On a conjecture about the exponent set of primitive matrices

AbstractM. 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 ⩽[(n2−2n+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