Vantieghems theorem

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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)


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


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


  • 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.