Pascal's triangle and constructible polygons

Abstract

For any positive integer k let, φ(k) be the Euler totient function of k. In this note, we find all binomial coefficients which can be the number of sides of a regular polygon which can be constructed using only the ruler and the compass; that is, we find all solutions of the equation (matrix presented) where k, n, α are positive integers such that n ≥ 2k.