Show that there is no pair s.t. $a^1+a^2+a^3+...+a^n=b^m$ Let $a,b,m$ and $n$ belong to integer. Is there any pair exist such that
$a^1+a^2+a^3+...+a^n=b^m$ where $a,b,m,n\ge 2$
Source code Pari/GP
for(m=2,10,for(n=2,10,for(a=2,100,for(b=2,100,if(sum(i=1,n,a^i)==b^m,print([a,n,b,m]))))))

Edit: more on observation
There always have at least one solution for $\sum_{i=1}^{n}a^i=b^m+k$ for every integer $k\ne 0$

Apologies for the short question and shows no effort towards solution. Thank you.
 A: Your identity can be expressed as $$a\frac{a^n-1}{a-1}=b^m\,.$$
Observing that $\gcd(a, \frac{a^n-1}{a-1})=1$, this reduces to $$\frac{x^{mn}-1}{x^m-1}=y^m$$ assuming $a=x^m$ and $b=xy$ with $\gcd(x,y)=1$.
Now, when $n=2$, we obtain $$y^m-x^m=1\,,$$ which cannot have any solution for nonzero $x,y$. Thus $n\ge 3$; this essentially becomes part of the equation $$\frac{a^n-1}{a-1}=y^m$$ with $n\ge 3, |a|\ge 2,|y|\ge 2,m\ge 2$ that was investigated by Nagell and Ljunggren in the first half of the last century, which is conjectured to have the only known solutions $$ (a^n,y^m)\in\left\lbrace (3^5,11^2), (7^4,20^2),(18^3,7^3),((-19)^3,7^3)\right\rbrace\,,$$ none of which has the values of $a\in\{3,7,18,-19\}$ being an $m$th power. So your problem has no solutions conditional on the Nagell-Ljunggren conjecture. In fact, thanks to the $abc$-conjecture (if Mochizuki’s proof is\has been confirmed by experts), then Shorey (1986) showed that there are only finitely many solutions. (It may well be that your problem is easier to solve in general because the $a$ variable is an $m$th power; however, I have not considered this possibility carefully in its own right).
A: (This is incomplete)
$$a ( 1+ a^2 + \ldots a^{n-1}) = b^m$$
Since $ \gcd(a, 1+ a^2 + \ldots a^{n-1}) = 1$, both of these terms must be powers of $m$. Let $ a = c^m$ and $b = cd$, which gives us
$$ d^ m = \frac{ c^{mn} -1 } { c^m - 1}  > \frac{ c^{mn}}{c^m}   \Rightarrow d > c^{n-1}. $$
If $ n < m$, then $ (c^{n-1} +1)^m (c^m-1) > c^{mn} -1 \Rightarrow  c^{n-1} + 1 > d$. Hence, there are no integer solutions for $d$.
So we must have $ n \geq m$.
