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  "body": "\n## Statement\n\nLet $p_n$ denote the $n$-th [[prime number|01826KMG2H]].\nThere is a constant $H$ such that $p_{n+1} - p_n \\le H$ for infinitely many $n$.\nEquivalently,\n$$\\liminf_{n \\to \\infty} \\, (p_{n+1} - p_n) \u003c \\infty,$$\nwhere $\\liminf$ is the [[limit inferior|033RZ32T4Q]].\n\n## Remarks\n\nYitang Zhang proved the statement in 2013 with $H = 7 \\cdot 10^7$.\nJames Maynard, and independently Terence Tao, gave a simpler proof with $H = 600$, and the Polymath project lowered the bound to $H = 246$.\nMaynard's method also shows that for every $m \\ge 1$, $\\liminf_{n \\to \\infty} (p_{n+m} - p_n) \u003c \\infty$.\nThe proof builds on the method of Goldston, Pintz and Yıldırım: a [[Selberg sieve|5252P3NHT7]] with multidimensional weights detects a tuple $n + h_1, \\ldots, n + h_k$ that contains at least two primes.\nThe input on primes in arithmetic progressions is the [[Bombieri–Vinogradov theorem|B3JMY54GJF]] in Maynard's version and a strengthening of it for smooth moduli in Zhang's version.\nThe [[twin prime conjecture|0FER36N8SB]] is the claim that $H = 2$ works; it remains open.\n\n## Explanations\n\n### L1 human\n\nThe primes become rarer as numbers grow: on average the gap after $p$ is about $\\log p$.\nStill, this theorem says that the primes keep coming in close pairs forever: some fixed distance $H$ is reached again and again.\nThe twin prime conjecture predicts the smallest possible distance $H = 2$, as in $11, 13$ or $41, 43$; the best proven value is $246$.\n\n### L2 human\n\nThe statement is the first unconditional step towards the twin prime conjecture.\nGoldston, Pintz and Yıldırım had shown that bounded gaps would follow from a level of distribution of the primes greater than $1/2$, which is beyond the [[Bombieri–Vinogradov theorem|B3JMY54GJF]].\nZhang proved such a level for a restricted class of moduli; Maynard changed the sieve weights so that the level $1/2$ is already enough.\nUnder the Elliott–Halberstam conjecture and its generalisations, the method gives smaller values of $H$, down to $6$.\n",
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