Why does 2 result in the same value regardless of whether it is added to itself, multiplied by itself, or put to the power of itself? [closed]
I'm inferring that any hyperoperation you could apply here using two for every value would result in four. Why is this?
Solution 1:
Suppose $n$ is an integer and $n+n=n\cdot n$.
Then $2n=n^2$ so $n^2-2n=0$. Factoring gives $n(n-2)=0$.
So either $n=0$ or $n=2$.
But if we also have that $n+n=n\cdot n=n^n$ then we have $n^2=n^n$ which gives $n=1$ or $n=2$, so $n=2$ is the common solution to the two equations.