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Italian Mathematical Olympiad, 2003

Algebra Difficulty 4.6 AIME Prove it Italy

Problem:

Let $x_{0}, x_{1}, x_{2}, \ldots$ be the sequence defined by $x_{0}=2$ and $x_{n+1}=5+\left(x_{n}\right)^{2}$ for every $n \geq 0$ . Prove that this sequence does not contain any prime numbers other than 2.

Solution

Solution:

Note that the numbers in question are alternately even and odd, because $5$ is added to the square of the previous term. Therefore only the terms in odd position can give other primes. But these are all multiples of $3$ and greater than or equal to $x_{1}=9$ . Indeed the sequence is clearly increasing, and in passing from $x_{n}$ to $x_{n+2}=5+\left(5+x_{n}\right)^{2}=30+10\left(x_{n}\right)^{2}+\left(x_{n}\right)^{4}$ divisibility by $3$ is preserved.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.