Prove that for a natural number $k≥2$ , there are a minimum of two perfect squares in the interval $(k^3,(k+1)^3)$
I tried induction , supposing that there are $m^2<n^2$ in the interval , and to prove that there exists two perfect squares $q^2<r^2$ in $((k+1)^3,(k+2)^3))$.
Let $m$ be the largest integer such that $m^2\le k^3$
$m\le k^{\frac 32}$
if $k>1$
$4m \le 4k^{\frac 32} < 3k^2$ and $4<3k+1$
$k^3<(m+1)^2<(m+2)^2 = m^2 + 4m + 4 < k^3 + 3k^2 + 3k + 1 = (k+1)^3$