Erdős Problem #700

The question

erdos_700.parts.i. Let $f(n)={\min }_{1<k\le n/2}\gcd (n,(\genfrac{}{}{0ex}{}{n}{k}))$ 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.

This follows the largest-prime reading on erdosproblems.com. In [ErSz78, p. 98], the first prose line below (7) reads: "where $P(n)$ is the greatest prime power which divides $n$." The paper's reading is stated with $Q(n)$ in erdos_700.variants.prime_power_divisor.

erdos_700.parts.ii. Let $f(n)={\min }_{1<k\le n/2}\gcd (n,(\genfrac{}{}{0ex}{}{n}{k}))$.

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

Erdős–Szekeres [ErSz78] could not prove this. (Since $f(n)\ge p(n)$, the least prime factor of $n$, there are infinitely many $n$ — those of the form $p^2$ — with $f(n)\ge n^{1/2}$; 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 $f(n)={\min }_{1<k\le n/2}\gcd (n,(\genfrac{}{}{0ex}{}{n}{k}))$.

(c) Is it true that, for every composite $n$, $f(n){\ll }_{A}n/(\log n{)}^A$ for every $A>0$?

Erdős–Szekeres [ErSz78] prove the weaker bound $f(n)\le (1+o(1))n/\log n$ (the case $A=1$). Here $f(n){\ll }_{A}n/(\log n{)}^A$ 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).

theorem erdos_700.parts.i :
    {n : ℕ | ¬ n.Prime ∧ 1 < n ∧ f n = n / P n} = erdos_700.parts.i.answer

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

theorem erdos_700.parts.iii :
    (∀ 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.

Variants

• erdos_700.variants.prime_power_divisor — Characterise those composite n such that f(n) = n/Q(n), where Q(n) is the largest prime-power divisor of n.

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 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.