Who can prove that a triangular number cannot be a cube, fourth power or fifth power? Triangular numbers (See  https://en.wikipedia.org/wiki/Triangular_number )
are numbers of the form $$\frac{n(n+1)}{2}$$
In ProofWiki I found three claims about triangular numbers. The three claims are that a triangular number cannot be a cube, not a fourth power and not a fifth power. Unfortunately, neither was a proof given nor did I manage to do it myself. Therefore my qeustions :

Does someone know a proof that a triangular number cannot be a cube, a fourth power or a fifth power ?


First, notice $n$ and $n+1$ are coprime. And if the product of coprime numbers is a n-th power then both are also n-th powers. Now divide the problem into the cases where $n$ is odd and even.
$$n=2t$$
$$t(2t+1)=a^b$$
Then $t$ and $2t+1$ are b-th powers. Let $t=y^b$, $2t+1=x^b$. Then 
$$x^b-2y^b=1$$
Applying the same substitutions to the case where $n$ is odd you find $$x^b-2y^b=-1$$
In this answer Keith Conrad proves the only solution is $x=1$, $y=0$, which mean $n=0$.
