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 k have the property that 2 n k + 1 2^nk+1 is not a Carmichael number for any n ∈ N n\in \mathbb {N} ; this implies the existence of a set K \mathscr {K} of positive lower density such that for any k ∈ K k\in \mathscr {K} the number 2 n k + 1 2^nk+1 is neither prime nor Carmichael for every n ∈ N n\in \mathbb {N} . Next, using a recent result of Matomäki and Wright, we show that there are ≫ x 1 / 5 \gg x^{1/5} Carmichael numbers up to x x that are also Sierpiński and Riesel. Finally, we show that if 2 n k + 1 2^nk+1 is Lehmer, then n ⩽ 150 ω ( k ) 2 log ⁡ k n\leqslant 150\,\omega (k)^2\log k , where ω ( k ) \omega (k) is the number of distinct primes dividing k k .