{
  "aliases": [
    "unique-factorization-theorem",
    "unique-prime-factorization-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0C3PN4075R"
    },
    {
      "field": "uses",
      "target": "0N10YRNNWV"
    }
  ],
  "body": "\n## Statement\n\nEach positive integer $n$ has exactly one [[prime factorization|0AEKDFCYFA]].\nThat is, there are [[primes|0SN0MAEKYX]] $p_1 \\le p_2 \\le \\dots \\le p_k$ with $n = p_1 p_2 \\cdots p_k$.\nIf also $n = q_1 q_2 \\cdots q_l$ with primes $q_1 \\le q_2 \\le \\dots \\le q_l$, then $k = l$ and $p_i = q_i$ for each $i$.\n\n## Remarks\n\nThe [[existence of prime factorization|020K0X7HVR]] gives the first part. The uniqueness follows from [[Euclid's lemma|051VYKA1EX]].\nThe statement uses the convention that $1$ is not prime. If $1$ were prime, $6 = 2 \\cdot 3 = 1 \\cdot 2 \\cdot 3$ would have two factorizations.\nUnique factorization fails in some rings. In $\\mathbb{Z}[\\sqrt{-5}]$, $6 = 2 \\cdot 3 = (1 + \\sqrt{-5})(1 - \\sqrt{-5})$ gives two different factorizations into irreducible elements.\n\n## History\n\nEuclid's Elements contains the main steps of the proof, in Book VII, Propositions 30 to 32, and in Book IX, Proposition 14.\nGauss proved the uniqueness in his Disquisitiones Arithmeticae of 1801.\n\n## Explanations\n\n### L1 human\n\nEach positive integer has one and only one recipe as a product of primes.\nYou can multiply the primes in any order, but the list of primes stays the same.\nFor example, $12 = 2 \\cdot 2 \\cdot 3$, and no other list of primes has the product $12$.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "aliases": [
      "unique-factorization-theorem",
      "unique-prime-factorization-theorem"
    ],
    "kind": "statement",
    "msc": [
      "11A51"
    ],
    "name": "Fundamental theorem of arithmetic",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q670235",
      "wikipedia": "Fundamental theorem of arithmetic"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic"
      }
    ],
    "tag": "084HH1NDCB",
    "uses": [
      "0AEKDFCYFA",
      "020K0X7HVR",
      "051VYKA1EX"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/084HH1NDCB/",
  "links": [
    {
      "field": "uses",
      "target": "0AEKDFCYFA"
    },
    {
      "field": "uses",
      "target": "020K0X7HVR"
    },
    {
      "field": "uses",
      "target": "051VYKA1EX"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/084HH1NDCB.md",
  "msc": [
    "11A51"
  ],
  "name": "Fundamental theorem of arithmetic",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/084/084HH1NDCB.md",
  "status": null,
  "tag": "084HH1NDCB",
  "uri": "m2://084HH1NDCB",
  "url": "https://math2.tamnd.com/wiki/fundamental-theorem-of-arithmetic/",
  "writing": {
    "tag": "084HH1NDCB",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}