Let $m, n \geq 1$ be integers that are coprime. Determine the GCD of $5^{m}+7^{m}$ and $5^{n}+7^{n}$

## Order
Let $s_{n}=5^{n}+7^{n}$. If $n \geq 2 m$, we have

$$
s_{n}=s_{m} s_{n-m}-5^{m} 7^{m} s_{n-2 m}
$$

so that the GCD of $s_{m}$ and $s_{n}$ is equal to the GCD of $s_{m}$ and $s_{n-2 m}$. Similarly, if $m < n < 2 m$, we have

$$
s_{n}=s_{m} s_{n-m}-5^{n-m} 7^{n-m} s_{2 m-n}
$$

so that the GCD of $s_{m}$ and $s_{n}$ is equal to the GCD of $s_{m}$ and $s_{2 m-n}$.

Thus, if $m+n$ is even, the desired GCD is that of $s_{1}$ and $s_{1}$, which is 12, and otherwise that of $s_{0}$ and $s_{1}$, which is 2.