Abstract. We consider the set [Formula: see text] of [Formula: see text]-matrices with integer elements of size at most [Formula: see text] and obtain upper bounds on the number of matrices from [Formula: see text], for which the characteristic polynomial has a fixed discriminant [Formula: see text]. When [Formula: see text], this corresponds to counting matrices with a repeated eigenvalue and thus is related to counting nondiagonalizable matrices. For [Formula: see text], this problem seems not to have been studied previously, while for [Formula: see text], both our approach and the final result improve on those of A. J. Hetzel, J. S. Liew, and K. Morrison [ Amer. Math. Monthly, 114 (2007), pp. 491–499].

We obtain some theoretical and experimental results concerning various properties (the number of fixed points, image distribution, cycle lengths) of the dynamical system naturally associated with Fermat quotients acting on the set $\{0,\dots,p-1\}$. In particular, we improve the lower bound of Vandiver [Bull. Amer. Math. Soc., 22 (1915), pp. 61–67] on the image size of Fermat quotients on the above set (from $p^{1/2}-1$ to $(1+o(1))p(\log p)^{-2}$). We also consider pseudorandom properties of Fermat quotients such as uniform distribution and linear complexity.