On the Number of Witnesses ( a mod n) Providing A Proper Divisor of n

Abstract

Some of the witnesses ( a mod n) for an odd composite integer n > 1, in the probabilistic primality test of Miller and Rabin, provide a non-trivial divisor of n of the form gcd( a 2 kn ′ − 1, n), where n − 1 = 2 ν 2( n − 1) n′ and 0 ≤ k ≤ ν 2( n − 1). Considered here are extreme values of both the number of such withnesses for n and the ratio between this number and the total number of witnesses for n.