Let $n \in \mathbb{N}$, show that $$f(n) = \frac{(2+\sqrt{3})^{1+2n}+(2-\sqrt{3})^{1+2n}+2}{6}$$ is a square.

For $n=1$ to $5$ we have ($n, \ \sqrt{f(n)}, \ f(n)$):

$\begin{array}{lll}

1. & 3. & 9. \\

2. & 11. & 121. \\

3. & 41. & 1681. \\

4. & 153. & 23409. \\

5. & 571. & 326041.

\end{array}$

2-2018 Teaching by misleading

2 months ago

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