{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "084HH1NDCB"
    }
  ],
  "body": "\n## Statement\n\nEach positive integer $n$ has a [[prime factorization|0AEKDFCYFA]].\nThat is, there are [[prime numbers|0SN0MAEKYX]] $p_1, \\dots, p_k$, with $k \\ge 0$, such that $n = p_1 p_2 \\cdots p_k$.\n\n## Remarks\n\nThe proof uses strong induction on $n$. If $n = 1$, take $k = 0$. If $n$ is prime, take $k = 1$. Otherwise $n = a b$ with $1 \u003c a, b \u003c n$, and we join the factorizations of $a$ and $b$.\nThe [[fundamental theorem of arithmetic|084HH1NDCB]] adds that the factorization is unique, up to the order of the factors.\n\n## Explanations\n\n### L1 human\n\nSplit each number that is not prime into two smaller factors. Continue until all the pieces are prime.\nThe pieces always get smaller, so the splitting stops.\nFor example, $84 = 2 \\cdot 42 = 2 \\cdot 2 \\cdot 21 = 2 \\cdot 2 \\cdot 3 \\cdot 7$.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "11A51"
    ],
    "name": "Existence of prime factorization",
    "role": "theorem",
    "sources": [
      {
        "locator": "Existence",
        "url": "https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"
      }
    ],
    "tag": "020K0X7HVR",
    "uses": [
      "0AEKDFCYFA",
      "0SPBEDVNW1"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/020K0X7HVR/",
  "links": [
    {
      "field": "uses",
      "target": "0AEKDFCYFA"
    },
    {
      "field": "uses",
      "target": "0SPBEDVNW1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/020K0X7HVR.md",
  "msc": [
    "11A51"
  ],
  "name": "Existence of prime factorization",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/020/020K0X7HVR.md",
  "status": null,
  "tag": "020K0X7HVR",
  "uri": "m2://020K0X7HVR",
  "url": "https://math2.tamnd.com/wiki/existence-of-prime-factorization/",
  "writing": {
    "tag": "020K0X7HVR",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}