Question 19
Let $n$ be a positive integer that is not a perfect cube. Define real numbers $a,b,c$ by
$$ a~=~\sqrt[3]{n},\qquad b~=~\frac{1}{a~-~[a]},\qquad c~=~\frac{1}{b~-~[b]}, $$
where $[x]$ denotes the integer part of $x$.
Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra + sb + tc = 0$.
$$ a~=~\sqrt[3]{n},\qquad b~=~\frac{1}{a~-~[a]},\qquad c~=~\frac{1}{b~-~[b]}, $$
where $[x]$ denotes the integer part of $x$.
Prove that there are infinitely many such integers $n$ with the property that there exist integers $r$, $s$, $t$, not all zero, such that $ra + sb + tc = 0$.