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Number theory
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Erdős problem 1095 - lower conjecture
Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that
g(k)≥exp(clogkk)
for some constant
c>0
.
Pin rotation in progress; nothing can be submitted right now.
[EES74] Ecklund, Jr., E. F. and Erd\H{o}s, P. and Selfridge, J. L., A new function associated with the prime factors of {(\spn\sbk)}. Math. Comp. (1974), 647--649.
[ELS93] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.
[GrRa96] Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika (1996), 73--107.
[Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika (1999), 41--55.
[SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster, Jonathan, An algorithm and estimates for the {E}rdős-{S}elfridge function. (2020), 371--385.
No one has attempted this yet.
Formal statement
Lean type
∃ c > 0, ∀ᶠ (k : ℕ) in Filter.atTop, ↑(Erdos1095.g k) ≥ Real.exp (c * ↑k / Real.log ↑k)
What you must prove
import FormalConjectures.ErdosProblems.«1095»
import TaskSupport
namespace Bounty
theorem target : fcTypeOfName% "Erdos1095.erdos_1095.variants.lower_conjecture" := by
sorry
end Bounty
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.
Erdős problem 1095 - lower conjecture · Conjectures.io