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Catalan numbers modulo a prime power. (English) Zbl 1283.11050

Summary: Let \(C_n = (2n)!/((n + 1)!n!)\) be the \(n\)-th Catalan number. It is proved that for any odd prime \(p\) and integers \(a, k\) with \(0 \leq a < p\) and \(k > 0\), if \(0 \leq a < (p + 1)/2\), then the Catalan numbers \(C_{p^1 -a}, \dots, C_{p^k -a}\) are all distinct modulo \(p^k\), and the sequence \((C_{p^n -a})_{n \geq 1}\) modulo \(p^k\) is constant from \(n = k\) on; if \((p + 1)/2 \leq a < p\), then the Catalan numbers \(C_{p^1 -a}, \dots, C_{p^{k+1} -a}\) are all distinct modulo \(p^k\), and the sequence \((C_{p^n -a})_{n\geq 1}\) modulo \(p^k\) is constant from \(n = k + 1\) on. A similar conclusion is proved for \(p = 2\) recently by H.-Y. Lin [Integers 11, A55, 5 p. (2011; Zbl 1251.11010), print version 12, No. 2, 161–165 (2012)].

MSC:

11B75 Other combinatorial number theory

Citations:

Zbl 1251.11010
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