Mathegoras

Complex numbers › Roots of unity and complex roots › Counting complex solutions

Counting z with integer parts of f(z)

High school · ●●●●○ 100 pts

Let f:C→Cf : \mathbb{C} \to \mathbb{C} be defined by f(z)=z2+iz+1f(z) = z^2 + iz + 1. How many complex numbers zz are there such that Im(z)>0\text{Im}(z) > 0 and both the real and the imaginary parts of f(z)f(z) are integers with absolute value at most 1010?

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quadratic equationscountinglattice points