Question 15
Find the largest prime number $p \lt 1000$ for which there exists a complex number $z$ satisfying
• the real and imaginary part of $z$ are both integers;
• $|z| = \sqrt{p},$ and
• there exists a triangle whose three side lengths are $p,$ the real part of $z^3,$ and the imaginary part of $z^3.$
• the real and imaginary part of $z$ are both integers;
• $|z| = \sqrt{p},$ and
• there exists a triangle whose three side lengths are $p,$ the real part of $z^3,$ and the imaginary part of $z^3.$