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- Euler's_four-square_identity abstract "In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Specifically:Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the and are real numbers, a more elegant proof is available: the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers.The identity was used by Lagrange to prove his four square theorem. More specifically, it implies that it is sufficient to prove the theorem for prime numbers, after which the more general theorem follows. The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any to , to , or by changing the signs inside any of the squared terms on the right hand side.Hurwitz's theorem states that an identity of form,where the are bilinear functions of the and is possible only for n = {1, 2, 4, 8}. However, the more general Pfister's theorem allows that if the are just rational functions of one set of variables, hence has a denominator, then it is possible for all . Thus, a different kind of four-square identity can be given as,where,Note also the incidental fact that,".
- Euler's_four-square_identity wikiPageExternalLink OO0841.pdf.
- Euler's_four-square_identity wikiPageExternalLink 005b.
- Euler's_four-square_identity wikiPageID "222004".
- Euler's_four-square_identity wikiPageRevisionID "594858785".
- Euler's_four-square_identity hasPhotoCollection Euler's_four-square_identity.
- Euler's_four-square_identity subject Category:Elementary_algebra.
- Euler's_four-square_identity subject Category:Elementary_number_theory.
- Euler's_four-square_identity subject Category:Mathematical_identities.
- Euler's_four-square_identity type Abstraction100002137.
- Euler's_four-square_identity type Attribute100024264.
- Euler's_four-square_identity type Identity104618070.
- Euler's_four-square_identity type MathematicalIdentities.
- Euler's_four-square_identity type Personality104617562.
- Euler's_four-square_identity comment "In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four squares, is itself a sum of four squares. Specifically:Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach (but he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring.".
- Euler's_four-square_identity label "Euler's four-square identity".
- Euler's_four-square_identity label "Identidad de los cuatro cuadrados de Euler".
- Euler's_four-square_identity label "Identità dei quattro quadrati di Eulero".
- Euler's_four-square_identity label "Identité des quatre carrés d'Euler".
- Euler's_four-square_identity label "Tożsamość czterech kwadratów Eulera".
- Euler's_four-square_identity label "Vier-kwadratenidentiteit van Euler".
- Euler's_four-square_identity label "Тождество четырёх квадратов".
- Euler's_four-square_identity label "متطابقة المربعات الأربع لأويلر".
- Euler's_four-square_identity label "欧拉四平方和恒等式".
- Euler's_four-square_identity sameAs Identidad_de_los_cuatro_cuadrados_de_Euler.
- Euler's_four-square_identity sameAs Identité_des_quatre_carrés_d'Euler.
- Euler's_four-square_identity sameAs Identità_dei_quattro_quadrati_di_Eulero.
- Euler's_four-square_identity sameAs 오일러의_네_제곱수_항등식.
- Euler's_four-square_identity sameAs Vier-kwadratenidentiteit_van_Euler.
- Euler's_four-square_identity sameAs Tożsamość_czterech_kwadratów_Eulera.
- Euler's_four-square_identity sameAs m.01gdkc.
- Euler's_four-square_identity sameAs Q1149225.
- Euler's_four-square_identity sameAs Q1149225.
- Euler's_four-square_identity sameAs Euler's_four-square_identity.
- Euler's_four-square_identity wasDerivedFrom Euler's_four-square_identity?oldid=594858785.
- Euler's_four-square_identity isPrimaryTopicOf Euler's_four-square_identity.