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Updated Oct 8

Problem 1057

Statement. Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$ . Is it true that $C(x)=x^{1-o(1)}$ ?

Status. Open. The site's label is OPEN, and no claim page is recorded.

Source. erdosproblems.com/1057, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1057, https://www.erdosproblems.com/1057.

References.

[AGP94] [[../library/integer_sequences/alford_1994_infinitely_many_carmichael_numbers/_index|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, 3rd ed. Problem Books in Mathematics, Springer (2004), xviii+437 pp. A13 "Carmichael numbers", printed p. 50: the report of Alford, Granville and Pomerance's infinitely many Carmichael numbers, more than $x^\beta$ below $x$ with $\beta>0.290306>2/7$ , and "Erdős had conjectured that $(\ln C(x))/\ln x$ tends to 1 as $x$ tends to infinity", with his upper bound improving Knödel; no proofs. Library home: guy_2004_unsolved_problems_number_theory.
[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. In R. A. Mollin (ed.), Number Theory and Applications, Kluwer Academic Publishers (1989), 135-161.

Formalization. Statement in formal-conjectures.

Current assessment

Scope. The question, status and literature are not assessed here: the page carries the site's label (OPEN) and its references, and no status search is recorded. The only authored note is the one below.

Two release manuscripts on smooth shifted primes. Two manuscripts of the OpenAI mathematics release, both dated 24 September 2026, Weighted dilation graphs, smooth shifted primes and totient fibers and The Poisson-Dirichlet law for prime predecessors, recorded on the cards openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers and openai_2026_poisson_dirichlet_law_prime_predecessors, claim respectively that for every fixed $\delta>0$ there are $x^{1-o(1)}$ primes $p\in(2x,5x]$ with $P^+(p-1)\le x^\delta$ (Theorem 1.2 of the first), and a Poisson-Dirichlet law for the large prime factors of $p-1$ , under which for every fixed $u$ a positive proportion of the primes $p\le x$ have $P^+(p-1)\le p^{1/u}$ . Both name the construction of Carmichael numbers by Alford, Granville and Pomerance [AGP94], which needs a positive proportion of primes $p$ with smooth $p-1$ together with a theorem on primes in arithmetic progressions, as an application of such estimates; neither states a bound on $C(x)$ , and the release claims nothing on this problem. The first card records that its count is weaker than the positive-proportion hypothesis the theorem of [AGP94] needs, so no Carmichael bound follows from it alone; the second card records that the manuscript's consequence (1.3) would supply that hypothesis for every exponent, so that the exponent of $C(x)$ would come to rest on the progression exponent alone, a deduction the manuscript does not make. A lower bound for $C(x)$ obtained by feeding these claims into the method of [AGP94] would be a new result, not one the release claims; nothing in either manuscript is verified in this corpus, and neither has a claim page.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (27)

Sequences and Densities of Integers arithmetic_functions/lichtman_2022_primes_arithmetic_progressions_large_moduli_shifted Corollary 1.2 (p. 2): at least x^0.3389 Carmichael numbers up to x Theorem 1.1 (p. 1): shifted primes p - a with no prime factor above x^0.2844 Theorem 1.4 (p. 2): primes in progressions to quadrilinear moduli up to x^(17/32)arithmetic_functions/openai_2026_poisson_dirichlet_law_prime_predecessors Theorem 1.1: the ordered prime factors of p-1 follow the Poisson-Dirichlet law PD(1)Theorem 7.1: over primes, interior divisor counts of p-1 on long prime slots match their reciprocal-sum mean arithmetic_functions/openai_2026_weighted_dilation_graphs_smooth_shifted_primes_totient_fibers Theorem 1.2: x^(1-o(1)) primes p in (2x,5x] with P^+(p-1) ≤ x^δ Sequences and Densities of Integers integer_sequences/alford_1994_infinitely_many_carmichael_numbers Theorem 1 (p. 705): C(x) ≥ x^{EB} for each E in the set ℰ and B in the set ℬ, hence C(x) > x^{2/7} for all large x Theorem 3 (p. 707): for each B in the set ℬ, the interval (0, B) lies in the set ℰ Theorem 4 (p. 707): a lower bound for primes ≡ 1 mod d up to x^{1−ε} gives C(x) ≥ x^{1−2ε}Theorem 5 (p. 708): for each α with 0 < α < 25/144 there is a computable x(α) with C(x) ≥ x^α for all x ≥ x(α)integer_sequences/erdos_1956_pseudoprimes_carmichael_numbers Conjecture (p. 201) with the heuristic of p. 206: C(x) > x^{1-eps} for every eps > 0 and x > x(eps)Inequality (6) (p. 201) with Lemma 2 (p. 203): C(x) < x exp(-c_5 log x log log log x / log log x)Lemma 1 (p. 202): integers up to x composed of k given primes number fewer than x exp(-c_6 u log u), where k^u = x integer_sequences/pomerance_1989_two_methods_elementary_analytic_number_theory Theorem 5.1 (p. 152), with Lemma 5.2: C(x) <= x/L(x)^{1+o(1)} for the count of Carmichael numbers up to x integer_sequences/sorenson_webster_2015_strong_pseudoprimes_twelve_bases Theorem 1.1 (p. 2): the least strong pseudoprimes to the first 12 and the first 13 prime bases Theorem 1.2 (p. 2): all two-prime strong pseudoprimes to the first m prime bases up to B in B^(2/3+o(1)) operations Theorem 3.8 (p. 13): assuming Conjecture 3.6, all strong pseudoprimes to the first m prime bases up to B in B^(2/3+o(1)) time number_theory/guy_2004_unsolved_problems_number_theory

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