On a conjecture about the exponent set of primitive matrices

Abstract

M. Lewin and Y. Vitek conjecture [7] that every integer ⩽[ (n> 2−2n+2) 2 ]+1 is an exponent of some n× n primitive matrix. In this paper, we prove three results related to Lewin and Vitek's conjecture: (1) Every integer ⩽[ (n 2−2n+2) 4 ]+1 is an exponent of some n× n primitive matrix. (2) The conjecture is true when n is sufficiently large. (3) We give a counterexample to show that the conjecture is not true in the case when n=11.