{
  "aliases": [
    "lte-lemma"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $p$ be an odd prime, let $x$ and $y$ be integers with $p \\nmid x$, $p \\nmid y$ and $p \\mid x - y$, and let $n$ be a positive integer.\nThen $\\nu_p(x^n - y^n) = \\nu_p(x - y) + \\nu_p(n)$, where $\\nu_p$ is the [[p-adic valuation|04VCPS30ZE]].\n\n## Remarks\n\nFor $p = 2$ the formula needs a stronger hypothesis. If $x$ and $y$ are odd and $4 \\mid x - y$, then $\\nu_2(x^n - y^n) = \\nu_2(x - y) + \\nu_2(n)$.\nFor $p = 2$, odd $x$ and $y$, and even $n$, the general formula is $\\nu_2(x^n - y^n) = \\nu_2(x - y) + \\nu_2(x + y) + \\nu_2(n) - 1$.\nFor example, $\\nu_3(10^9 - 1) = \\nu_3(9) + \\nu_3(9) = 4$, and indeed $10^9 - 1 = 3^4 \\cdot 37 \\cdot 333667$.\n\n## History\n\nÉdouard Lucas had some of the ideas by 1878. Mihai Manea published the modern form in 2006.\n\n## Explanations\n\n### L1 human\n\nWhen $p$ divides $x - y$, the lemma tells you exactly how many factors $p$ are in $x^n - y^n$.\nYou add the factors $p$ of $x - y$ and the factors $p$ of $n$.\nOlympiad problems often use the lemma.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "lte-lemma"
    ],
    "kind": "statement",
    "msc": [
      "11A51"
    ],
    "name": "Lifting-the-exponent lemma",
    "role": "lemma",
    "same_as": {
      "wikidata": "Q97704248",
      "wikipedia": "Lifting-the-exponent lemma"
    },
    "sources": [
      {
        "locator": "Statements",
        "url": "https://en.wikipedia.org/wiki/Lifting-the-exponent_lemma"
      }
    ],
    "tag": "0JYF6PTKEW",
    "uses": [
      "04VCPS30ZE"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0JYF6PTKEW/",
  "links": [
    {
      "field": "uses",
      "target": "04VCPS30ZE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0JYF6PTKEW.md",
  "msc": [
    "11A51"
  ],
  "name": "Lifting-the-exponent lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0JY/0JYF6PTKEW.md",
  "status": null,
  "tag": "0JYF6PTKEW",
  "uri": "m2://0JYF6PTKEW",
  "url": "https://math2.tamnd.com/wiki/lifting-the-exponent-lemma/",
  "writing": {
    "tag": "0JYF6PTKEW",
    "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
      }
    ]
  }
}