History   Help on query formulation   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