The board / number theory / #440
Erdős Problem #440
SOLVEDdatabase updated 2025-12-27
Resolved in some way other than a straight proof or disproof — usual for open-ended questions, or for multi-part problems whose parts went different ways.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
For a strictly increasing sequence `a` of positive integers, `count a x` is the number of indices `i` with `lcm(aᵢ, aᵢ₊₁) ≤ x`.
Let \(A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}\) be infinite and let \(A(x)\) count the number of indices for which \(\mathrm{lcm}(a_i,a_{i+1})\leq x\). Is it true that \(A(x) \ll x^{1/2}\)? The answer is yes: Tao has given a simple proof, and Erdős and Szemerédi [ErSz80] proved the sharp bound \(A(x)\leq (c+o(1))x^{1/2}\) (see `erdos_440.variants.erdos_szemeredi`).
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/440.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/440
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