1) If p is not 2 or 5, then the order of the matrix (0,1;1,1) is the multiplicative order of any eigenvalue (i.e. (1+/-sqrt(5))/2), since, for those p's the matrix is diagonalizable. 2) Again, for those p, by Hensel's lemma, fp(p^n) = p^(n-1) fp(p) 3) Since we're asking for the multipicative order of an element of a finite field, this is much like the question of Artin's conjecture which asks how often 2 is a primitive root (or more generally, how often a is a primitive root). The really interesting quantity is how far it deviates from the maximum value. So one would want to define fq(p) = (p-1)/fp(p) if 5 is a quadratic residue of p, and fq(p) = (p^2-1)/fp(p) if 5 is a quadratic non-residue of p. fq wouldn't be bounded, but would tend to be small. Victor