There are infinitely many Carmichael numbers

  title={There are infinitely many Carmichael numbers},
  author={W. R. Alford and Andrew Granville and Carl Pomerance},
  journal={Annals of Mathematics},

Carmichael numbers and the sieve

Carmichael Meets Chebotarev

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

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
    Canadian Mathematical Bulletin
  • 2009
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

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

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)

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.


  • 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



On Numbers Analogous to the Carmichael Numbers

A base a pseudoprime is an integer n such that 1 A Carmichael number is a composite integer n such that (1) is true for all a such that (a, n ) = l. It was shown by Carmichael [1] that, if n is a

On Linnik's constant.