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 Distinguishing Prime Numbers from Composite Numbers
A new algorithm for testing primality is presented. The algorithm is distinguishable from the lovely algorithms of Solvay and Strassen [36], Miller [27] and Rabin [32] in that its assertions of
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.