Catalan and Apéry numbers in residue classes

Authors:
Moubariz Z. Garaev;Florian Luca;Igor E. Shparlinski
Affiliations:
Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México;Instituto de Matemáticas, Universidad Nacional Autónoma de México, Morelia, Michoacán, México;Department of Computing, Macquarie University, Sydney, NSW, Australia
Venue:
Journal of Combinatorial Theory Series A
Year:
2006

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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 p13/2(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 pO(p) 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.