod_{1\leq i\leq k}(n+i). Is it true that there exists some c>0c>0 such that, for all large nn,

q(n,log⁡n)<(1−c)(log⁡n)2?q(n,\log n)<(1-c)(\log n)^2?
MSC 11 · Number theory

Inferred from the community field; finer classification has not been reviewed.

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Problem statement

Let q(n,k)q(n,k) denote the least prime which does not divide ∏1≤i≤k(n+i)\prod_{1\leq i\leq k}(n+i). Is it true that there exists some c>0c>0 such that, for all large nn,

q(n,log⁡n)<(1−c)(log⁡n)2?q(n,\log n)<(1-c)(\log n)^2?

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Background

The upper bound q(n,log⁡n)≤(1+o(1))(log⁡n)2q(n,\log n)\leq (1+o(1))(\log n)^2 follows from observing that the primorial of q(n,k)q(n,k) is at most ∏1≤i≤k(n+i)\prod_{1\leq i\leq k}(n+i).

Probabilistic heuristics described by Tao in the comments of [457] suggest that

q(n,log⁡n)≪log⁡log⁡nlog⁡log⁡log⁡nlog⁡nq(n,\log n)\ll \frac{\log\log n}{\log\log\log n}{\log n}

for all nn.

[457] concerns lower bounds for q(n,log⁡n)q(n,\log n) (and in particular the constructions described there proved that the above upper bound would be best possible).

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1181 link

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