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Friday, 27 March 2015

Key ingredient for the cyclotomic primality test

Let n be a natural number. Let q be a prime not dividing n, let r be a power of a prime number l not dividing n and let map the character \chi:(\mathbb{Z}/q\mathbb{Z})^*\to\mu_r.
If:
  1. for every prime p dividing n there exists \lambda_p in l-adic integers such that p^{l-1}=n^{(l-1)\lambda_p} in multiplicative group of l-adic integers;
  2. the Gaussian sum \tau(\chi) satisfies \tau(\chi)^{\sigma_n-n}\in \langle\xi_r\rangle, \text{ in the ring } \mathbb{Z}[\xi_r,\xi_q]/(n) . there \langle\xi_r\rangle= cyclic subgroup of (\mathbb{Z}[\xi_r]/(n))^* of order r generated by  \xi_r.
Then we have \chi(p)= \chi(n)^{\lambda_l} for every prime divisor p of n.
By \mu_r we denote subgroup of r-th roots of unity of \bar{\mathbb{Q}}^*.

Four primality testing algorithms

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