MathLinks上的题目

11 December 2005

Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by

$$ a = \sqrt[3]{n}, b = \frac{1}{a - \lfloor a \rfloor}, c = \frac{1}{b - \lfloor b \rfloor}, $$

where $\lfloor x \rfloor$ denotes the integer part of $x$. Prove that there are infinitely many such integers $n$ with the property that there exist integers $x,y,z$, not all zero, such that

$$ ax + by + cz = 0. $$

郁闷啊，做到现在……

至此已经尝试了不下5种方法了，头痛了啦！