Question 60
Let the sequences $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ satisfy $a_0 = b_0 = 1$, $a_n = 9a_{n-1} - 2b_{n-1}$ and $b_n = 2a_{n-1} + 4b_{n-1}$ for all positive integers $n$. Let $c_n = a_n + b_n$ for all positive integers $n$. Prove that there do not exist positive integers $k, r, m$ such that $c_r^2 = c_k c_m$.