id: 05986917
dt: j
an: 2012d.00340
au: Alaca, Şaban; Williams, Kenneth S.
ti: Nonexistence of a composition law.
so: Math. Mag. 80, No. 2, 142-144 (2007).
py: 2007
pu: Mathematical Association of America (MAA), Washington, DC
la: EN
cc: F65
ut: representation as sums of squares; not composition law for the polynomial
ci:
li:
ab: 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.
rv: