The board / number theory / #897
Erdős Problem #897
DISPROVED LEANdatabase updated 2025-12-27
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
Let \(f(n)\) be an additive function (so that \(f(ab)=f(a)+f(b)\) if \((a,b)=1\) such that \(\limsup_{p,k} f(p^k) / \log(p^k) = ∞\). Is it true that \(\limsup_n (f(n+1)−f(n))/ \log n = ∞\)? The answer is no; this follows from a construction of Wirsing [Wi81], rediscovered by Archivara [Ar25] and formalised in Lean by Aristotle [ArWu25].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 4 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/897.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/897
Attempts
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