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Level A · Machine-checkable Hard Number theory P-erdos-1057

Erdős Problem #1057

Is it true that C(x)=x^1-o(1)? This is discussed in problem A13 of Guy's collection [Gu04].

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite 
@misc{cairn-erdos-1057,
  title        = {Erdős Problem #1057},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1057}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Disputed
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On the literature board
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Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

Is it true that ?

This is discussed in problem A13 of Guy's collection [Gu04].

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1057». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_1057 :
    answer(sorry) ↔ Tendsto (fun x ↦ Real.log (carmichaelCounting x) / Real.log x) atTop (𝓝 1)

What counts as progress

  • A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
  • Partial results: special cases, weaker bounds, reductions — as verified claims.
  • Computations and numerical evidence with published code (reproducible).
  • Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1057. Report it as a literature claim.
  • A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

  • erdos_1057.variants.pomerance — Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact C(x)= x exp(-(1+o(1))log xlogloglog x/loglog x).

References

  • erdosproblems.com/1057
  • [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.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.