On the Residues of Binomial Coefficients and Their Products Modulo Prime Powers

Abstract

In this paper, we show several arithmetic properties on the residues of binomial coefficients and their products modulo prime powers, e.g.,

$$ {\left( {\begin{array}{*{20}c} {{pq - 1}} \\ {{{{\left( {pq - 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {pq - 1} \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right)} \equiv {\left( {\begin{array}{*{20}c} {{p - 1}} \\ {{{{\left( {p - 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {p - 1} \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right)}{\left( {\begin{array}{*{20}c} {{q - 1}} \\ {{{{\left( {q - 1} \right)}} \mathord{\left/ {\vphantom {{{\left( {q - 1} \right)}} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right)}{\left( {\bmod pq} \right)}, $$

for any distinct odd primes p and q. Meanwhile, we discuss the connections with the prime recognitions.