Shifting nth root algorithm not useful for computing roots with a computer, as it is always outperformed by much simpler binary search, and has the same memory complexity. 1. 0 1 1 0 1 1. 7 3 2 0 5 1. 7 0 9 9 7 1. 6 2 6 5 7 Shifting nth root algorithm The shifting "n"th root algorithm is an algorithm for extracting the "n"th root of a positive real number which proceeds iteratively by shifting in "n" digits of the radicand, starting with the most significant, and produces one digit of the root on each iteration, in a manner similar
Root of unity theorem. Every th root of unity is a primitive th root of unity for exactly one positive divisor of . This implies that This formula represents the factorization of the polynomial into irreducible factors. Applying Möbius inversion to the formula gives where is the Möbius function. So the first few cyclotomic polynomials are If is a prime number, then all the th roots of unity except 1 are primitive th roots, and we have Substituting any positive integer ≥ 2 for , this sum becomes a base repunit. Thus a necessary (but not sufficient) condition for a repunit to be