Infinitely Many Carmichael Numbers for a Modified Miller-Rabin Prime Test

Abstract

We study a variant of the Miller-Rabin primality test, which only looks at the last (z+1) powers of the base. This test is between Miller-Rabin and Fermat in terms of strength. For (z=1) the test can be thought of as a variant of the Solovay-Strassen test. We show that for every (z ≥ 0) this test has infinitely many "Carmichael" numbers. We also give empirical results on the rate of growth of the test's "Carmichael" numbers, noting that the growth rate decreases geometrically with increasing (z). We provide some heuristic evidence for this pattern. We also extend our existence result to some generalizations of Miller-Rabin that use (b)-th powers instead of squares.