The board / number theory / #826
Erdős Problem #826
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).
Are there infinitely many \(n\) such that, for all \(k\geq 1\) \[ \tau(n + k) \ll k? \]
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open. The authoritative wording is on erdosproblems.com/826.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/826
Attempts
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exactly ten sharp-constant solutions \(n\leq10^9\), with a rigorous all-\(k\) checker and \((q,2q-1)\) prime-pair obstruction; #826 remains open at the near-critical \(\log\tau\) concentration lemma.
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