Carrie Finch

A curious connection between fermat numbers and finite groups

1. INTRODUCTION. In the seventeenth century, Fermat defined the sequence of numbers F n = 2 2 n + 1 for n ≥ 0, now known as Fermat numbers. If F n happens to be prime, F n is called a Fermat prime. Fermat showed that F n is prime for each n ≤ 4, and he conjectured that F n is prime for all n (see Brown [1] or Burton [2, p. 271]). Almost one hundred years passed before Euler demonstrated in 1732 that F 5 is in fact composite. Ironically, it is now known that F n is composite for many values of n and, as of the date this article was written, no new Fermat primes had been discovered. In this paper we solve a problem in finite groups whose solution relies heavily on techniques from elementary number theory. While it is not unusual for this phenomenon to occur, the main result is surprisingly a direct consequence of the fact that F 5 is composite.

Sierpinski and Carmichael numbers

Abstract : 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 2(exp n)k + 1 is not a Carmichael number for any n epilson N; this implies the existence of a set K of positive lower density such that for any k epsilon K the number 2(exp n)k + 1 is neither prime nor Carmichael for every K epilson N. Next, using a recent result of Matomaki we show that there are x1/5 Carmichael numbers up to x that are also Sierpinski and Riesel. Finally, we show that if 2(exp n)k+1 is Lehmer then n 150 omega(k)2 log k, where omega(k) is the number of distinct primes dividing k.