Given integers $x_1,\dots,x_n>1$.

Let's assume WLOG that ${x_1}\leq\ldots\leq{x_n}$.

I want to prove that the only integer solutions to any equation of this type are:

- $x_{1,2,3 }=3\implies\prod\limits_{i=1}^{3}x_i=\sum\limits_{i=1}^{3}x_i^2=27$
- $x_{1,2,3,4}=2\implies\prod\limits_{i=1}^{4}x_i=\sum\limits_{i=1}^{4}x_i^2=16$

I have a partial proof below, but there are a few holes in it that I would be happy to get help with.

For $n=1$:

- $x_1<x_1^2$
- $\color{green}{\text{Done.}}$

For $n=2$:

- ${x_1x_2}\leq{x_2x_2}={x_2^2}<{x_1^2+x_2^2}$
- $\color{green}{\text{Done.}}$

For $n=3$ and $x_1=2$:

- $\color{red}{\text{This seems to be the most difficult case.}}$
- $\color{red}{\text{I suspect that there might even be a solution here.}}$

For $n=3$ and $x_{1,2,3}=3,3,3$:

- $\prod\limits_{i=1}^{3}x_i=27=\sum\limits_{i=1}^{3}x_i^2$
- $\color{blue}{\text{This is a solution.}}$

For $n=3$ and $x_{1,2,3}=3,3,4$:

- $\prod\limits_{i=1}^{3}x_i=36>34=\sum\limits_{i=1}^{3}x_i^2$
- $\color{red}{\text{How do we prove that it holds for every }x_i\text{ being replaced with a larger value?}}$

For $n=4$ and $x_{1,2,3,4}=2,2,2,2$:

- $\prod\limits_{i=1}^{4}x_i=16=\sum\limits_{i=1}^{4}x_i^2$
- $\color{blue}{\text{This is a solution.}}$

For $n=4$ and $x_{1,2,3,4}=2,2,2,3$:

- $\prod\limits_{i=1}^{4}x_i=24>21=\sum\limits_{i=1}^{4}x_i^2$
- $\color{red}{\text{How do we prove that it holds for every }x_i\text{ being replaced with a larger value?}}$

For $n>4$ and $x_{1,\dots,n}=2,\dots,2$:

- $\prod\limits_{i=1}^{n}x_i=2^n>4n=\sum\limits_{i=1}^{n}x_i^2$
`can prove by induction`

- $\color{red}{\text{How do we prove that it holds for every }x_i\text{ being replaced with a larger value?}}$

**UPDATE:**

Based on @sciona's comments:

For $n=3$ and $x_1=2$, we can show that there are no solutions:

$\sum\limits_{i=1}^{n}x_i^2-\prod\limits_{i=1}^{n}x_i=4+x_2^2+x_3^2-2x_2x_3=4+(x_2-x_3)^2>0$

$\sum\limits_{i=1}^{n}x_i^2-\prod\limits_{i=1}^{n}x_i>0\implies\sum\limits_{i=1}^{n}x_i^2>\prod\limits_{i=1}^{n}x_i\implies\sum\limits_{i=1}^{n}x_i^2\neq\prod\limits_{i=1}^{n}x_i$

We can use the solution $(3,3,3)$ in order to generate infinitely many more:

If $(a,b,c)$ is a solution, then so is $(b,c,bc-a)$

For example, $(3,3,3)\rightarrow(3,3,6)\rightarrow(3,6,15)\rightarrow\dots$

We can use the solution $(2,2,2,2)$ in order to generate infinitely many more:

If $(a,b,c,d)$ is a solution, then so is $(b,c,d,bcd-a)$

For example, $(2,2,2,2)\rightarrow(2,2,2,6)\rightarrow(2,2,6,22)\rightarrow\dots$

So my question narrows-down to proving that there are no other types of solutions.