Erdős Problem #935

The question

erdos_935.parts.i. For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $$Q_2(n)=\prod _{\begin{array}{c}p\\ k_p\ge 2\end{array}}{p}^{k_p}.$$ Is it true that, for every $ϵ>0$ and $\ell \ge 1$, if $n$ is sufficiently large then $$Q_2(n(n+1)\cdots (n+\ell ))<n^{2+ϵ}?$$

A problem of Erdős [Er76d, p. 32], who writes that if this is true then it seems very difficult to prove. This is the first part.

erdos_935.parts.iii. For any integer $n=\prod p^{k_p}$ let $Q_2(n)$ be the powerful part of $n$, so that $$Q_2(n)=\prod _{\begin{array}{c}p\\ k_p\ge 2\end{array}}{p}^{k_p}.$$ If $\ell \ge 2$ then is $$\underset{n\to \infty }{\lim }\frac{Q_2(n(n+1)\cdots (n+\ell ))}{n^{\ell +1}}=0?$$

This is the third part. The restriction $\ell \ge 2$ is needed: for $\ell =1$ the Pell solutions of erdos_935.variants.mahler give $Q_2(n(n+1))=n(n+1)$ infinitely often.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«935» (2 statements).

theorem erdos_935.parts.i :
    ∀ ε : ℝ, 0 < ε → ∀ ℓ : ℕ, 1 ≤ ℓ → ∀ᶠ n : ℕ in atTop,
      ((Erdos367.B 2 (∏ m ∈ Finset.Icc n (n + ℓ), m) : ℕ) : ℝ) < (n : ℝ) ^ (2 + ε)

theorem erdos_935.parts.iii :
    ∀ ℓ : ℕ, 2 ≤ ℓ → Tendsto
      (fun n : ℕ ↦ ((Erdos367.B 2 (∏ m ∈ Finset.Icc n (n + ℓ), m) : ℕ) : ℝ) / (n : ℝ) ^ (ℓ + 1))
      atTop (nhds 0)

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/935. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/935
• [Er76d] Erdős, P., *Problems and results on number theoretic properties of consecutive integers and related questions*. Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) (1976), 25-44. PDF
• [Fe26] Feng, T. et al., *Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems*. arXiv:2601.22401 (2026).

The powerful part $Q_2(m)$ of $m$ is Erdos367.B 2 m, the product of $p^a$ over the primes $p$ with $p^a\parallel m$ and $a\ge 2$. The second question is essentially the same as erdosproblems.com/367.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.