The board / number theory / #678
Erdős Problem #678
PROVED LEANdatabase updated 2026-01-07
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Write \(M(n, k)\) be the least common multiple of \(\{n+1, \dotsc, n+k\}\). Let \(k\) be sufficiently large. Are there infinitely many \(m, n\) with \(m \geq n + k\) such that \[ M(n, k) > M(m, k + 1) \]? The answer is yes, as proved in a strong form by Cambie [Ca24]. [Ca24] S. Cambie, Resolution of an Erdős' problem on least common multiples. arXiv:2410.09138 (2024). This was formalized in Lean by Alexeev using Aristotle, on top of the PNT+ project. For a fixed \(k\) there are only finitely many such pairs, so "infinitely many" is read here as ranging over \(k\) as well: for every sufficiently large \(k\) at least one pair \((m, n)\) occurs. See `erdos_678.variants.infinitely_many_triples` for the reading in which the infinitude is stated directly, and `erdos_678.variants.not_infinitely_many_pairs` for why it cannot be asked of a single \(k\).
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/678.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/678
Attempts
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