一组二次方程与斐波那契类序列

给定两个互素数$a$和$b$，已知恰好有一个二次方程在非负整数解$(x,y)$：$$ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}.$$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.