Catalan and Apéry numbers in residue classes

Abstract: We estimate character sums with Catalan numbers and middle binomial coefficients modulo a prime p. We use this bound to show that the first at most elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations like the Apéry numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p. [Copyright &y& Elsevier]