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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nA primitive root modulo $n$ is an integer $g$ coprime to $n$ whose [[multiplicative order|0SBRX5CZ5R]] modulo $n$ is $\\varphi(n)$, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\n\n## Remarks\n\nEquivalently, $g$ is a primitive root modulo $n$ when the powers $g, g^2, \\dots, g^{\\varphi(n)}$ represent all the residue classes coprime to $n$.\nSo a primitive root exists exactly when the group $(\\mathbb{Z}/n\\mathbb{Z})^\\times$ is cyclic, and then $\\overline{g}$ generates this group.\nFor example, $3$ is a primitive root modulo $7$: its powers are $3, 2, 6, 4, 5, 1$ modulo $7$.\nThere is no primitive root modulo $8$: each odd $a$ satisfies $a^2 \\equiv 1 \\pmod 8$, but $\\varphi(8) = 4$.\n\n## Explanations\n\n### L1 human\n\nA primitive root is a number whose powers run through all the residues coprime to $n$ before they come back to $1$.\nModulo $5$, the powers of $2$ are $2, 4, 3, 1$. So $2$ is a primitive root modulo $5$, but $4$ is not, since $4^2 \\equiv 1$.\n",
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