A necessary and sufficient condition for primality, and its source (Q2553461)

Consider Pascal's triangle with each row beginning two places to the right from the preceding row. The entry in the \(n\)th row and \(k\)th column is then the binomial coefficient \(\binom{n}{k-2n}\), \(k,n=0,1,\ldots\). The authors show that \(k\) is a prime if and only if all entries in the \(k\)th column are divisible by their row number \(n\). Hence, if \(\binom{n}{k-2n}\equiv 0\bmod n\) for all \(n\).