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Self-similar values of quadratic forms: A remark on pattern and duality. (English)
J. Recreat. Math. 35(2006), No. 3, 202-212 (2010).
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 $10^ka+b=f(a,b)$.
Classification: F60
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