Erdős Problem #887

The question

erdos_887.parts.i. Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}},n^{\frac{1}{2}}+C{n}^{\frac{1}{4}})$.

erdos_887.parts.ii. Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{\frac{1}{2}},n^{\frac{1}{2}}+C{n}^{\frac{1}{4}})$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«887» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_887.parts.i : ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
    #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ answer(sorry)

theorem erdos_887.parts.ii : ∃ K, ∀ C > (0 : ℝ), ∀ᶠ n in atTop,
    #{ d ∈ Ioo ⌊√n⌋₊ ⌈√n + C * n^((1 : ℝ) / 4)⌉₊ | d ∣ n } ≤ K

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/887. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• erdos_887.variants.rosenfeld_4 — Erdős and Rosenfeld, ask whether 4 is the best possible K for the infinitude of n with (at least) K divisors in (n^1/2, n^1/2 + n^1/4).

References

• erdosproblems.com/887
• [ErRo97] Erdős, Paul and Rosenfeld, Moshe, The factor-difference set of integers. Acta Arith. (1997), 353--359.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.