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Erdős Problem 700: erdos_700.parts.i

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Added 1 Nov 2026 · not yet checked against the literature · importance 1.70

What the prompt says

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Statement

Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$ 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.

Lean statement

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

Sources(2)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/ErdosProblems/700.lean.

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