# The least witness of a composite number

## Abstract

We consider the problem of finding the least witness of a composite number. If *n* 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*) < 2log^{2} 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 show *w*(*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*_{∈}(*n*^{1/(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.

## Keywords

Deterministic Algorithm Average Running Time Composite Number Generalize Riemann Hypothesis Deterministic Polynomial Time## Preview

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