Let $m$ be a natural number. Can $m(m+1)$ be written as a seventh power of a natural number? If it is true, is it possible to generalize?
I can use only Euclidean Division. I am not suppose to use the Fundamental theorem of arithmetic.
I think the answer is no, then I have assumed that $m(m+1)=a^{7}$ and I would like to get a contradiction. I know that $m(m+1)=2(1+2+\cdots+m)$, but I got stucked here. I have tried to use the Euclidean Division and write $m=aq+r$, for $0\leq r , and this doesn't help me as well.
I would appreciate your help.