Pseudoprime numbers.

Abstract

Our objective in this paper is to study pseudoprime numbers. In the course of its development, we discuss pseudoprime to the base 2, and subsequently generalize it to any base a. In chapter 1, we provide readers with a short account of the necessary background from elementary number theory that is needed throughout the paper. We answer questions regarding the number of pseudoprimes, recognition of pseudoprimes and the distribution of pseudoprimes for both the base 2 and a. Necessary and sufficient conditions for an integer to be pseudoprime is established and several sequences generating infinitely many pseudoprimes are given. We discuss some special kinds of pseudoprimes including absolute pseudoprimes(or Carmichael numbers), Euler pseudoprimes, and strong pseudoprimes. We conclude the paper with a brief discussion of two probabilistic primality tests, one based on the concept of Euler pseudoprime and the other on strong pseudoprimes.