Vantieghems theorem

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

:$prod\_\{1\; leq\; k\; leq\; n-1\}\; left(\; 2^k\; -\; 1\; ight)\; equiv\; n\; mod\; left(\; 2^n\; -\; 1\; ight).$

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

:$prod\_\{1\; leq\; k\; leq\; n-1\}\; left(\; X^k\; -\; 1\; ight)\; equiv\; n-\; left(\; X^n\; -\; 1\; ight)/left(\; X\; -\; 1\; ight)\; mod\; left(\; X^n\; -\; 1\; ight)$

or:

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

References

* L. J. P. Kilford, "A generalization of a congruence due to Vantieghem only holding for primes", 2004, arxiv|math|0402128. An article with proof and generalizations.