Further investigations with the strong probable prime test

Recently, Damgard, Landrock and Pomerance described a procedure in which a k-bit odd number is chosen at random and subjected to t random strong probable prime tests. If the chosen number passes all t tests, then the procedure will return that number ; otherwise, another k-bit odd integer is selected and then tested. The procedure ends when a number that passes all t tests is found. Let p kt denote the probability that such a number is composite. The authors above have shown that p k,t ≤ 4 −t when k ≥ 51 and t ≥ 1. In this paper we will show that this is in fact valid for all k ≥ 2 and t ≥ 1.