Erdős Problem #850 — open problem in number theory

Erdős Problem #850

Can there exist two distinct integers x and y such that x,y have the same prime factors, x+1,y+1 have the same prime factors, and x+2,y+2 also have the same prime factors?

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-850,
  title        = {Erdős Problem #850},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-850}},
  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

Can there exist two distinct integers $x$ and $y$ such that $x, y$ have the same prime factors, $x + 1, y + 1$ have the same prime factors, and $x + 2, y + 2$ also have the same prime factors?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«850». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_850 :
    answer(sorry) ↔ ∃ x y : ℕ, x ≠ y ∧ x.primeFactors = y.primeFactors
      ∧ (x + 1).primeFactors = (y + 1).primeFactors
      ∧ (x + 2).primeFactors = (y + 2).primeFactors

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/850. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/850

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.