Ostrowski–Hadamard gap theorem

In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n is prime if and only if

$\prod _{1\le k\le n-1}(2^k-1)\equiv nmod(2^n-1).$

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

$\prod _{1\le k\le n-1}(X^k-1)\equiv n-(X^n-1)/(X-1)mod(X^n-1)$

or:

$\prod _{1\le k\le n-1}(X^k-1)\equiv nmod(X^n-1)/(X-1).$

References

• L. J. P. Kilford, A generalization of a congruence due to Vantieghem only holding for primes, 2004, Template:Arxiv. An article with proof and generalizations.