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Euler’s four-square identity

In a letter to Goldbach on April 15, 1750 L.Euler mentioned the following polynomial identity :

Euler’s four-square  identity: If typeset structure and typeset structure can be written as a sum of four squares, then also typeset structure can be written in this form. More precisely,

(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 _ 1 b _ 1 ... 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 .(1)

The identity can be written in many forms. For instance, the substitution typeset structure, typeset structure brings it into the form:

(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 _ 1 b _ 1 ... 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 .(2)

Euler’s four-square identity is covered by Hurwitz theorem saying that except for typeset structure, there exists no identity of the form

(x _ 1^2 + x _ 2^2 + ... + x _ n^2) (y _ 1^2 + y _ 2^2 + ... + y _ n^2) = z _ 1^2 + z _ 2^2 + ... + z _ n^2(3)

where the typeset structure’s are bilinear functions of  typeset structure  and  typeset structure.

1st proof: Expand both sides and compare them. In both cases the sides are equal to

a _ 1^2 b _ 1^2 + a _ 2^2 b _ 1^2 + a _ 3^2 b _ 1^2 + a _ 4^2 b _ 1^2 + a _ 1^2 b _ 2^2 + a _  ...  _ 3^2 + a _ 4^2 b _ 3^2 + a _ 1^2 b _ 4^2 + a _ 2^2 b _ 4^2 + a _ 3^2 b _ 4^2 + a _ 4^2 b _ 4^2 .

2nd proof:  If we take the existence and properties of quaternions for granted, the second identity follows from the fact that the norm of the product of two quaternions is the product of their norms .

3rd proof: The following matrix approach is actually another form of the previous proof. If typeset structure are complex numbers then

(              _    ) (                 _   ) = (                                              ...                              βγ + α δ       αγ - β δ(4)

Taking the determinants on both sides gives

(N(α) + N(β)) (N(γ) + N(δ)) = N(αγ - Overscript[β, _] δ) + N(βγ + Overscript[α, _] δ),

where typeset structure denotes the norm.  Since the norm of a complex number is a sum of two squares, the result follows (the idea to use the last identity for the proof of Euler Four-Square identity goes back to  C.F.Gauß, Posthumous manuscript, Werke 3, 1876, 383-384).

To experiment with the formula go to   .

Lagrange ( [1] ,  p.65) gave the following generalization of Euler’s four squares identities

(p _ 1^2 - B q _ 1^2 - C r _ 1^2 + B C s _ 1^2) (p _ 2^2 - B q _ 2^2 - C r _ 2^2 + B C s _ 2^2 ... bsp;  s _ 1 q _ 2)^2 + B C(q _ 1 r _ 2 - p _ 1 s _ 2 ± p _ 2 s _ 1 -+ r _ 1 q _ 2)^2 .(5)

For typeset structure we get Euler’s four squares identities.

References

[1]  Le Besgue, V. A. (1862). Introduction a la théorie des nombres. Paris: Mallet-Bachelier.

Cite this web-page as:

Štefan Porubský: Euler's Four-Square Identity.

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