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Erdős problem 1108 - part i For each k ≥ 2 k \geq 2 k ≥ 2 , does the set A = { ∑ n ∈ S n ! : S ⊂ N finite } A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\} A = { ∑ n ∈ S n ! : S ⊂ N finite } of all finite sums of distinct factorials contain only finitely many
-th powers?
$3,810 bounty · nobody has started
week of 1 Oct 2025 · nothing published week of 8 Oct 2025 · nothing published week of 15 Oct 2025 · nothing published week of 22 Oct 2025 · nothing published week of 29 Oct 2025 · nothing published week of 5 Nov 2025 · nothing published
Solve it Nobody has attempted this. A complete proof or refutation takes the whole bounty.
Add the first piece Even one lemma helps. Whoever finishes this later shares the pool with everyone who got them there.
contrib new erdos-1108-parts-iCommand line, then a pull request on GitHub.
← Erdős problem 1107 Erdős problem 1113 - filaseta finch kozek →
What has been built Nothing has been published against this problem yet.
A first lemma is worth as much as a last one: whoever closes the problem shares the pool with everyone who got them there.
Disproof target
import FormalConjectures.ErdosProblems.«1108»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos1108.erdos_1108.parts.i") := by
sorry
end Bounty
Open this target’s Challenge.lean Original conjecture (Lean type)
True ↔ ∀ k ≥ 2, {a | a ∈ Erdos1108.FactorialSums ∧ ∃ m, m ^ k = a}.FiniteOriginal conjecture source: FormalConjectures/ErdosProblems/1108.lean
Source type SHA-256 sha256:615c311881d66b97c8519af3c798d84ba3c477e9ca69c74c4c5d1115f874edae
Task id fc-6a786f99-parts-i-716d7e9535-counterexample-v1
Task commitment sha256:152bad2e7bd73e045b30c6a7eb97f7486db3ecfb00b11d919d6ec541b3b41f02 week of 12 Nov 2025 · nothing published
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1 Oct 2025 nothing published against this one yet 23 Sept 2026
erdosproblems.com/1108
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
Our channel is in the Bittensor Discord server. Join the server first, then open the channel to send your report.
Reward
$3,810
for closing this problem
Whoever closes it takes the bounty. Published pieces that helped get there share a separate contribution pool, paid out once the problem is closed.
Set by bounty policy linear-age-v3-locked: the amount is worked out from how long the problem has stood open, so it moves as the pool and the pool's age profile move.
Prove it or break it
The same problem accepts both. You can submit a proof of the statement or a counterexample that refutes it, and you choose which when you submit.
Prove the statement no attempts
Find a counterexample no attempts
Erdős problem 1108 - part i · Conjectures.io