Vantieghem's theorem

The vantieghems theorem is a prime criterion of number theory . It says that a natural number n is prime if and only if

${\displaystyle \prod _{1\leq k\leq n-1}\left(2^{k}-1\right)\equiv n\mod \left(2^{n}-1\right).}$

Similarly, n is prime if and only if the following congruence of polynomials in X holds:

${\displaystyle \prod _{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n-\left(X^{n}-1\right)/\left(X-1\right)\mod \left(X^{n}-1\right)}$

or:

${\displaystyle \prod _{1\leq k\leq n-1}\left(X^{k}-1\right)\equiv n\mod \left(X^{n}-1\right)/\left(X-1\right).}$

literature

• LJP Kilford: A generalization of a congruence due to Vantieghem only holding for primes. 2004, arxiv : math / 0402128 . An article with evidence for this prime number criterion.