parmenides51 wrote:
Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$
Entrepreneur wrote:
$n=1$ appears to be the only solution but I am unable to prove it.
For \( n \geq 2 \), we have \( n^2 \geq 2n \), from which it follows that
\[
9^{n^2} - 3^{n^2} = 3^{n^2} \left(3^{n^2} - 1\right) \geq 3^{2n} \left(3^{2n} - 1\right) = 81^n - 9^n
\]Since \( 81^n - 9^n > 23^n - 17^n \), the equation has no solutions for \( n \geq 2 \).