A question of computer science?

By the Binomial Theorem, $(3+\sqrt{8})^{101} + (3-\sqrt{8})^{101} = 2\sum_{n=0}^{50}\binom{101}{2n}\times 3^{101-2n}\times8^n$ Is a positive integer, and so $\big\{(3+\sqrt{8})^{101}\big\} + (3-\sqrt{8})^{101}$ Is also a positive integer. But $3-\sqrt{8} = (3+\sqrt{8})^{-1}$ , and so both $\big\{(3+\sqrt{8})^{101}\big\}$ and $(3-\sqrt{8})^{101}$ lie strictly between $0$ and $1$ . Since their sum is an integer, that sum must be $\boxed{1}$ .