Let us define $s(n)$ as any solution to the equation $x^2 = n$, such that $x \neq n$. I am looking for a numeric system such that $s(s(s\cdots s(1)))$ is always an integer or is composed of integer parts.
For example: on the set of Natural Numbers, there is no solution to $s(1)$. However, if we extend the set with negative integers, we have $s(1) = -1$. Now, what about $s(s(1))$? There is no solution either. But if we extend with Gaussian integers, we have $s(s(1)) = i$. Now, what about $s(s(s(1)))$? The solution to this equation is $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2}i$, but that number is not on the set of Gaussian integers, since it has fractional parts. My question is: is it possible to meaningfully extend Gaussian integers (say, with some other element) such that $s(s(s(1)))$ can be represented with only integer parts?
One idea I had is to jump from complex integers to quaternion integers, but I realized there is no quaternion $Q$ such that $Q^2 = i$, and such that $Q$ has only integer parts.
Is there any numeric system that satisfies the desired property?
The only solutions in quaternions are $1,-1,i,-i,j,-j,k,-k$.
Notice that your solution has at the same time to have modulus $1$ and to have integer parts. This means $\sqrt{a^2+b^2+c^2+d^2}=1$ and $a,b,c,d\in \mathbb{Z}$ If modulus of any of ${a,b,c,d}$ is $2$ or greater, the value of the root is greater than $1$. If at least $2$ of the components have modulus $1$ then also the root is greater than $1$. So, the only solutions are when only one of the components is $\pm1$ and the others are zero.