Find four prime numbers less than $100$ which are factors of $3^{32}-2^{32}$.

## Solution

Factorizing,we have

\begin{align} 3^{32}-2^{32} &=&(3^{16}+2^{16})((3^{16}-2^{16})\ &=&(3^{16}+2^{16})(3^{8}+2^{8})(3^{4}+2^{4})(3^{2}+2^{2})(3^{2}-2^{2}) \label{y2006r1p1e2} \end{align}

From Fermat’s Little Theorem, we have $3^{17-1}-1 - (2^{17-1}-1)$ is divisible by $17$.

From $\ref{y2006r1p1e2}$, we see that $(3^{32}-2^{32})$ is divisible by $5$, $13$ and $97$.