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  "body": "\n## Statement\n\nFor every integer $k \\ge 1$, there are positive integers $a$ and $d$ such that the $k$ numbers $a, a + d, a + 2d, \\dots, a + (k-1)d$ are all [[prime numbers|01826KMG2H]].\n\n## Remarks\n\nGreen and Tao proved more: every subset $A$ of the set $\\mathcal{P}$ of primes with\n$$\\limsup_{N \\to \\infty} \\frac{\\abs{A \\cap \\set{1, \\dots, N}}}{\\abs{\\mathcal{P} \\cap \\set{1, \\dots, N}}} \u003e 0$$\ncontains arithmetic progressions of every length.\n[[Szemerédi's theorem|B23B0QEMN5]] does not apply directly, because the primes have density zero by the [[prime number theorem|0EHBY5Y84N]].\nThe proof is a transference principle.\nA relative Szemerédi theorem holds for dense subsets of a pseudorandom set.\nThe primes sit densely inside such a set, built from sieve weights of Goldston and Yıldırım.\nFor example, $5, 11, 17, 23, 29$ is a progression of five primes with difference $6$.\nThe primes contain no infinite arithmetic progression: for $a \\ge 2$ the term $a + ad = a(1 + d)$ is composite, and for $a = 1$ the first term is not prime.\n\n## Explanations\n\n### L1 human\n\nYou can find evenly spaced prime numbers in long rows, as long as you like.\nFor example, $3, 5, 7$ is a row of three and $5, 11, 17, 23, 29$ is a row of five.\nThe theorem says rows of every length exist, even though primes get rarer.\n\n### L2 human\n\nThe Green–Tao theorem combines [[additive combinatorics|3NM362A2JG]] with analytic number theory.\nIts idea is that Szemerédi's theorem survives inside sparse but random-looking sets.\nIt is a special case of the open [[Erdős conjecture on arithmetic progressions|FN978TDX67]], since the sum of the reciprocals of the primes diverges.\n",
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