New results on the exponent set of primitive nearly reducible matrices

AbstractFor a primitive, nearly reducible matrix, J.A. Ross defined e(n) to be the least integer greater than 5 that no n × n matrix has as its exponent, and he proved n+2⩽e(n)⩽n2−4n+6. Recently, J.Y. Shao proved e(n)⩾n2−2n+109+1 and Yang Shangjun and J.P. Barker proved e(n)>n24−n232 if a certain conjecture in number theory is true. In this paper we obtain the following new results: e(n)⩾n2n2+1+1 for n⩾5, and e(n)⩾m2nm+12 where m is any given positive integer and n is large enough. In fact we have determined e(n) asymptotically: limn→∞e(n)n2=