Question

(a) If x = 1 + i √3 : y = 1 – i √3 & z = 2, prove that x^p + y^p = z^p for every prime p > 3. 

(b) Find the sum of the series 1(2– ω) (2– ω²) + 2(3– ω) (2– ω²) ……… (n – 1) (n – ω) (n – ω² ) where ω is one of the imaginary cube root of unity. 

Answer 1

x = 1 + i√3 y = 1 – i √3 Z = 2

x = –2wy = –2w²Z = 2

x^p + y^pp > 3 prime (P is odd)

= –2^pw^p – 2^pw^2p = –2^p (w^p + w^2p) = 2^p = z^p