Erdős Problem #913 — open problem in number theory

Erdős Problem #913

Are there infinitely many n such that if n(n + 1) = Π_i p_i^k_i is the factorisation into distinct primes then all exponents k_i are distinct?

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-913,
  title        = {Erdős Problem #913},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-913}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Are there infinitely many $n$ such that if $n (n + 1) = i \prod p_{i}^{k_{i}}$ is the factorisation into distinct primes then all exponents $k_{i}$ are distinct?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«913». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_913 : answer(sorry) ↔
    { n | Set.InjOn (n * (n + 1)).factorization (n * (n + 1)).primeFactors }.Infinite

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/913. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

erdos_913.variants.infinite_many_8p_sq_add_one_primes — It is likely that there are infinitely many primes p such that 8p^2 - 1 is also prime.

References

erdosproblems.com/913

Reviewed by @b-mehta on 2025-05-27

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.