Competition problem — Number theory

Library / /354 of 520

Number theory Difficulty 6.3 National olympiad Prove it

Theorem 5 Let $m \geqslant 1$ . The reduced residue system modulo $m$ can be represented as
$g^{0}, g^{1}, \cdots, g^{\rho(m)-1},$

where $g$ is some integer, if and only if $m=1,2,4, p^{\alpha}, 2 p^{\alpha}(p$ an odd prime, $\alpha \geqslant 1)$ , i.e., if and only if there exists a primitive root modulo $m$ .

Solution

None

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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.