Euler's four-square identity
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In mathematics, Euler's four-square identity says that the product of two numbers, each of which being a sum of four squares, is itself a sum of four squares. Specifically:
- <math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
- <math>(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +\,</math>
- <math>(a_1 b_2 + a_2 b_1 + a_3 b_4 - a_4 b_3)^2 +\,</math>
- <math>(a_1 b_3 - a_2 b_4 + a_3 b_1 + a_4 b_2)^2 +\,</math>
- <math>(a_1 b_4 + a_2 b_3 - a_3 b_2 + a_4 b_1)^2.\,</math>
Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach[1][1] (but note that he used a different sign convention from the above). It can be proven with elementary algebra and holds in every commutative ring. If the <math>a_k</math> and <math>b_k</math> 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 sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any <math>a_k</math> to <math>-a_k</math>, <math>b_k</math> to <math>-b_k</math>, or by changing the signs inside any of the squared terms on the right hand side. For example, changing <math>a_1</math> to <math>-a_1</math>, <math>b_1</math> to <math>-b_1</math>, and changing the signs of the second, third, and fourth terms on the right hand side yields the alternate form:
- <math>(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2)=\,</math>
- <math>(a_1 b_1 - a_2 b_2 - a_3 b_3 - a_4 b_4)^2 +\,</math>
- <math>(a_1 b_2 + a_2 b_1 - a_3 b_4 + a_4 b_3)^2 +\,</math>
- <math>(a_1 b_3 + a_2 b_4 + a_3 b_1 - a_4 b_2)^2 +\,</math>
- <math>(a_1 b_4 - a_2 b_3 + a_3 b_2 + a_4 b_1)^2.\,</math>
The identity was used by Lagrange to prove his four square theorem. More specifically, it allows the theorem to be proven only for prime numbers.
See also
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External links
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