# Euler four-square identity

The Euler four-square identity simply states that

$({x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2})({y}_{1}^{2}+{y}_{2}^{2}+{y}_{3}^{2}+{y}_{4}^{2})=$ ${({x}_{1}{y}_{1}+{x}_{2}{y}_{2}+{x}_{3}{y}_{3}+{x}_{4}{y}_{4})}^{2}+{({x}_{1}{y}_{2}-{x}_{2}{y}_{1}+{x}_{3}{y}_{4}-{x}_{4}{y}_{3})}^{2}$ $+{({x}_{1}{y}_{3}-{x}_{3}{y}_{1}+{x}_{4}{y}_{2}-{x}_{2}{y}_{4})}^{2}+{({x}_{1}{y}_{4}-{x}_{4}{y}_{1}+{x}_{2}{y}_{3}-{x}_{3}{y}_{2})}^{2}$

It may be derived from the property of quaternions that the norm of the product is equal to the product of the norms.

Title | Euler four-square identity |
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Canonical name | EulerFoursquareIdentity |

Date of creation | 2013-03-22 12:35:20 |

Last modified on | 2013-03-22 12:35:20 |

Owner | vitriol (148) |

Last modified by | vitriol (148) |

Numerical id | 6 |

Author | vitriol (148) |

Entry type | Theorem |

Classification | msc 11N32 |

Related topic | Quaternions |

Related topic | MultiplicativityOfSumsOfSquares |

Related topic | LagrangesFourSquareTheorem |

Related topic | SumsOfTwoSquares |