# Vantieghems theorem

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

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

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

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

or:

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

## Example

Let n=7 forming the product 1*3*7*15*31*63 = 615195. 615195 = 7 mod 127 and so 7 is prime
Let n=9 forming the product 1*3*7*15*31*63*127*255=19923090075. 19923090075 != 9 mod 511 and so 9 is composite

## 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.