(m,k)-perfect numbers

formal-conjectures · wikipedia · ams-11

An integer n : ℤ is (m,k)-perfect if σᵐ(n) = kn where σᵐ is the mᵗʰ iterate of the
sum of divisors function.

There does not exist a $(2,5)$-perfect number

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Superperfect_number#Generalizations
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#General
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/Superperfectnumbers.lean
Local target: Corpus.WikipediaSuperperfectnumbers.twoFivePerfect
Source SHA-256: 538d6355d91122784956bb39b2893bde9f5e174083598130b3b1956cd79a33cb
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record