Question 27
Let k be a fixed integer greater than 1, and let $ m = 4k^{2} - 5 $. Show that there exist positive integers a and b such that the sequence $ (x_{n}) $ defined by
$$ x_{0}=a,\quad x_{1}=b,\quad x_{n+2}=x_{n+1}+x_{n}\quad for\quad n=0,1,2,\ldots $$
has all of its terms relatively prime to $ m $.
$$ x_{0}=a,\quad x_{1}=b,\quad x_{n+2}=x_{n+1}+x_{n}\quad for\quad n=0,1,2,\ldots $$
has all of its terms relatively prime to $ m $.