"quadratic reciprocity" meaning in English

{
  "etymology_text": "The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a² = b in modular arithmetic. It was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss.",
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "quadratic reciprocity (uncountable)",
      "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": "topical",
          "langcode": "en",
          "name": "Number theory",
          "orig": "en:Number theory",
          "parents": [
            "Mathematics",
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "examples": [
        {
          "ref": "2007, Paul B. Garrett, Abstract Algebra, Taylor & Francis (Chapman Hall/CRC Press), page 287",
          "text": "Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this.",
          "type": "quotation"
        },
        {
          "ref": "2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153",
          "text": "Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity.",
          "type": "quotation"
        },
        {
          "ref": "2013, J. Voight, “Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms”, in Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, Springer, page 284",
          "text": "An interesting consequence of the above algorithm is that one can evaluate the Jacobi symbol in deterministic polynomial time in certain cases analogous to the way (“reduce and flip”) that one computes this symbol using quadratic reciprocity in the case F#x3D;#x5C;mathbb#x7B;Q#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p."
      ],
      "id": "en-quadratic_reciprocity-en-noun-Q48aefmf",
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "theorem",
          "theorem"
        ],
        [
          "prime number",
          "prime number"
        ],
        [
          "square",
          "square"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "raw_glosses": [
        "(number theory) The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p."
      ],
      "related": [
        {
          "word": "Legendre symbol"
        },
        {
          "word": "Jacobi symbol"
        },
        {
          "word": "quadratic residue"
        }
      ],
      "synonyms": [
        {
          "word": "law of quadratic reciprocity"
        }
      ],
      "tags": [
        "uncountable"
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Adrien-Marie Legendre",
        "Carl Friedrich Gauss",
        "Leonhard Euler",
        "quadratic reciprocity"
      ]
    }
  ],
  "word": "quadratic reciprocity"
}

{
  "etymology_text": "The theorem highlights a particular form of reciprocity in the solvability of the quadratic equation a² = b in modular arithmetic. It was conjectured by Leonhard Euler and Adrien-Marie Legendre and first proved by Carl Friedrich Gauss.",
  "head_templates": [
    {
      "args": {
        "1": "-"
      },
      "expansion": "quadratic reciprocity (uncountable)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "related": [
    {
      "word": "Legendre symbol"
    },
    {
      "word": "Jacobi symbol"
    },
    {
      "word": "quadratic residue"
    }
  ],
  "senses": [
    {
      "categories": [
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "English uncountable nouns",
        "en:Number theory"
      ],
      "examples": [
        {
          "ref": "2007, Paul B. Garrett, Abstract Algebra, Taylor & Francis (Chapman Hall/CRC Press), page 287",
          "text": "Yes, but we need not only Quadratic Reciprocity but also Dirichlet's theorem on primes in arithmetic progressions to see this.",
          "type": "quotation"
        },
        {
          "ref": "2009, Sam Vandervelde, Circle in a Box, American Mathematical Society, page 153",
          "text": "Gauss studied these sorts of numbers while attempting to formulate and prove higher order reciprocity laws, following his success with quadratic reciprocity.",
          "type": "quotation"
        },
        {
          "ref": "2013, J. Voight, “Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms”, in Krishnaswami Alladi, Manjul Bhargava, David Savitt, Pham Huu Tiep, editors, Quadratic and Higher Degree Forms, Springer, page 284",
          "text": "An interesting consequence of the above algorithm is that one can evaluate the Jacobi symbol in deterministic polynomial time in certain cases analogous to the way (“reduce and flip”) that one computes this symbol using quadratic reciprocity in the case F#x3D;#x5C;mathbb#x7B;Q#x7D;.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p."
      ],
      "links": [
        [
          "number theory",
          "number theory"
        ],
        [
          "theorem",
          "theorem"
        ],
        [
          "prime number",
          "prime number"
        ],
        [
          "square",
          "square"
        ],
        [
          "modulo",
          "modulo"
        ]
      ],
      "raw_glosses": [
        "(number theory) The mathematical theorem which states that, for given odd prime numbers p and q, the question of whether p is a square modulo q is equivalent to the question of whether q is a square modulo p."
      ],
      "tags": [
        "uncountable"
      ],
      "topics": [
        "mathematics",
        "number-theory",
        "sciences"
      ],
      "wikipedia": [
        "Adrien-Marie Legendre",
        "Carl Friedrich Gauss",
        "Leonhard Euler",
        "quadratic reciprocity"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "law of quadratic reciprocity"
    }
  ],
  "word": "quadratic reciprocity"
}