Structure and asymptotics for Catalan numbers modulo primes using automata

2017/01/11 by Rob Burns, Burns, Rob
Computer Science · Mathematics · #11B65 #Advanced Combinatorial Mathematics #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1701.02975

openalex publication_date 2017/01/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C_n be the nth Catalan number. We show that the asymptotic density of the set \n: C_n ≡ 0 \mod p \ is 1 for all primes p, We also show that if n = p^k -1 then C_n ≡ -1 \mod p. Finally we show that if n ≡ \ (p+1)/(2), (p+3)/(2), ..., p-2 \ \mod p then p divides C_n. All results are obtained using the automata method of Rowland and Yassawi.

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