# There are infinitely many Carmichael numbers

@article{Alford1994ThereAI,
title={There are infinitely many Carmichael numbers},
author={W. R. Alford and Andrew Granville and Carl Pomerance},
journal={Annals of Mathematics},
year={1994},
volume={139},
pages={703-722}
}
• Published 1 May 1994
• Mathematics
• Annals of Mathematics
356 Citations
Carmichael Meets Chebotarev
• Mathematics
• 2013
Abstract. For any finite Galois extension $K$ of $\mathbb{Q}$ and any conjugacy class $C$ in $\text{Gal}\left( {K}/{\mathbb{Q}}\; \right)$ , we show that there exist infinitely many Carmichael
Piatetski-Shapiro sequences
• Mathematics
• 2012
We consider various arithmetic questions for the Piatetski-Shapiro sequences bncc (n = 1, 2, 3, . . .) with c > 1, c 6∈ N. We exhibit a positive function θ(c) with the property that the largest prime
Carmichael Numbers with a Square Totient
• W. Banks
• Mathematics
Abstract Let $\varphi$ denote the Euler function. In this paper, we show that for all large $x$ there are more than ${{x}^{0.33}}$ Carmichael numbers $n\,\le \,x$ with the property that $\varphi On the difficulty of finding reliable witnesses • Mathematics ANTS • 1994 It is shown that there are finite sets of odd composites which do not have a reliable witness, namely a common witness for all of the numbers in the set. On Types of Elliptic Pseudoprimes • Mathematics journal of Groups, Complexity, Cryptology • 2021 We generalize the notions of elliptic pseudoprimes and elliptic Carmichael numbers introduced by Silverman to analogues of Euler-Jacobi and strong pseudoprimes. We investigate the relationships among Carmichael Numbers for GL(m) • Mathematics • 2020 We propose a generalization of Carmichael numbers, where the multiplicative group Gm = GL(1) is replaced by GL(m) for m ≥ 2. We prove basic properties of these families of numbers and give some Pseudoprosti brojevi Pitanje je li odre deni veliki prirodni broj n prost ili složen jedno je od najvažnijih u teoriji brojeva, a često se javlja na primjer u kriptografiji3 . U primjenama se najčešće zadovoljavamo Millions of Perrin pseudoprimes including a few giants It is thought that well over 90% of all 20-digit Perrin pseudoprimes are found, compared to the previously known just over 100,000, and two new sequences are proposed that do not provide any pseudopRimes up to$10^9$at all. FACTORS OF CARMICHAEL NUMBERS AND AN EVEN WEAKER$k$-TUPLES CONJECTURE • Thomas Wright • Mathematics Bulletin of the Australian Mathematical Society • 2019 One of the open questions in the study of Carmichael numbers is whether, for a given$R\geq 3$, there exist infinitely many Carmichael numbers with exactly$R\$ prime factors. Chernick [‘On Fermat’s