"Pell's equation" meaning in All languages combined

{
  "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"
          ],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "English entries with language name categories using raw markup",
          "parents": [
            "Entries with language name categories using raw markup",
            "Entry maintenance"
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          "kind": "other",
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          "parents": [
            "Terms with non-redundant non-automated sortkeys",
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        },
        {
          "kind": "other",
          "name": "Terms with German translations",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Terms with Italian translations",
          "parents": [],
          "source": "w"
        },
        {
          "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": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "ref": "2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409",
          "text": "We introduced Pell's equation\nx5C;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": "quotation"
        }
      ],
      "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": [
    {
      "args": {},
      "expansion": "Pell's equation (plural Pell's equations)",
      "name": "en-noun"
    }
  ],
  "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 entries with language name categories using raw markup",
        "English eponyms",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with non-redundant non-automated sortkeys",
        "English terms with quotations",
        "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": "quotation"
        },
        {
          "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": "quotation"
        },
        {
          "ref": "2013, John J. Watkins, Number Theory: A Historical Approach, Princeton University Press, page 409",
          "text": "We introduced Pell's equation\nx5C;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": "quotation"
        }
      ],
      "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"
}