Sierpiński and Carmichael numbers

Abstract

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2 n k + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2 n k + 1 is neither prime nor Carmichael for every n ∈ N. Next, using a recent result of Matomäki, we show that there are x 1/5 Carmichael numbers up to x that are also Sierpiński and Riesel. Finally, we show that if 2 n k + 1 is Lehmer, then n 150 ω(k) 2 log k, where ω(k) is the number of distinct primes dividing k.

There are no Carmichael numbers of the form 2 n p+1 with p prime

“…This is the main theorem in [3]. Letting K := k odd : 2 n k + 1 n ≥ 0 contains some Carmichael number , the set K is of asymptotic density zero (see [2]). The smallest element of K is 27 (see [3,Theorem 2]), and a representation indicating 27 as a member of K is given by 1729 = 27 × 2 6 + 1 with the Carmichael number 1729 being known as the Ramanujan taxicab number!…”

Carmichael numbers in the sequence $(2^n k+1)_{n\geq 1}$

“…Besides k = 1 there are other values of k for which the sequence 2 n k + 1 does not contain any Carmichael numbers. Indeed, in [2], it has been shown, among other things, that if we put K = {k : (2 n k + 1) n≥0 contains some Carmichael number}, then K is of asymptotic density zero. This contrasts with the known fact, proved by Erdős and Odlyzko [9], that the set {k : (2 n k + 1) n≥0 contains some prime number} is of positive lower density.…”

Carmichael numbers and the sieve

“…The conjecture was proved unconditionally by Matomäki [20] in the special case that a is a quadratic residue modulo b, and using an extension of her methods Wright [23] established the conjecture in full generality. The techniques introduced in [1] have led to many other investigations into the arithmetic properties of Carmichael numbers; see [2][3][4][5]7,8,10,[14][15][16][17][18][19]21,24] and the references therein.…”