Problem

Source: 2010 Saudi Arabia IMO TST V p2

Tags: number theory, algebra, Sequence



a) Prove that for each positive integer $n$ there is a unique positive integer $a_n$ such that $$(1 + \sqrt5)^n =\sqrt{a_n} + \sqrt{a_n+4^n} . $$b) Prove that $a_{2010}$ is divisible by $5\cdot 4^{2009}$ and find the quotient