The board / number theory / #828
Erdős Problem #828
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).
Is it true that, for any \(a \in \mathbb{Z}\), there are infinitely many \(n\) such that \[\phi(n) | n + a\]?
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/828.Prizenone
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/828
Attempts
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classified all possible bounded-\(\omega\) infinite families, reproduced the exact \(a=1\) list through seven prime factors, and isolated the unresolved unbounded-prime-support product-divisibility le
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