In mathematics, Euler's four-square identity says that the product of two numbers, each of which is a sum of four
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length ad ...
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 in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
, the following expressions are equal:
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in m ...
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 encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
.
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 to , and/or any to .
If the and 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 distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
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. Hamilton defined a quate ...
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,
where the are bilinear functions of the and 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 and be a pair of quaternions. Their quaternion conjugates are and . Then
and
The product of these two is , where is a real number, so it can commute with the quaternion , yielding
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
where is the Hamilton product of and :
Then
If where is the scalar part and is the vector part, then so
So,
University of Connecticut
The University of Connecticut (UConn) is a public land-grant research university in Storrs, Connecticut, a village in the town of Mansfield. The primary 4,400-acre (17.8 km2) campus is in Storrs, approximately a half hour's drive from Hart ...
If the are just rational functions of one set of variables, so that each has a
denominator
A fraction (from la, fractus, "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 ...
, then it is possible for all .
Thus, another four-square identity is as follows:
where and are given by
Incidentally, the following identity is also true:
See also
*
Brahmagupta–Fibonacci identity
In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity say ...
Latin square
In combinatorics and in experimental design, a Latin square is an ''n'' × ''n'' array filled with ''n'' different symbols, each occurring exactly once in each row and exactly once in each column. An example of a 3×3 Latin s ...