An extension of Babbage's criterion for primality

Romeo Meštrović
2013 Mathematica Slovaca  
AbstractLet n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n − 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.
doi:10.2478/s12175-013-0164-8 fatcat:rxxzp5nadbdr3pmuhwljr3c35y