Further investigations with the strong probable prime test

Recently, Damgård, 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 $p_{k,t}$ denote the probability that such a number is composite. The authors above have shown that $p_{k,t}\le 4^{-t}$ when $k\ge 51$ and $t\ge 1$. In this paper we will show that this is in fact valid for all $k\ge 2$ and $t\ge 1$.