Catalan and Apéry numbers in residue classes

Moubariz Z. Garaev*, Florian Luca, Igor E. Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

11 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)851-865
Number of pages15
JournalJournal of Combinatorial Theory: Series A
Issue number5
Publication statusPublished - Jul 2006


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