(a) It is an interesting fact that if p is a prime number such that p- 1 is divi

Question

(a) It is an interesting fact that if

p is a prime number such that p- 1 is divisible by 4, then p can be written as the sum of two squares. (By a square, we mean the square of an integer.) Find a complex number z whose real and imaginary parts are both integers and such that zz(bar)= 29.

I know zz= a2 + b2

(b) Find four complex numbers

z with the property that Re(z), Im(z),Re(z-1) and Im(z-1

) are all integers, where Re and Im denote the real and imaginary parts respectively of a complex number.

Explanation / Answer

a) z=5+2i b)5+2i, 5-2i,4+2i,4-2i

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