I am currently in high school and we just finished learning about complex numbers. While looking at a problem called the Centrifuge problem: https://mattbaker.blog/2018/06/25/the-balanced-centrifuge-problem/. I came across 2 conjectures and I wanted to know how the roots of unity are related to linear combinations of prime numbers:
this is the conjecture:
For any integers $n ≥ 2$ and $1 ≤ k ≤ n$, one may find k distinct $n^{\rm th}$ roots of unity whose sum is 0 if and only if both $k$ and $n − k$ are expressible as linear combinations of prime factors of $n$ with nonnegative coefficients.
- can someone explain this relation?
- is there a proof for this which can be understood using elementary knowledge of number theory and complex numbers.