<p>Find the largest prime number <img alt="$p<1000$" height="17" src="https://latex.artofproblemsolving.com/a/e/6/ae61db26a6250a8d16b8528d5db5a4cd137b41de.png" width="72" /> for which there exists a complex number <img alt="$z$" height="10" src="https://latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png" width="11" /> satisfying</p> <ul> <li>the real and imaginary part of <img alt="$z$" height="10" src="https://latex.artofproblemsolving.com/b/1/3/b13f21416d84e13708696f34dea81026cda583c9.png" width="11" /> are both integers;</li> <li><img alt="$|z|=\sqrt{p},$" height="21" src="https://latex.artofproblemsolving.com/b/d/f/bdf15a55b497216b8a54bab09331495376ae42d3.png" width="75" /> and</li> <li>there exists a triangle whose three side lengths are <img alt="$p,$" height="11" src="https://latex.artofproblemsolving.com/9/c/6/9c699725218958e044a4f271042b0b40b9ed713f.png" width="14" /> the real part of <img alt="$z^{3},$" height="18" src="https://latex.artofproblemsolving.com/8/b/f/8bf3c8309a5f00c95e1794cdefd4b164ee95142a.png" width="20" /> and the imaginary part of <img alt="$z^{3}.$" height="17" src="https://latex.artofproblemsolving.com/9/d/5/9d57c6551a6972f3de7fa7d6f8bbbf27086e155d.png" width="24" /></li> </ul>

Question No: 342

Find the largest prime number $p<1000$ 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 $z^{3}.$

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Tags: Algebra Geometry

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