Erdős Problem #700

The question

erdos_700.parts.i. Let $f (n) = min_{1 < k \leq n /2} g cd (n, (k n))$ and let $P (n)$ be the largest prime dividing $n$ .

(a) Characterise those composite $n$ such that $f (n) = n / P (n)$ .

Erdős–Szekeres [ErSz78] note that $f (n) = n / P (n)$ when $n$ is a product of two primes (erdos_700.variants.prime_mul), with $n = 30$ a further example. The characterisation itself is open; we state it as the (unknown) set of all composite n with f n = n / P n.

erdos_700.parts.ii. Let $f (n) = min_{1 < k \leq n /2} g cd (n, (k n))$ .

(b) Are there infinitely many composite $n$ such that $f (n) > n^{1/2}$ ?

Erdős–Szekeres [ErSz78] could not prove this. (Since $f (n) \geq p (n)$ , the least prime factor of $n$ , there are infinitely many $n$ — those of the form $p^{2}$ — with $f (n) \geq n^{1/2}$ ; the question asks for the strict inequality.) Here $f (n) > n^{1/2}$ is written as (f n) ^ 2 > n.

erdos_700.parts.iii. Let $f (n) = min_{1 < k \leq n /2} g cd (n, (k n))$ .

(c) Is it true that, for every composite $n$ , $f (n) ≪_{A} n / (lo g n)^{A}$ for every $A > 0$ ?

Erdős–Szekeres [ErSz78] prove the weaker bound $f (n) \leq (1 + o (1)) n / lo g n$ (the case $A = 1$ ). Here $f (n) ≪_{A} n / (lo g n)^{A}$ is spelled out as: for every A > 0 there is a constant C (depending on A) with f(n) ≤ C · n/(log n)^A for every composite n.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«700» (3 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_700.parts.i :
    {n : ℕ | ¬ n.Prime ∧ 1 < n ∧ f n = n / P n} = answer(sorry)

theorem erdos_700.parts.ii :
    answer(sorry) ↔ {n : ℕ | ¬ n.Prime ∧ 1 < n ∧ (f n) ^ 2 > n}.Infinite

theorem erdos_700.parts.iii :
    answer(sorry) ↔ (∀ A : ℝ, 0 < A → ∃ C : ℝ, 0 < C ∧ ∀ n : ℕ, ¬ n.Prime → 1 < n →
      (f n : ℝ) ≤ C * (n : ℝ) / (Real.log n) ^ A)

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/700. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/700

A problem of Erdős and Szekeres [ErSz78].

References:

[ErSz78] Erdős, P. and Szekeres, G., _Some number theoretic problems on binomial coefficients_, Austral. Math. Soc. Gaz. (1978), 97-99.
OEIS A091963
Guy, R. K., _Unsolved Problems in Number Theory_, B31, B33.

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.

Current state

The problem