On the number of false witnesses for a composite number

Abstract

If a is not a multiple of n and a n − 1 ≢ 1 mod n {a^{n - 1}}\;equiv \;1\bmod \,n , then n must be composite and a is called a "witness" for n . Let F ( n ) F(n) denote the number of "false witnesses" for n , that is, the number of a mod n a\bmod n with a n − 1 ≡ 1 mod n {a^{n - 1}} \equiv 1\bmod n . Considered here is the normal and average size of F ( n ) F(n) for n composite. Also considered is the situation for the more stringent Euler and strong pseudoprime tests.