An extension of Babbage’s criterion for primality

Mathematica Slovaca

Romeo Meštrović¹ 

关键词: Babbage’s criterion for primality;    Lucas’ theorem;    congruence;    prime power;   

DOI  :  10.2478/s12175-013-0164-8

学科分类：数学（综合）

来源: Slovenska Akademia Vied * Matematicky Ustav / Slovak Academy of Sciences, Mathematical Institute

【 摘 要 】

Let n > 1 and k > 1 be positive integers. We show that if $$left( {egin{array}{*{20}c} {n + m} \ n \ end{array} } ight) equiv 1 (mod k)$$ 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.

【 授权许可】

Unknown