{
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  "body": "\n## Statement\n\nLet $a$ and $d$ be positive integers that are [[coprime|0GFKP8F8AG]].\nThen there are infinitely many [[prime numbers|0SN0MAEKYX]] of the form $a + n d$, with $n$ a natural number.\nThat is, there are infinitely many primes $p$ with $p \\equiv a \\pmod{d}$.\n\n## Remarks\n\nThe hypothesis $\\gcd(a, d) = 1$ is necessary. Each term of $a + n d$ is a multiple of $\\gcd(a, d)$, so at most one term can be prime when $\\gcd(a, d) \u003e 1$.\nFor $d = 4$, the theorem gives infinitely many primes $p \\equiv 1 \\pmod 4$ and infinitely many primes $p \\equiv 3 \\pmod 4$.\nA special case with an elementary proof is that there are [[infinitely many primes congruent to 1 modulo n|0T1BSD7H3M]].\nThe primes are in fact equally distributed among the $\\varphi(d)$ possible classes, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\n\n## History\n\nDirichlet proved the theorem in 1837. His proof introduced Dirichlet characters and Dirichlet $L$-functions.\n\n## Explanations\n\n### L1 human\n\nThe sequence $3, 7, 11, 15, 19, 23, \\dots$ adds $4$ at each step. It contains infinitely many primes.\nIn general, a sequence $a, a + d, a + 2d, \\dots$ contains infinitely many primes, unless $a$ and $d$ have a common factor that blocks them all.\n",
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  "name": "Dirichlet's theorem on arithmetic progressions",
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