"Pell's equation" meaning in English

{
  "etymology_text": "Named by Leonhard Euler after the 17th-century mathematician John Pell, whom Euler mistakenly believed to be the first to find a general solution.",
  "forms": [
    {
      "form": "Pell's equations",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "Pell's equation (plural Pell's equations)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
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          "source": "w"
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        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "derived": [
        {
          "word": "generalised Pell's equation"
        }
      ],
      "examples": [
        {
          "ref": "1974, Allan M. Kirch, Elementary Number Theory: A Computer Approach, Intext Educational Publishers, page 212:",
          "text": "However, due to Euler's mistake in attributing the equation to English mathematician John Pell (1610-1585), Equation (27.14) is called Pell's equation. Results concerning Pell's equation will be stated without proof.",
          "type": "quote"
        },
        {
          "ref": "1989, Mathematics Magazine, Volume 62, Mathematical Association of America, page 258:",
          "text": "Thus (P,Q) satisfies Pell's equation and so by Lemma 1, P#x2F;Q is a convergent to #x5C;sqrtd.",
          "type": "quote"
        },
        {
          "ref": "2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409:",
          "text": "We introduced Pell's equation\n#x5C;qquad#x5C;qquad#x5C;qquad#x5C;qquadx²-ny²#x3D;1\nin Chapter 4 as an example of a Diophantine equation. The solution x#x3D;577,#x5C;y#x3D;408 of the Pell equation x²-2y²#x3D;1 was used in India in the fourth century to produce the fraction #x5C;tfrac#x7B;577#x7D;#x7B;408#x7D; as an excellent rational approximation for #x5C;sqrt 2.\nIt is easy to see why solutions to Pell's equation can be used to approximate solutions to #x5C;sqrtn—this was known to Archimedes, who used this method to approximate square roots.",
          "type": "quote"
        }
      ],
      "glosses": [
        "The Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y."
      ],
      "id": "en-Pell's_equation-en-noun-EImLm3nU",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "Diophantine equation",
          "Diophantine equation"
        ]
      ],
      "raw_glosses": [
        "(number theory) The Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y."
      ],
      "related": [
        {
          "word": "Pell number"
        }
      ],
      "synonyms": [
        {
          "word": "Pell-Fermat equation"
        },
        {
          "word": "Pell equation"
        }
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "translations": [
        {
          "code": "de",
          "lang": "German",
          "sense": "Diophantine equation",
          "tags": [
            "feminine"
          ],
          "word": "Pellsche Gleichung"
        },
        {
          "code": "it",
          "lang": "Italian",
          "sense": "Diophantine equation",
          "tags": [
            "feminine"
          ],
          "word": "equazione di Pell"
        }
      ]
    }
  ],
  "word": "Pell's equation"
}

{
  "derived": [
    {
      "word": "generalised Pell's equation"
    }
  ],
  "etymology_text": "Named by Leonhard Euler after the 17th-century mathematician John Pell, whom Euler mistakenly believed to be the first to find a general solution.",
  "forms": [
    {
      "form": "Pell's equations",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
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      "expansion": "Pell's equation (plural Pell's equations)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Pell number"
    }
  ],
  "senses": [
    {
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        "English countable nouns",
        "English entries with incorrect language header",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
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        "Entries with translation boxes",
        "Pages with 1 entry",
        "Pages with entries",
        "Quotation templates to be cleaned",
        "Terms with German translations",
        "Terms with Italian translations",
        "en:Number theory"
      ],
      "examples": [
        {
          "ref": "1974, Allan M. Kirch, Elementary Number Theory: A Computer Approach, Intext Educational Publishers, page 212:",
          "text": "However, due to Euler's mistake in attributing the equation to English mathematician John Pell (1610-1585), Equation (27.14) is called Pell's equation. Results concerning Pell's equation will be stated without proof.",
          "type": "quote"
        },
        {
          "ref": "1989, Mathematics Magazine, Volume 62, Mathematical Association of America, page 258:",
          "text": "Thus (P,Q) satisfies Pell's equation and so by Lemma 1, P#x2F;Q is a convergent to #x5C;sqrtd.",
          "type": "quote"
        },
        {
          "ref": "2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409:",
          "text": "We introduced Pell's equation\n#x5C;qquad#x5C;qquad#x5C;qquad#x5C;qquadx²-ny²#x3D;1\nin Chapter 4 as an example of a Diophantine equation. The solution x#x3D;577,#x5C;y#x3D;408 of the Pell equation x²-2y²#x3D;1 was used in India in the fourth century to produce the fraction #x5C;tfrac#x7B;577#x7D;#x7B;408#x7D; as an excellent rational approximation for #x5C;sqrt 2.\nIt is easy to see why solutions to Pell's equation can be used to approximate solutions to #x5C;sqrtn—this was known to Archimedes, who used this method to approximate square roots.",
          "type": "quote"
        }
      ],
      "glosses": [
        "The Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "Diophantine equation",
          "Diophantine equation"
        ]
      ],
      "raw_glosses": [
        "(number theory) The Diophantine equation x²-my²=1 for a given integer m, to be solved in integers x and y."
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "Pell-Fermat equation"
    },
    {
      "word": "Pell equation"
    }
  ],
  "translations": [
    {
      "code": "de",
      "lang": "German",
      "sense": "Diophantine equation",
      "tags": [
        "feminine"
      ],
      "word": "Pellsche Gleichung"
    },
    {
      "code": "it",
      "lang": "Italian",
      "sense": "Diophantine equation",
      "tags": [
        "feminine"
      ],
      "word": "equazione di Pell"
    }
  ],
  "word": "Pell's equation"
}