Finitely many perfect powers (and powerful numbers) among sums of distinct factorials? (Erdős #1108)
Statement
Let $$A = \left\{ \sum_{n\in S} n! : S\subset \mathbb{N}\text{ finite} \right\}$$ be the set of all sums of distinct factorials. For a fixed integer $k\ge 2$, does $A$ contain only finitely many perfect $k$th powers? More generally, does $A$ contain only finitely many powerful numbers, where an integer $m\ge 1$ is powerful if $p\mid m\Rightarrow p^2\mid m$ for every prime $p$?
Acceptance. FULLY RESOLVES: a complete proof (machine-checkable in Lean/Coq preferred, otherwise a full written proof) that for each fixed $k\ge 2$ the set $A$ contains only finitely many perfect $k$th powers, and/or that $A$ contains only finitely many powerful numbers — OR a disproof exhibiting an explicit infinite family of perfect $k$th powers (respectively powerful numbers) that are sums of distinct factorials. ADVANCES (each independently checkable): (a) a reproducible exhaustive search certifying that no sum of distinct factorials up to a new record bound $N$ is a nontrivial $k$th power or powerful number (beyond the finitely many small known ones), delivered with the enumeration code, an exhaustiveness certificate, and any extension of the associated OEIS sequences; (b) a proof for a specific exponent $k$, or for a restricted class of subsets $S$ (for instance bounded $|S|$), that strengthens the Brindza–Erdős bound stated in the background; (c) a new explicit powerful or perfect-power sum of distinct factorials, if one exists, with a full factorisation certificate. Deliver the proof file, the search code plus attained bound and updated sequence data, or the witness with its factorisation.
Background
Asked by Erdős at Oberwolfach in 1988 [Ob1]; tagged 'number theory | factorials'. Context on the source page: it is open even whether there are infinitely many squares of the form $1+n!$, a special case tightly connected to the Brocard–Ramanujan problem $n!=x^2-1$, which appears on this venue as Erdős #398 (erdosproblems.com/398). The question was motivated by a problem of Mahler, discussed with Erdős days before Mahler's death in 1988: for $k\ge 5$, does $A_k=\{\sum_{n\in S}k^n:S\text{ finite}\}$ contain only finitely many squares? Mahler showed there are infinitely many squares in $A_k$ for $k\le 4$, and found only a single square for $k\ge 5$, namely $1+7+7^2+7^3=400$. The strongest partial result toward #1108 is that of Brindza and Erdős [BrEr91], who proved that for every fixed $r$, if $n_1!+\cdots+n_r!$ is powerful then the largest index is bounded, $n_1\ll_r 1$. Related OEIS sequences listed on the source page are A051761, A115645, and A025494 (factorial-sum sequences; exact definitions to be confirmed against OEIS). A Lean formalisation exists in DeepMind's formal-conjectures repository. Listed as open on erdosproblems.com/1108 (fetched 2026-07-13, status 'open'); no Erdős prize is attached. Attacker's tool: enumerate subset-sums of factorials up to a large bound, test each candidate for perfect-power and powerful status via factorisation and exponent checks, extend the verified range and any associated OEIS sequences, and combine with the Brindza–Erdős bound (which makes the search over a bounded number $r$ of summands effectively finite) to push toward a proof.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #1108 (T. F. Bloom) | website |
| REF-02 | OEIS A051761 — factorial-sum sequence related to Erdős #1108 | website |
| REF-03 | OEIS A115645 — factorial-sum sequence related to Erdős #1108 | website |
| REF-04 | OEIS A025494 — factorial-sum sequence related to Erdős #1108 | website |
| REF-05 | Lean formalisation of Erdős #1108 (DeepMind formal-conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.