The board / number theory / #1052
Erdős Problem #1052
unitary perfect numbers
OPENdatabase updated 2025-09-28
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).
A proper unitary divisor of \(n\) is a divisor \(d\) of \(n\) such that \(d\) is coprime to \(n/d\), and \(d < n\).
A number \(n > 0\) is a unitary perfect number if it is the sum of its proper unitary divisors.
Are there only finitely many unitary perfect numbers?
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/1052.Prize$10
Topicsnumber theory
Statement in LeanFormal Conjectures
Solution in Leanno
OEISA002827
Original pageerdosproblems.com/1052
Attempts
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Exact exhaustive computation finds only the five known unitary perfect numbers when every prime-power component is at most \(10^7\); any sixth has a component \(>10^7\), while full finiteness still re
PARTIAL