{
  "aliases": [
    "diophantus-identity"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "04NPWJMA6H"
    }
  ],
  "body": "\n## Statement\n\nLet $a$, $b$, $c$ and $d$ be elements of a commutative ring. Then\n$$(a^2 + b^2)(c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2 = (ac + bd)^2 + (ad - bc)^2.$$\n\n## Remarks\n\nSo the product of two sums of two squares is again a sum of two squares.\nFor example, $5 \\cdot 13 = 65$, and the identity gives $65 = 1^2 + 8^2 = 7^2 + 4^2$.\nFor complex numbers, the identity says $|z|^2 |w|^2 = |zw|^2$ with $z = a + bi$ and $w = c + di$.\nIt is a step in the proof of the [[sum of two squares theorem|04NPWJMA6H]].\n\n## History\n\nThe identity appears in the Arithmetica of Diophantus, of the third century.\nBrahmagupta found it again in the seventh century, and Fibonacci published it in Europe in 1225.\n\n## Explanations\n\n### L1 human\n\nIf two numbers are sums of two squares, then so is their product.\nYou can check the identity by expanding both sides: the cross terms $2abcd$ cancel.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "diophantus-identity"
    ],
    "kind": "statement",
    "msc": [
      "11E25"
    ],
    "name": "Brahmagupta–Fibonacci identity",
    "role": "proposition",
    "same_as": {
      "wikidata": "Q279090",
      "wikipedia": "Brahmagupta–Fibonacci identity"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity"
      },
      {
        "locator": "History",
        "url": "https://en.wikipedia.org/wiki/Brahmagupta%E2%80%93Fibonacci_identity"
      }
    ],
    "tag": "0K2DMACEZ4"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0K2DMACEZ4/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/0K2DMACEZ4.md",
  "msc": [
    "11E25"
  ],
  "name": "Brahmagupta–Fibonacci identity",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0K2/0K2DMACEZ4.md",
  "status": null,
  "tag": "0K2DMACEZ4",
  "uri": "m2://0K2DMACEZ4",
  "url": "https://math2.tamnd.com/wiki/brahmagupta-fibonacci-identity/",
  "writing": {
    "tag": "0K2DMACEZ4",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}