{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "3601Z7V838"
    }
  ],
  "body": "\n## Summary\n\nA Thue equation is a [[Diophantine equation|08B1RTW7VZ]] $f(x, y) = m$ with $m$ a nonzero integer.\nHere $f$ is an irreducible homogeneous polynomial in two variables, with integer coefficients and degree at least $3$.\nAxel Thue proved in 1909 that such an equation has only finitely many integer solutions.\nHis proof came from a sharpening of Liouville's bound on rational approximation of algebraic numbers, the line that led to the [[Thue–Siegel–Roth theorem|R9JZA3G2NX]].\nLater, [[Baker's theorem|WMBJSNG57F]] gave explicit bounds on the solutions.\n\n## Explanations\n\n### L1 human\n\nTake an equation such as $x^3 - 2y^3 = 1$ and look for whole number solutions.\nIf $(x, y)$ is a large solution, then $x/y$ is an extremely good rational approximation of $\\sqrt[3]{2}$.\nThue showed that algebraic numbers like $\\sqrt[3]{2}$ cannot have too many such good approximations.\nSo there are only finitely many solutions.\nFor degree $2$ this fails: the Pell equation $x^2 - 2y^2 = 1$ has infinitely many solutions.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11D59",
      "11D41"
    ],
    "name": "Thue equation",
    "same_as": {
      "wikidata": "Q2378270",
      "wikipedia": "Thue equation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Thue_equation"
      }
    ],
    "specializes": [
      "08B1RTW7VZ"
    ],
    "tag": "3Q6CFPQZDF"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/3Q6CFPQZDF/",
  "links": [
    {
      "field": "specializes",
      "target": "08B1RTW7VZ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/3Q6CFPQZDF.md",
  "msc": [
    "11D59",
    "11D41"
  ],
  "name": "Thue equation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/3Q6/3Q6CFPQZDF.md",
  "status": null,
  "tag": "3Q6CFPQZDF",
  "uri": "m2://3Q6CFPQZDF",
  "url": "https://math2.tamnd.com/wiki/thue-equation/",
  "writing": {
    "tag": "3Q6CFPQZDF",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}