## Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?

Let S be the multiplicative semigroup of numbers generated by B={2cos(πn)∣n≥4}. Question: Does every number of S factorize uniquely (up to perm.) as a product of elements in B? Note that 2cos(πn)=eiπn+e−iπn, so we can reformulate the question as follows. Let 4≤n1≤⋯≤nr and 4≤m1≤⋯≤ms such that ∑(ϵ1,…,ϵr)∈{±1}reiπ(∑rk=1ϵknk)=∑(ϵ1,…,ϵs)∈{±1}seiπ(∑sk=1ϵkmk) Then, is it true that r=s and nk=mk,∀k? … Read more