Vantieghems theorem

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In number theory, Vantieghem's theorem is a primality criterion. It states that a natural number n≥3 is prime if and only if ∏ 1 ≤ k ≤ n − 1 ( 2 k − 1 ) ≡ n mod ( 2 n − 1 ) . {\displaystyle \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: ∏ 1 ≤ k ≤ n − 1 ( X k − 1 ) ≡ n − ( X n − 1 ) / ( X − 1 ) mod ( X n − 1 ) {\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)} or: ∏ 1 ≤ k ≤ n − 1 ( X k − 1 ) ≡ n mod ( X n − 1 ) / ( X − 1 ) . {\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).}