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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, Euler's four-square identity says that the product of two numbers, each of which is a sum of four
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s, is itself a sum of four squares.


Algebraic identity

For any pair of quadruples from a
commutative ring In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
, the following expressions are equal: \begin & \left(a_1^2+a_2^2+a_3^2+a_4^2\right) \left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\ mu&\qquad = \left(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4\right)^2 + \left(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3\right)^2 \\ mu &\qquad\qquad+ \left(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2\right)^2 + \left(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1\right)^2. \end
Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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 verified with
elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ...
. The identity was used by
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiafour square theorem. More specifically, it implies that it is sufficient to prove the theorem for
prime numbers A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
, 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 a_k to -a_k, and/or any b_k to -b_k. If the a_k and b_k are
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, the identity expresses the fact that the absolute value of the product of two
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for
complex numbers In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a ...
. This property is the definitive feature of
composition algebra In mathematics, a composition algebra over a field is a not necessarily associative algebra over together with a nondegenerate quadratic form that satisfies :N(xy) = N(x)N(y) for all and in . A composition algebra includes an involution ...
s. Hurwitz's theorem states that an identity of form, \left(a_1^2+a_2^2+a_3^2+\dots+a_n^2\right)\left(b_1^2+b_2^2+b_3^2+\dots+b_n^2\right) = c_1^2+c_2^2+c_3^2+ \dots + c_n^2 where the c_i are bilinear functions of the a_i and b_i is possible only for ''n'' = 1, 2, 4, or 8.


Proof of the identity using quaternions

Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: . This defines the quaternion multiplication rule , which simply reflects Euler's identity, and some mathematics of quaternions. Quaternions are, so to say, the "square root" of the four-square identity. But let the proof go on: Let \alpha = a_1 + a_2 i + a_3 j + a_4 k and \beta = b_1 + b_2 i + b_3 j + b_4 k be a pair of quaternions. Their quaternion conjugates are \alpha^* = a_1 - a_2 i - a_3 j - a_4 k and \beta^* = b_1 - b_2 i - b_3 j - b_4 k. Then A := \alpha \alpha^* = a_1^2 + a_2^2 + a_3^2 + a_4^2 and B := \beta \beta^* = b_1^2 + b_2^2 + b_3^2 + b_4^2. The product of these two is A B = \alpha \alpha^* \beta \beta^*, where \beta \beta^* is a real number, so it can commute with the quaternion \alpha^*, yielding A B = \alpha \beta \beta^* \alpha^*. No parentheses are necessary above, because quaternions associate. The conjugate of a product is equal to the commuted product of the conjugates of the product's factors, so A B = \alpha \beta (\alpha \beta)^* = \gamma \gamma^* where \gamma is the Hamilton product of \alpha and \beta: \begin \gamma &= \left( a_1 + \langle a_2, a_3, a_4 \rangle\right) \left(b_1 + \langle b_2, b_3, b_4 \rangle\right) \\ mu& = a_1 b_1 + a_1 \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle b_1 + \langle a_2, \ a_3, \ a_4\rangle \langle b_2, \ b_3, \ b_4\rangle \\ mu& = a_1 b_1 + \langle a_1 b_2, \ a_1 b_3, \ a_1 b_4\rangle + \langle a_2 b_1, \ a_3 b_1, \ a_4 b_1\rangle \\ &\qquad - \langle a_2,\ a_3, \ a_4\rangle \cdot \langle b_2, \ b_3, \ b_4\rangle + \langle a_2, \ a_3, \ a_4\rangle \times \langle b_2, \ b_3, \ b_4\rangle \\ mu& = a_1 b_1 + \langle a_1 b_2 + a_2 b_1, \ a_1 b_3 + a_3 b_1, \ a_1 b_4 + a_4 b_1\rangle \\ &\qquad - a_2 b_2 - a_3 b_3 - a_4 b_4 + \langle a_3 b_4 - a_4 b_3, \ a_4 b_2 - a_2 b_4, \ a_2 b_3 - a_3 b_2\rangle \\ mu& = (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) \\ &\qquad + \langle a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3, \ a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4, \ a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2\rangle \\ mu\gamma &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\ &\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k. \end Then \begin \gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4) - (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3) i \\ &\qquad - (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4) j - (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2) k . \end If \gamma = r + \vec u where r is the scalar part and \vec u = \langle u_1, u_2, u_3\rangle is the vector part, then \gamma^* = r - \vec u so \begin \gamma \gamma^* &= (r + \vec u) (r - \vec u) = r^2 - r \vec u + r \vec u - \vec u \vec u = r^2 + \vec u \cdot \vec u - \vec u \times \vec u \\ &= r^2 + \vec u \cdot \vec u = r^2 + u_1^2 + u_2^2 + u_3^2. \end So, \begin A B = \gamma \gamma^* &= (a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 + (a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 \\ &\qquad + (a_1 b_3 + a_3 b_1 + a_4 b_2 - a_2 b_4)^2 + (a_1 b_4 + a_4 b_1 + a_2 b_3 - a_3 b_2)^2. \end


Pfister's identity

Pfister found another square identity for any even power:Keith Conra
Pfister's Theorem on Sums of Squares
from
University of Connecticut The University of Connecticut (UConn) is a public land-grant research university system with its main campus in Storrs, Connecticut, United States. It was founded in 1881 as the Storrs Agricultural School, named after two benefactors. In 1893, ...
 If the c_i are just rational functions of one set of variables, so that each c_i has a
denominator A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
, then it is possible for all n = 2^m. Thus, another four-square identity is as follows: \begin &\left(a_1^2+a_2^2+a_3^2+a_4^2\right)\left(b_1^2+b_2^2+b_3^2+b_4^2\right) \\ mu &\quad= \left(a_1 b_4 + a_2 b_3 + a_3 b_2 + a_4 b_1\right)^2 + \left(a_1 b_3 - a_2 b_4 + a_3 b_1 - a_4 b_2\right)^2 \\ &\quad\qquad+ \left(a_1 b_2 + a_2 b_1 + \frac - \frac\right)^2 + \left(a_1 b_1 - a_2 b_2 - \frac - \frac\right)^2 \end where u_1 and u_2 are given by \begin u_1 &= b_1^2 b_4 - 2 b_1 b_2 b_3 - b_2^2 b_4 \\ u_2 &= b_1^2 b_3 + 2 b_1 b_2 b_4 - b_2^2 b_3 \end Incidentally, the following identity is also true: u_1^2+u_2^2 = \left(b_1^2+b_2^2\right)^2\left(b_3^2+b_4^2\right)


See also

* Brahmagupta–Fibonacci identity (sums of two squares) * Degen's eight-square identity * Pfister's sixteen-square identity *
Latin square Latin ( or ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken by the Latins in Latium (now known as Lazio), the lower Tiber area around Rome, Italy. Through the expansion o ...


References


External links


A Collection of Algebraic Identities


Lettre CXV from Euler to Goldbach {{DEFAULTSORT:Euler's Four-Square Identity Algebraic identities Elementary number theory Squares in number theory Leonhard Euler Quaternions