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A generalization of a necessary and sufficient condition for primality due to Vantieghem. (English) Zbl 1126.11307

The author present a family of congruences which hold if and only if a natural number \(n\) is prime. For any positive integers \(m\) and \(n\), the congruence \[ \prod_{d=1}(1-m^d)\equiv n\pmod{\frac{m^n-1}{m-1}} \] is shown to hold if and only if \(n\) is prime, thus generalizing a result of E. Vantieghem [Indag. Math., New Ser. 2, No. 2, 253–255 (1991; Zbl 0734.11003)]

MSC:

11A51 Factorization; primality
11A07 Congruences; primitive roots; residue systems

Citations:

Zbl 0734.11003
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