Sorry for the long title. I recently watched this video on youtube: https://youtu.be/C7A3uFC76G0 . After solving the problem I wanted to see if I could make some kind of generalization, as a challenge to myself. I will try to briefly describe my process here.
So the problem goes as follows: Solve for all $x\in \mathbb{R}$ \begin{equation} \label{eqpolyexp} (x^{2}-7x+11)^{x^{2}-13x+42}=1. \end{equation} The problem boils down to considering three cases:
- $k^{o}=1$ for all $k\neq 0$.
- $(-1)^{2i}=1$ for all $i\in \mathbb{Z}$.
- $1^{j}=1$ for all $j\in \mathbb{Z}$.
After some simple calculations we find the following solutions: $x=2~,3~,4~,5~,6$ and $7$.
So, I thought it was rather interesting that the ''solution set'' ($x=2~,3~,4~,5~,6$ and $7$) to the equation looked like it did and wondered how the coefficients and constants of two quadratic polynomials $f_{2}(x)=x^{2}+ax+b$ and $g_{2}(x)=x^{2}+cx+d$ had to look like in order for the equation $$f_{2}(x)^{g_{2}(x)}=1$$ to have solutions $x\in\{m ,m +1,m+2,m+3,m+4,m+5\}$ where $m$ is some integer. I could derive the following:
Let $m\in \mathbb{Z}$ be a given positive integer. If $$f(x)=x^{2}-(2m+3)x+(m^{2}+3m+1)$$ and $$g(x)=x^{2}-(2m+9)x+(m^{2}+9m+20),$$ then the equation $$f(x)^{g(x)}=1$$ has solutions $$x\in \{m ,m+1, m+2, m+3, m+4, m+5\}.$$
My question is whether this can be generalized further? I would also like to know if there are any efficient methods of finding the coefficients and constants of an $n$th degree polynomial, given some specific solutions? I would like to give an answer to the following problem
What $n$:th-degree polynomials $f_{n}(x),g_{n}(x)$ have solutions $x\in \{\nu _{n} ,\nu _{n} +1,\ldots , \nu _{n} +(3n-1)\}$ to the equation $$f_{n}(x)^{g_{n}(x)}=1$$ for some positive integer $\nu _{n}\in \mathbb{Z}$?