The board / number theory / #461
Erdős Problem #461
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The \(t\)-smooth component \(s_t(n)\) of \(n\): the product of all primes \(p < t\) dividing \(n\), counted with multiplicity.
The number \(f(n,t)\) of distinct values of \(s_t(m)\) for \(m \in [n+1, n+t]\).
Let \(s_t(n)\) be the \(t\)-smooth component of \(n\) - that is, the product of all primes \(p\) (with multiplicity) dividing \(n\) such that \(p<t\). Let \(f(n,t)\) count the number of distinct possible values for \(s_t(m)\) for \(m\in [n+1,n+t]\). Is it true that \[f(n,t)\gg t\] (uniformly, for all \(t\) and \(n\))? Erdős and Graham report they can show \[f(n,t) \gg \frac{t}{\log t}.\]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/461.Prizenone
Topicsnumber theory, primes
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/461
Attempts
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Explicit CRT arithmetic gives f(n,500)=218 (hence any uniform constant is at most 0.436), and exhaustive periodic enumeration gives the exact minima for every t<=13; Erdős #461 remains open.
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