2023 AIME I Problem 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.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.