Conjecture: The only integer solutions to the function $\displaystyle f(x)=\sqrt{x^4+x^3+x^2+x+1}$ are $\displaystyle (-1,1)$,$\displaystyle (0,1)$, and $\displaystyle (3,11)$. No others exist. Furthermore, $\displaystyle \lim_{|n|\rightarrow \infty} frac(f(n))=\frac{7}{8}$ for n odd and $\displaystyle \frac{3}{8}$ for n even, when $\displaystyle n \in \mathbb{Z}$.
I agree, it is an interesting problem, and looking at it numerically, I believe it to be true. But why these three solutions should exist uniquely, I do not know.
UPDATE: http://www.mathhelpforum.com/math-he...duction-3.html Proof for general case found a posteriori.