Here :
Who can prove that a triangular number cannot be a cube, fourth power or fifth power?
I asked how it can be proven that a triangular number (a number of the form $\frac{n(n+1)}{2}$) cannot be a cube, fourth power or fifth power and lulu posted very useful results which probably solve the problem completely.
But even more seems to be true :
CONJECTURE : A triangle number cannot be a power $k^s$ with $k\ge 2$ , $s\ge 3$
The conjecture is true for $n\le 10^9$ and I currently verify the range $[10^9,10^{10}]$ without having found a counter-example yet.
Is it known whether this conjecture is true ?