id: 05800663
dt: j
an: 2010e.00418
au: Breu, Raymund; van der Poorten, Alf
ti: Self-similar values of quadratic forms: A remark on pattern and duality.
so: J. Recreat. Math. 35(2006), No. 3, 202-212 (2010).
py: 2010
pu: Baywood Publishing Company, Amityville, NY
la: EN
cc: F60
ut:
ci:
li:
ab: Summary: Our text is Hendrik Lenstra’s nice observation that
$12^2+33^2=1233$. One notices readily that such a cute decomposition of
$10^ka+b$ as a sum of two squares $a^2+b^2$ is no more than rewriting
the sum of squares $10^{2k}+1$ as a different sum $(10^k-2a)^2+(2b-1)$.
But that does not hint at more radical generalization. We show here
that much the same can be told with the sum of squares $x^2+y^2$
replaced by an arbitrary quadratic form $f(x,y)=ux^2+ vxy+wy^2$, where
$u,v$, and $w$ are integers; and the cute representations now are of
the shape $10^ka+b=f(a,b)$.
rv: