Competition problem — Number theory

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Number theory Difficulty 5.8 AIME, harder Prove it Romania

Find all natural numbers $m, n$ so that $85^m - n^4 = 4$ .

Solution

$(n - 1)^2 = 5^m - 1 \quad \text{and} \quad (n + 1)^2 = 17^m - 1.$
For $m > 1$ there are many (more than one) perfect squares between $5^m - 1$ and $17^m - 1$ (e.g. $9^m$ and $16^m$ ), therefore $m = 1$ and $n = 3$ .

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