Prove that \(5^n-3^n\) is not divisible by \(2^n+65\) for any positive integer \(n\).
Suppose again that \(5^{n} \equiv 3^{n} \pmod {m=2^{n}+65}\). Like in the first solution, we conclude that \(n\) must be odd, and \(n \geqslant 3\), so \(8 \mid 2^{n}\).
Using Jacobi symbols,
\[-1=\left(\frac{2^{n}+65}{5}\right)=\left(\frac{5}{2^{n}+65}\right)=\left(\frac{5^{n}}{2^{n}+65}\right)=\left(\frac{3^{n}}{2^{n}+65}\right)=\left(\frac{3}{2^{n}+65}\right)=\left(\frac{2^{n}+65}{3}\right)=1,\]
contradiction.