Erdős Problem #726 — open problem in number theory

Erdős Problem #726

As n→ ∞ ranges over integers Σ_p≤ n1_n∈ (p/2,p)pmodp1/p∼ loglog n/2? A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75]. By n∈ (p/2,p)pmodp we mean n≡ rpmodp for some integer r with p/2<r<p. The remainder n % p is computed in ℕ before casting to ℝ.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-726,
  title        = {Erdős Problem #726},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-726}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

As $n \to \infty$ ranges over integers $\sum_{p \leq n} 1_{n \in (p /2, p) (mod p)} \frac{1}{p} \sim \frac{l o g l o g n}{2}$ ?

A conjecture of Erdős, Graham, Ruzsa, and Straus [EGRS75].

By $n \in (p /2, p) (mod p)$ we mean $n \equiv r (mod p)$ for some integer $r$ with $p /2 < r < p$ . The remainder n % p is computed in ℕ before casting to ℝ.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«726». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_726 :
    answer(sorry) ↔
      (fun n : ℕ ↦ ∑ p ∈ (range (n + 1)).filter
          (fun p : ℕ ↦ p.Prime ∧ (p : ℝ) / 2 < ((n % p : ℕ) : ℝ)),
        (1 : ℝ) / (p : ℝ))
      ~[atTop] (fun n : ℕ ↦ Real.log (Real.log (n : ℝ)) / 2)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
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/726. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/726
[EGRS75] Erdős, P., and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $(\sp 2 n \sb n)$ . Math. Comp. (1975), 83-92.

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.