let-give-w-e-i-2pi-n-and-S-k-0-n-1-w-k-2-1-prove-that-S-k-0-n-1-w-q-k-2-2-find-S-

Question Number 28546 by abdo imad last updated on 26/Jan/18

l e t g i v e w = e^{i \frac{2 π}{n}} a n d S = \sum_{k = 0}^{n - 1} w^{k^{2}}

1) p r o v e t h a t S = \sum_{k = 0}^{n - 1} w^{{(q + k)}^{2}}

2) f i n d ∣ S ∣ .