GULF 2016

Consider sequences $a_0$ , $a_1$ , $a_2$ , $\cdots$ of non-negative integers defined by selecting any $a_0$ , $a_1$ , $a_2$ (not all 0) and for each $n$ $\geq$ 3 letting $a_n$ = | $a_n-1$ - $a_n-3$ |

1-In the particular case that $a_0$ = 1, $a_1$ = 3 and $a_2$ = 2, calculate the beginning of the sequence, listing $a_0$ , $a_1$ , $\cdots$ , $a_{19}$ , $a_{20}$ .

2-Prove that for each sequence, there is a constant $c$ such that $a_i$ $\leq$ $c$ for all $i$ $\geq$ 0. Note that the constant $c$ my depend on the numbers $a_0$ , $a_1$ and $a_2$ 3-Prove that, for each choice of $a_0$ , $a_1$ and $a_2$ , the resulting sequence is eventually periodic.

4-Prove that, the minimum length p of the period described in (3) is the same for all permitted starting values $a_0$ , $a_1$ , $a_2$ of the sequence

Let $x$ be a real number that satisfies $x^1 + x^{-1} = 3$ Prove that $x^n + x^{-n}$ is an positive integer , then prove that the positive integer $x^{3^{1437}}+x^{3^{-1437}}$ is divisible by at least $1439 \times 2^{1437}$ positive integers