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Number theory
Date added 13 Aug 2026
Erdős problem 269 - irrational Let P P P be a finite set of primes with ∣ P ∣ ≥ 2 |P| \ge 2 ∣ P ∣ ≥ 2 and let
{ a 1 < a 2 < … } \{a_1 < a_2 < \dots\} { a 1 < a 2 < … } be the set of positive integers whose prime factors
are all in
. Is the sum
∑ n = 1 ∞ 1 [ a 1 , … , a n ] \sum_{n=1}^\infty \frac{1}{[a_1,\ldots,a_n]} n = 1 ∑ ∞ [ a 1 , … , a n ] 1
irrational?
$2,390 bounty · nobody has started
week of 18 Sept 2025 · nothing published week of 25 Sept 2025 · nothing published week of 2 Oct 2025 · nothing published week of 9 Oct 2025 · nothing published week of 16 Oct 2025 · nothing published week of 23 Oct 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-269-variants-irrationalCommand line, then a pull request on GitHub.
← Erdős problem 264 - part ii Erdős problem 272 - szabo strong →
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.
Proof target
import FormalConjectures.ErdosProblems.«269»
import TaskSupport
namespace Bounty
theorem target : fcTypeOfName% "Erdos269.erdos_269.variants.irrational" := by
sorry
end Bounty
Open this target’s Challenge.lean Original conjecture (Lean type)
True ↔ ∀ (P : Finset ℕ), (∀ p ∈ P, Nat.Prime p) → P.card ≥ 2 → Irrational (Erdos269.series ↑P)Original conjecture source: FormalConjectures/ErdosProblems/269.lean
Source type SHA-256 sha256:2a63529741eab05474bb3f1f4ee24cb6baee35189869a74d1b7a0b616410a9c9
Task id fc-8432eac9-variants-irrational-39590f47c3-formalized-v1
Task commitment sha256:9ae9f1b141ebbe7c019ccee5ad180a611134c3e833d8ecb93df4a620cafd01a5 week of 30 Oct 2025 · nothing published
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18 Sept 2025 nothing published against this one yet 10 Sept 2026
erdosproblems.com/269
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.
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Reward
$2,390
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 dynamic-age-v2-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 269 - irrational · Conjectures.io