On the number of residues of certain second-order linear recurrences

For every monic polynomial f E 77[X ] with deg(f) >= 1, let (f ) be the set of all linear recurrences with values in 77 and characteristic polynomial f , and let { r ( f ) := rho ( x ; m ) : x E (f ), m E 77 + , where rho ( x ; m ) is the number of distinct residues of x modulo m . Dubickas and Novikas proved that r ( X 2 - X - 1) = 77 + . We generalize this result by showing that r ( X 2 - a 1 X -1) = 77 + for every nonzero integer a 1 . As a corollary, we deduce that for all integers a 1 >= 1 and ) iJ ) k >= 2 there exists 4 E D8 such that the sequence of fractional parts frac(4 alpha n) n >= 0 , where alpha := a 1 + a 2 1 + 4 /2, has exactly k limit points. Our proofs are constructive and employ some results on the existence of special primitive divisors of certain Lehmer sequences.