MathNet 2025 — Part 02 (100 complete bilingual questions) Question 10

Consider a sequence of integers $a_1, a_2, a_3, \dots$ such that $a_1 \gt 1$ and $(2^{a_n} - 1)a_{n+1}$ is a square for all positive integers $n$. Is it possible that two terms of such a sequence be equal?