Rabin-miller Primality Test: Composite Numbers Which Pass It

The Rabin-Miller primality test is a probabilistic test which can be found in several algebraic computing systems (such as Pari, Maple, ScratchPad) because it is very easy to implement and, with a reasonable amount of computing, indicates whether a number is composite or "probably prime" with a very low probability of error. In this paper, we compute composite numbers which are strong pseudoprimes to several chosen bases. Because these bases are those used by the Scratchpad implementation of the test, we obtain, by a method which differs from a recent one by Jaeschke, composite numbers which are found to be "probably prime" by this test. 1. Preliminaries First, we recall the following definitions: 1.1. Definitions. Let ¿> e N*. A number n e N* is a pseudoprime to base b if bn~x = 1 modulo n. It is a strong pseudoprime to base b if it is odd and if one of the following conditions is satisfied, with n 1 = 2kq and q odd: b9 = 1 modulo n or there exists an integer i such that 0 < i < k and b2q = -1 modulo n . The Rabin-Miller test consists in, given an odd number n, checking if n is a strong pseudoprime to several bases which are either chosen randomly or taken in a predetermined set, depending on the implementation. If n is not a strong pseudoprime to some of the chosen bases, n is proved to be composite. Conversely, Rabin has shown in [12] that if « is a strong pseudoprime to k bases, it is "probably prime" with an error probability of less than 1/4* . This test is implemented in the Computing Algebra System Scratchpad, in which the bases used are the first ten prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. In this paper, we compute a composite number which is a strong pseudoprime to these bases. So, this number passes the Scratchpad test. In order to show that Received by the editor February 7, 1992 and, in revised form, October 15, 1992. 1991 Mathematics Subject Classification. Primary 11A15, 11A51, 11Y11.