1. ## Complex Numbers

(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(this z has a bar above it) = 29.

I know zz(bar)= a^2 + b^2 so,
29=a^2 +b^2
but i dont know where to go from there

(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.

2. Originally Posted by Arita
(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(this z has a bar above it) = 29.

I know zz(bar)= a^2 + b^2 so,
29=a^2 +b^2
but i dont know where to go from there

(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.

If $z=x+yi\,,\,\,x,y,\in\mathbb{R}\,,\,\,then\,\,z\ove rline{z}=x^2+y^2=29=5^2+2^2\Longrightarrow$

$z=5+2i\,,\,\,or\,\,z=-2+5i\,,\,\,or\,\,z=-5-2i$ , etc.

Tonio

3. Thanks for the help! Do you know how I would go about solving the second part of the question?

4. Originally Posted by Arita
Thanks for the help! Do you know how I would go about solving the second part of the question?

Take any of the examples from the first part of my answer....or any other complex number with integer real and imaginary parts.

Tonio

5. Originally Posted by Arita
(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(this z has a bar above it) = 29.

I know zz(bar)= a^2 + b^2 so,
29=a^2 +b^2
but i dont know where to go from there

(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.
The "real part" of z- 1 is just the (real part of z)-1 and the "imaginary part" of z- 1 is just the imaginary part of z. And a x-1 is integer if and only if x is so this problem is just asking for four complex numbers whose real and imaginary parts are integers.

As tonio said, just pick four numbers, A+ Bi, where A and B are integers.

6. This one says Re(z-1) and Im(z-1),
My question is slighty different with Re(z^(-1)) and Im(z^(-1))
7. There are only two integers, $x$, such that $x^{-1}$ is also an integer. What are they?