MathNet 2019-part-01 — 184 problems (EN–VI) Question 66

Let $F_n$ be a sequence defined recursively by $F_1 = F_2 = 1$ and $F_{n+1} = F_n + F_{n-1}$ for $n \ge 2$. Find all pairs of positive integers $(x, y)$ such that
$$
5F_x - 3F_y = 1.
$$