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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.

 Apr 16, 2019
 #1
avatar+5655 
+2

\(|z^3|^2 = ||z|^2|^3 = 74^2 + c^2 = (x^2+y^2)^3\\ \text{Using software I get }x=1,~y=5,~c=-110\\ z = 1 +5i\)

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 Apr 16, 2019
 #2
avatar+7709 
+1

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.

 Apr 18, 2019

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