The least witness of a composite number

Abstract

We consider the problem of finding the least witness of a composite number. Ifn is a composite number then a number w for which n is not a strong pseudo-prime to the base w is called a witness for n. Let w(n) be the least witness for a composite n. Bach [7] assuming the Generalized Riemann Hypothesis (GRH) showed that w(n) < 2log2 n. In this paper we are interested in obtaining upper bounds for w(n) without assuming the GRH. Burthe [15) showed that w(n) = O ∈(n 1/(8√e)+ε) for all composite numbers n which are not a product of three distinct prime factors. For the three prime factor case he was able to show that w(n) = O ∈(n 1/(6√e)+∈). We improve his result to showw(n) = O ∈(n 1/(8√e)+∈) for all composite numbers n except Carmichael numbers n= pqr for which v 2(p − 1) = v 2 (q − 1) = v 2 (r − 1). For the special Carmichaels we use an argument due to Heath-Brown to get w(n) = O ∈(n1/(6.568√e)+∈). We conjecture w(n) = O ∈(n 1/(8√e)+∈) for every composite number n and look at open problems. It appears to be very difficult to settle our conjecture.