Igor Shparlinski
speaks on "Catalan and Ap{\'e}ry Numbers in Residue Classes" (joint work with Moubariz Garaev and Florian Luca) 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 p^{13/2} (log p)^6 elements of each sequence already fall in all residue classes modulo every sufficiently large p, which improves the previously known result requiring p^{O(p)} elements. We also study, using a different technique, similar questions for sequences satisfying polynomial recurrence relations, for example, for Apery numbers. We show that such sequences form a finite additive basis modulo p for every sufficiently large prime p.