Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Find the answer

9. Use primitive roots to find all cubic and quartic residues modulo $m$ :
$m=13,17,19,23,41,43,17^{2}, 23^{2}, 41^{2}, 43^{2} .$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

9. Using the first formula of Theorem 2, we can write down all cubic and quartic residues.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.