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  "body": "\n## Statement\n\nLet $n$ be a positive integer.\nThen there are infinitely many [[prime numbers|0SN0MAEKYX]] $p$ such that $p \\equiv 1 \\pmod{n}$.\n\n## Remarks\n\nThis is the case $a = 1$ of [[Dirichlet's theorem on arithmetic progressions|0GNVN3CA2P]].\nThe case $a = 1$ has an elementary proof with cyclotomic polynomials. A prime factor $p$ of $\\Phi_n(m)$ that does not divide $n$ satisfies $p \\equiv 1 \\pmod n$.\nFor example, the primes $p \\equiv 1 \\pmod{10}$ start with $11, 31, 41, 61, 71$.\n\n## Explanations\n\n### L1 human\n\nFix a number $n$, for example $n = 6$. Then infinitely many primes leave the remainder $1$ when you divide them by $6$.\nThe first ones are $7, 13, 19, 31, 37$.\n",
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