ACAC Seminar Abstract

ACAC Seminar Abstract

ACAC Seminars

ACAC Seminar Abstract

On Carmichael, Lehmer, Riesel and Sierpinski numbers

Speaker: Florian Luca
Date, Time: Fri, 25 Nov 2011 15:00

A Sierpinski number k is an odd positive integer k such that 2^n k+1 is composite for any positive integer n. It is known that Sierpinski numbers occupy a positive proportion of all odd numbers, but the proportion is not 1. That is, there is also a positive proportion of odd integers k such that 2^n k+1 is prime for some n. Similar to the Sierpinski numbers are Riesel numbers which are odd integers k such that 2^n k-1 is composite for all positive integers n. In the talk, we will survey various known facts about Sierpinski and Riesel numbers, their distribution and their presence in various interesting subsequences of integers and study what happens if in their definition we replace primes by Carmichael numbers which are composite positive integers n such that the expression a^n-a is a multiple of n for all integers a. Several connections of the Carmichael numbers with the elusive Lehmer numbers will also be presented.

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