Question 127
Let the general term of sequence $\{a_n\}$ be
$$
a_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right), \quad n = 1, 2, \dots
$$
Prove that there exist infinite positive integers $m$ such that $a_{m+4}a_m - 1$ are perfect squares.
$$
a_n = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right), \quad n = 1, 2, \dots
$$
Prove that there exist infinite positive integers $m$ such that $a_{m+4}a_m - 1$ are perfect squares.