MathDB

Let

k

be a positive integer, let m\equal{}2^k\plus{}1, and let

r\neq 1

be a complex root of z^m\minus{}1\equal{}0. Prove that there exist polynomials

P(z)

and

Q(z)

with integer coefficients such that (P(r))^2\plus{}(Q(r))^2\equal{}\minus{}1.

Problem Statement