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 = 301 mod 511 and so 9 is composite

References

• Kilford, L.J.P. (2004). "A generalization of a necessary and sufficient condition for primality due to Vantieghem". Int. J. Math. Math. Sci. (69-72): 3889–3892. arXiv:math/0402128. Zbl 1126.11307.. An article with proof and generalizations.
• Vantieghem, E. (1991). "On a congruence only holding for primes". Indag. Math., New Ser. 2 (2): 253–255. Zbl 0734.11003.