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Nonexistence of a composition law. (English)
Math. Mag. 80, No. 2, 142-144 (2007).
Summary: It is known that every positive odd integer can be expressed in the form $x^2 + y^2 + 2z^2$ for some integers $x, y$ and $z$. Clearly one of $x$ and $y$ must be odd and one must be even. Thus every positive odd integer is of the form $(2x_1 + 1)^2 + 2x_2^2 + 4x_3^2$ for some integers $x_1,x_2$, and $x_3$. Let $m$ and $n$ be positive odd integers. Then $mn$ is also a positive odd integer and there exist integers $x_1,x_2,x_3$, $y_1,y_2,y_3$, $z_1,z_2$ and $z_3$ such that $$m = (2x_1 + 1)^2 + 2x_2^2 + 4x_3^2n = (2y_1 + 1)^2 + 2y_2^2 + 4y_3^2mn = (2z_1 + 1)^2 + 2z_2^2 + 4z_3^2.$$ Hence $$((2x_1 + 1)^2 + 2x_2^2 + 4x_3^2)((2y_1 + 1)^2 + 2y_2^2 + 4y_3^2) = (2z_1 + 1)^2 + 2z_2^2 + 4z_3^2.$$ The question naturally arises: Is this equality a consequence of some underlying composition law for the polynomial $(2x_1 + 1)^2 + 2x_2^2 + 4x_3^2$? We show directly from first principles that it is not.
Classification: F65
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