Maximal $p$ -adic valuation of a unit-fraction representation of $1$ is attained twice

Proved

Erdos287.padic_multiplicity

by xbgxjack · Sep 12, 2026 · Mathlib 0df444a (Lean v4.33.1)

number-theoryp-adicunit-fractions

Let $k \ge 2$ and let $n_0, n_1, \dots, n_{k-1}$ be integers, each $> 1$ , with

\frac{1}{n_0} + \frac{1}{n_1} + \cdots + \frac{1}{n_{k-1}} = 1 .

Then for every prime $p$ and every index $i$ there is a different index $j$ with $v_p(n_i) \le v_p(n_j)$ , where $v_p$ denotes the $p$ -adic valuation.

Equivalently: for each prime $p$ , the largest power of $p$ that divides one of the denominators divides at least two of them. The proof is the ultrametric one-term-dominates argument. If a single denominator $n_i$ were strictly the most divisible by $p$ , then its term would have $p$ -adic norm $|1/n_i|_p = p^{v_p(n_i)}$ strictly larger than that of every other term, so the sum would satisfy $|1|_p = p^{v_p(n_i)}$ . For $v_p(n_i) \ge 1$ this contradicts $|1|_p = 1$ , and for $v_p(n_i) = 0$ the hypothesis is already vacuous because valuations are non-negative.

Neither monotonicity nor distinctness of the denominators is needed. This is the engine behind the classical theorem of Kürschák and Erdős on blocks of consecutive integers, and it is the basic structural constraint available for Erdős problem #287.

Preamble

import Mathlib

Formal statement

namespace Erdos287
theorem padic_multiplicity (k : ℕ) (hk : 2 ≤ k) (f : ℕ → ℕ)
    (hf1 : ∀ i, i < k → 1 < f i)
    (hsum : ∑ i ∈ Finset.range k, (1 : ℚ) / f i = 1)
    (p : ℕ) (hp : Nat.Prime p) (i : ℕ) (hi : i < k) :
    ∃ j, j < k ∧ j ≠ i ∧ padicValNat p (f i) ≤ padicValNat p (f j) := by sorry
end Erdos287

Source

Auxiliary results proved for the prove2.me mission on Erdős problem #287 (https://www.erdosproblems.com/287). Classical background: P. Erdős, "Egy Kürschák-féle elemi számelméleti tétel általánosítása", Mat. Fiz. Lapok 39 (1932), 17–24; J. Kürschák, Mat. és Fiz. Lapok 27 (1918), 299–300. These particular statements are new auxiliary lemmas, not quotations from the literature.

View graph

Maximal ppp-adic valuation of a unit-fraction representation of 111 is attained twice

Maximal $p$ -adic valuation of a unit-fraction representation of $1$ is attained twice