The least witness of a composite number

R Balasubramanian, SV Nagaraj - International Workshop on Information …, 1997 - Springer
R Balasubramanian, SV Nagaraj
International Workshop on Information Security, 1997•Springer
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 …
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) < 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 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.
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