The odd Catalan numbers modulo 2^k

TL;DR

This paper investigates the behavior of odd Catalan numbers modulo powers of two, revealing a limited set of congruences and providing explicit formulas for these congruences based on powers of two.

Contribution

It establishes that odd Catalan numbers modulo 2^k have only k-1 distinct congruences, which can be explicitly determined using specific indices related to powers of two.

Findings

Number of distinct congruences is k-1 for modulo 2^k.

Congruences are derived from C_{2^m - 1} for m=1,...,k-1.

Provides explicit formulas for these congruences.

Abstract

We study the properties of the odd Catalan numbers, C_n, modulo 2^k for k >= 2. We show that there exist only k - 1 different congruences of the odd Catalan numbers modulo 2^k. Moreover, these congruences can be obtained by C_{2^m - 1} (mod 2^k) for m = 1, 2, ..., k - 1.