Fermat’s Little Theorem, Pseudoprimes, and Superpseudoprimes

Abstract

In this chapter we show how to apply Fermat numbers to generate infinitely many pseudoprimes and superpseudoprimes. To define pseudoprimes and superpseudoprimes, we will need to make use of Fermat’s little theorem which is a centerpiece of number theory. It gives a fundamental property of primes and is the basis of most tests for primality.

Statement of Fermat’s little theorem by Pierre de Fermat in a letter to Bernhard Frénicle de Bessy, October 18, 1640, [Mahoney, p. 291].

Without exception, every prime number measures one of the powers —1 of any progression whatever, and the exponent of the said power is a submultiple of the given prime number —1.