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# Find a complex number z such that the real part and imaginary part of z are both integers, and such that zz = 89

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Find a complex number z such that the real part and imaginary part of z are both integers, and such that zz = 89

Oct 29, 2017

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Find a complex number z such that the real part and imaginary part of z are both integers,

and such that zz = 89

I assume, $$z\bar{z} = 89$$

89 has one solution, because 89 is a prime number and $$89 = 1 \pmod{4}$$

$$89 = 5^2 + 8^2$$

so

$$z = 5 + 8i \text{ and } \bar{z} = 5-8i \text{ so } z\bar{z} = 5^2+8^2 = 89\\ \text{or} \\ z= 8 + 5i \text{ and } \bar{z} = 8-5i \text{ so } z\bar{z} = 8^2+5^2 = 89$$

Oct 30, 2017