Sierpinski and Carmichael numbers
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We establish several related results on Carmichael, Sierpinski and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n 2 N; this implies the existence of a set K of positive lower density such that for any k 2 K the number 2nk + 1 is neither prime nor Carmichael for every n 2 N. Next, using a recent result of Matom aki, we show that there are x1=5 Carmichael numbers up to x that are also Sierpi nski and Riesel. Finally, we show that if 2nk+1 is Lehmer, then n 6 150 !(k)2 log k, where !(k) is the number of distinct primes dividing k.
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