Find the greatest $n$ such that $(z+1)^n = z^n + 1$ has all its non-zero roots in the unitary circumference, e.g. $(\alpha+1)^n = \alpha^n + 1, \alpha \neq 0$ implies $|\alpha| = 1.$
n equals 7. For $n=7$ it is not difficult to see that all roots have modulus 1 ( use well-known trick to factorize $ (x+y)^7-x^7-y^7$). To prove this one is best, use Gauss theorem: the root of the derivative of a polynomial is in the convex hull of the roots of the polynomial. Thus, if all roots of a polynomial have modulus at most 1, so does the derivative of the polynomial. Just compute the derivative and find it's roots.