Competition problem — Number theory

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Number theory Difficulty 4.8 AIME Prove it United States

Problem:
A sequence $a_{1}, a_{2}, \ldots$ of positive integers satisfies
$a_{n+1}=a_{n}^{3}+103$
for every positive integer $n$ . Prove that the sequence contains at most one perfect square.

Solution

Solution:
It's easy to check that no two consecutive terms can be perfect squares, since the only squares which differ by $103$ are $51^{2}$ and $52^{2}$ .

Now, note that squares are $0$ , $1$ , or $4$ mod $8$ . After a perfect square appears, the next term must be $-1$ or $0 \bmod 8$ , and thereafter all terms are $-1$ , $-2$ modulo $8$ , so no more squares.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.