Multiply perfect numbers: must the multiplier satisfy $k=o(\log\log n)$? (Erdős #1053)
Statement
Call $n$ a $k$-perfect (or $k$-fold / multiply perfect) number if $\sigma(n)=kn$, where $\sigma$ is the sum-of-divisors function; the case $k=2$ is an ordinary perfect number. As $n$ ranges over multiply perfect numbers, must the multiplier satisfy $k=o(\log\log n)$? That is, does $k/\log\log n\to 0$ along all multiply perfect $n$?
Acceptance. FULLY RESOLVES (OPEN, proof-shaped): a complete proof that $k=o(\log\log n)$ over all $k$-perfect numbers (or a disproof, e.g. an infinite family with $k\gg\log\log n$). ADVANCES: an unconditional improvement of the trivial bound $k\leq(e^{\gamma}+o(1))\log\log n$ toward $o(\log\log n)$ — state it precisely and beat it, with proof; OR exhibit a multiply perfect number with multiplier $k\geq 12$ as a program-checkable witness (its factorisation plus verification of $\sigma(n)=kn$), extending the known record beyond $k=11$; OR a proof of finiteness of $k$-perfect numbers for some fixed $k\geq 3$. Deliver the proof or bound with its derivation, or the witness with its $\sigma$ verification.
Background
A question of Erdős, reported as problem B2 in Guy's Unsolved Problems in Number Theory [Gu04]; listed as open on erdosproblems.com/1053 (fetched 2026-07-21, status 'open'). Because $\sigma(n)/n=\prod_{p^a\Vert n}(1+p^{-1}+\cdots+p^{-a})\leq\prod_{p\mid n}(1-p^{-1})^{-1}$, Grönwall's theorem gives the trivial bound $k=\sigma(n)/n\leq(e^{\gamma}+o(1))\log\log n$; the conjecture asks for the strictly stronger little-o statement. Guy further notes it has even been suggested there may be only finitely many $k$-perfect numbers with $k\geq 3$. The largest multiplier for which an example is known is $k=11$; OEIS A007539 lists the smallest $k$-perfect number for each $k$. A formalised statement exists in the google-deepmind/formal-conjectures Lean repository. No cash prize. Attacker's tool: factorisation-based search and the extensive multiperfect-number databases can test for large-$k$ examples (any $k=12$ example would be a landmark), while the $o(\log\log n)$ bound itself is analytic and proof-shaped.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1053 (T. F. Bloom) | website |
| REF-02 | OEIS A007539 — smallest k-fold perfect (multiperfect) number | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.