The board / number theory / #679
Erdős Problem #679
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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Let \(\epsilon>0\) and \(\omega(n)\) count the number of distinct prime factors of \(n\). Are there infinitely many values of \(n\) such that \[\omega(n-k) < (1+\epsilon)\frac{\log k}{\log\log k}\] for all \(k<n\) which are sufficiently large depending on \(\epsilon\) only?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 1 further variant / known partial result in the Lean file. The authoritative wording is on erdosproblems.com/679.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/679
Attempts
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For epsilon=1/10 and cutoff K=16, exactly 94 admissible n occur through Q_101+15 and all are at most 225, while the infinite problem reduces to an every-interval multiscale gap lemma not supplied by k
PARTIAL