### 35th Putnam 1974

Problem B3

Prove that (cos-1(1/3) )/π is irrational.

Solution

Let x = cos-1(1/3). If x = m/n π for some integers m, n, then cos nx = cos mπ = ±1. But we show that cos nx cannot be ±1. It follows that x/π must be irrational as required.

As usual, we have cos nx = nC0 cn - nC2 cn-2s2 + nC4 cn-4s4 - ... , where c = cos x, s = sin x. We may put s2 = 1 - c2 to get cos nx = a polynomial of degree n in c with integer coefficients. The coefficient of cn = nC0 + nC2 + nC4 + ... = 2n-1. But c = 1/3, so cos nx = 2n-1/3n + k/3n-1 = (2n-1 + 3k)/3n for some integer k. This must be in its lowest terms since 2n-1 is not divisible by 3. In particular, it cannot be ±1.

[A variant on this is to consider cos(2nx). By a simple induction using cos 2y = 2 cos2y - 1, we show that cos(2nx) = an/bn, where an is not a multiple of 3 and bnis 3 to the power of 2n. It follows that as n runs through the natural numbers, all the values cos(2nx) are distinct. But if x/π was rational, there would only be finitely many distinct values.]