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

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

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:

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

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.