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Structure and asymptotics for Catalan numbers modulo primes using automata (1701.02975v1)

Published 11 Jan 2017 in math.NT

Abstract: Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set ${n: C_n \equiv 0 \mod p }$ is $1$ for all primes $p$, We also show that if $n = pk -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv { \frac{p+1}{2}, \frac{p+3}{2}, ..., p-2 } \mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi.

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