Let N = 2K+1 for some positive integer K. Evaluate explicitly

f(N) := Product (5 - 4*cos(2*j*pi/N))

where j runs through 1,2,...,K, and prove your answer.

over R , the polynomial x^(2K) + x^(2K-1) + ... + x^2 + x + 1 factors as Product (x^2 - 2 cos(2 j pi / (2K + 1)) x + 1) , the product being for j = 1, 2, ... , K . this is easy to see by factoring the polynomial completely over C , and then pairing up complex conjugate factors. now take x = 2 to get the answer: 2^(2K+1) - 1 . mike