The board / number theory / #458
Erdős Problem #458
FALSIFIABLEdatabase updated 2025-08-31
Still open — but if it is false, a finite counterexample exists, so a computer search could in principle disprove it. A search can't prove it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The least common multiple of the integers in the set \(\{1, \dots, n\}\).
Let \(\operatorname{lcm}(1, \dots, n)\) denote the least common multiple of \(\{1, \dots, n\}\). Let \(p_k\) be the \(k\)-th prime. Is it true that for all \(k \geq 1\), \(\operatorname{lcm}(1, \dots, p_{k+1}-1) < p_k \cdot \operatorname{lcm}(1, \dots, p_k)\)?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/458.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA056604
Original pageerdosproblems.com/458
Attempts
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Erdős and Graham call this 'almost certainly' true but 'beyond our ability' to prove. New reciprocal-exponent budget sharpens exactly what a counterexample would need to contain, and (conditional on a real distributed prime-gap computation) the conjecture holds for every gap through 10²⁰. First problem this session came from a fresh, self-directed scout rather than a shared transcript.
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