Vantieghems theorem

In number theory, Vantieghems theorem is a primality criterion. It states that a natural number n≥3 is prime if and only if

[math]\displaystyle{ \prod_{1 \leq k \leq n-1} \left( 2^k - 1 \right) \equiv n \mod \left( 2^n - 1 \right). }[/math]

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

[math]\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) }[/math]

or:

[math]\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). }[/math]

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. 2004 (69–72): 3889–3892. doi:10.1155/S0161171204403226. Bibcode: 2004math......2128K. . An article with proof and generalizations.
• Vantieghem, E. (1991). "On a congruence only holding for primes". Indag. Math.. New Series 2 (2): 253–255. doi:10.1016/0019-3577(91)90013-W.