OpenFrontiers
Join
← Back to the index
OP-9395OpenNumber theory

Erdős #1184 · number theory / primes

Let f(n,k)f(n,k) count the number of 1≤i≤k1\leq i\leq k such that P(n+i)>kP(n+i)>k (where P(m)P(m) is the largest prime divisor of mm). Is it true that, if α>1\alpha>1 is such that n=kα+o(1)n=k^{\alpha+o(1)}, then

f(n,k)=(1−ρ(α)+o…f(n,k)=(1-\rho(\alpha)+o…
MSC 11 · Number theory

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

Source reports OPENAs of 2026-09-08Source & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

Let f(n,k)f(n,k) count the number of 1≤i≤k1\leq i\leq k such that P(n+i)>kP(n+i)>k (where P(m)P(m) is the largest prime divisor of mm). Is it true that, if α>1\alpha>1 is such that n=kα+o(1)n=k^{\alpha+o(1)}, then

f(n,k)=(1−ρ(α)+o(1))k,f(n,k)=(1-\rho(\alpha)+o(1))k,

where ρ\rho is the Dickman function?

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

Erdős [Er76e] proved that, for every α>1\alpha>1, if kk is sufficiently large and n>kα−kn>k^\alpha-k then

f(n,k)>(1−1α+cα)kf(n,k) > \left(1-\frac{1}{\alpha}+c_\alpha\right)k

for some constant cα>0c_\alpha>0, and if 1<α<21<\alpha<2 and n≤kα−kn\leq k^\alpha-k then

f(n,k)<(α−1+o(1))k.f(n,k) < (\alpha-1+o(1))k.

He knew of no non-trivial bounds when α≥2\alpha \geq 2.

Ramachandra, Shorey, and Tijdeman [RST75b] proved that if n>exp⁡(c(log⁡k)2)n>\exp(c(\log k)^2) for a constant c>0c>0 then f(n,k)≥k−π(k)f(n,k)\geq k-\pi(k).

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 #1184 link

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.