There exists a complex number of the form z = x + yi, where x and y are positive integers, such that z^3 = -74 + ci, for some integer c. Find z.
\(|z^3|^2 = ||z|^2|^3 = 74^2 + c^2 = (x^2+y^2)^3\\ \text{Using software I get }x=1,~y=5,~c=-110\\ z = 1 +5i\)
z^3 = x^3 - iy^3 + 3xyi(x+yi) = x^3 - 3xy^2 + 3x^2yi-iy^3
Re(z^3) = x(x^2-3y^2)
x(x^2-3y^2) = 74.
Considering all possible values of x,
(x, y) = (1, 5).
c = Im(z^3) = 3(5) - 5^3 = -110.
