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  "body": "\n## Statement\n\nLet $T$ be the map on the positive integers given by $T(n) = n/2$ if $n$ is even and $T(n) = 3n + 1$ if $n$ is odd.\nFor every positive integer $n$, there is an integer $k \\ge 0$ with $T^k(n) = 1$, where $T^k$ is the $k$-fold iterate of $T$.\n\n## Remarks\n\nThe conjecture is open.\nIt has been checked by computer for all starting values up to very large bounds.\nRiho Terras showed in 1976 that the set of $n$ having some iterate $T^k(n) \u003c n$ has [[natural density|RVG9Z8NYA6]] $1$.\nTerence Tao showed in 2019 that almost all orbits, in the sense of logarithmic density, reach values below $f(n)$ for any function $f$ that tends to infinity.\nFor example, starting from $6$ the [[sequence|0G9EKM0E15]] is $6, 3, 10, 5, 16, 8, 4, 2, 1$.\nStarting from $27$ it takes $111$ steps to reach $1$ and climbs as high as $9232$ on the way.\nOnce the sequence reaches $1$, it cycles through $1, 4, 2, 1$; the conjecture says that this is the only cycle and that no orbit grows without bound.\n\n## Explanations\n\n### L1 human\n\nPick a positive whole number.\nIf it is even, halve it; if it is odd, triple it and add one.\nRepeat.\nThe conjecture says that you always end up at $1$, whatever number you start from.\nThe rule is simple enough for a child, yet nobody has found a proof, and many think current methods are not strong enough.\n",
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    "history": {
      "first_stated": "1937",
      "posed_by": "Lothar Collatz"
    },
    "kind": "conjecture",
    "msc": [
      "11B83",
      "11B37"
    ],
    "name": "Collatz conjecture",
    "same_as": {
      "wikidata": "Q837314",
      "wikipedia": "Collatz conjecture"
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    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Collatz_conjecture"
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    "tag": "6PCQEXEX4N",
    "uses": [
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      "0G9EKM0E15",
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/6PCQEXEX4N/",
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  "markdown": "https://math2.tamnd.com/node/6PCQEXEX4N.md",
  "msc": [
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    "11B37"
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  "name": "Collatz conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/6PC/6PCQEXEX4N.md",
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  "tag": "6PCQEXEX4N",
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  "url": "https://math2.tamnd.com/wiki/collatz-conjecture/",
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    "tag": "6PCQEXEX4N",
    "kind": "conjecture",
    "level": "W0",
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        "have": 0,
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