<s>[INST] $a^n+b^{n+2}=c^{n+1}$ (any solutions, other than this one?) I just made up an equation $a^n+b^{n+2}=c^{n+1}$ and chosen an exponents $n$ and $n+2$ and $n+1$ to be right where they are so that we have one solution, that is, a solution $(1,2,3)$ for $n=1$.
Are there any other solutions for $(a,b,c) \in \mathbb N^3$ and $n \in \mathbb N$?
 [/INST] If $n=1$, we can take arbitrary integers $b$ and $c$ and set $a=c^2-b^3$.
For any $n$, $(2^{n+2})^n+(2^n)^{n+2}=2^{n^2+2n+1}=(2^{n+1})^{n+1}$.
When $n=2$, $(46,3,13)$, $(75,10,25)$ and $(88,4,20)$ are also solutions (found with Excel).

For $n=2$, if $a^2+b^4=c^2$, then $(ac^2)^2+(bc)^4=(c^2)^3$.
$3^2+2^4=5^2$ $\implies$ $(75,10,25)$ is a solution.
$77^2+6^4=85^2$ $\implies$ $(556325,510,7225)$ is a solution.
$17^2+12^4=145^2$ $\implies$ $(357425,1740,21025)$ is a solution.
If $u^2-v^2$ or $2uv$ is a perfect square, the Pythagorean triple $(u^2-v^2,2uv,u^2+v^2)$ will lead to a solution for $n=2$.
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