How many perfect cubes are between $2^8+1$ to $2^{18}+1$ ( inclusively) How many perfect cubes are between $2^8+1$ to $2^{18}+1$ ( inclusively) $2^9,2^{12},2^{15},2^{18}$ are all perfect cube.there are many other. I try to use modulo 2 .but it won't work, and no other methods i tried get me nowhere Any ideas?

Hint:

For every positive integer $x$: $$2^8+1\le x^3\le2^{18}+1\iff\sqrt[3]{2^8+1}\le x\le \sqrt[3]{2^{18}+1}\iff 7\le x\le 2^6$$