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Erdős problem 1057 Is it true that C ( x ) = x 1 − o ( 1 ) C(x)=x^{1-o(1)} C ( x ) = x 1 − o ( 1 ) ?
This is discussed in problem A13 of Guy's collection [Gu04].
References
[AGP94] Alford, W. R. and Granville, Andrew and Pomerance, Carl, There are infinitely many Carmichael numbers. Ann. of Math. (2) (1994), 703--722. [Er56c] Erdős, P., On pseudoprimes and Carmichael numbers. Publ. Math. Debrecen (1956), 201--206.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Ha08] Harman, Glyn, Watt's mean value theorem and Carmichael numbers. Int. J. Number Theory (2008), 241--248.
[Li22] J. D. Lichtman, Primes in arithmetic progressions to large moduli and shifted primes without large prime factors. arXiv:2211.09641 (2022).
[Po89] Pomerance, Carl, Two methods in elementary analytic number theory. (1989), 135--161. Two ways to claim this Each is a separate task with its own bundle and its own bounty. Pick the one your proof argues for.
Bounty
$599
paid on an accepted proof
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.
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Lean type
True ↔ Filter.Tendsto (fun x => Real.log (Erdos1057.carmichaelCounting x) / Real.log x) Filter.atTop (nhds 1)What you must prove
import FormalConjectures.ErdosProblems.«1057»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos1057.erdos_1057") := by
sorry
end Bounty
Pinned source: FormalConjectures/ErdosProblems/1057.lean
Source type SHA-256 sha256:64d33ebee10cda22de1b2af0cc965bc01a26acb409de0026df24767fff99370e
Task id fc-379fc029-erdos1057-erdos-1057-1aafdd4cde-counterexample-v1
Task commitment sha256:b9efa922da1d6ef87faf5567ccbc8586c170ac4d9982acb1ec69c1b6b0740dc2 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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Erdős problem 1057 · Conjectures.io