Integrality From a Radical Recurrence

★ 9/10number theory Recursion Modular Arithmetic sequences Vieta's Formulas

Problem

A sequence has $a_0 = 1$ and

a_{n+1} = \frac{7a_n + \sqrt{45a_n^2 - 36}}{2}.

1. Prove that every $a_n$ is a positive integer. 2. Prove that $a_na_{n+1} - 1$ is always a perfect square.

Answer

Solution

Difficulty	9/10
Topics	number theory, Recursion, Modular Arithmetic, sequences, Vieta's Formulas

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