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Euler Four-Square Identity

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The amazing polynomial identity

(a_1^2+a_2^2+a_3^2+a_4^2)(b_1^2+b_2^2+b_3^2+b_4^2) =(a_1b_1-a_2b_2-a_3b_3-a_4b_4)^2+(a_1b_2+a_2b_1+a_3b_4-a_4b_3)^2+(a_1b_3-a_2b_4+a_3b_1+a_4b_2)^2+(a_1b_4+a_2b_3-a_3b_2+a_4b_1)^2,

communicated by Euler in a letter to Goldbach on April 12, 1749 (incorrectly given as April 15, 1705--before Euler was born--in Conway and Guy 1996, p. 232). The identity also follows from the fact that the norm of the product of two quaternions is the product of the norms (Conway and Guy 1996).

See also

Fibonacci Identity, Lagrange's Four-Square Theorem, Lebesgue Identity

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References

Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, p. 232, 1996.Nagell, T. Introduction to Number Theory. New York: Wiley, pp. 191-192, 1951.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, p. 8, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Referenced on Wolfram|Alpha

Euler Four-Square Identity

Cite this as:

Weisstein, Eric W. "Euler Four-Square Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulerFour-SquareIdentity.html

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