On a conjecture about the exponent set of primitive matrices

M. Lewin and Y. Vitek conjecture [7] that every integer $\text{?[}\text{(n>}{}^2\text{?2n+2)}\text{2}\text{]+1}$ 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 $\text{?[}\text{(n}{}^2\text{?2n+2)}\text{4}\text{]+1}$ 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.