Prove or disprove that for all positive integers $n$, $2^n +3^n$ and $5^n + 7^n$ are always coprime.
This is not a homework problem, it is merely a problem I set myself after doing some number theory drills.
I tried multiplying $2^n +3^n$ by $5^n$, and $5^n + 7^n$ by $2^n$. Then, I subtracted both to deduce that the gcd divides $15^n - 14^n$. But I don't know if this helps at all.
$2^5+3^5$ is divisible by $11$. So is $5^5+7^5$.