A Diophantine Story Of The Prime Civilization

We'll show there are no solutions to the first equation.

First note that we need $q>p$ , otherwise the left-hand side is positive and the right-hand side is not.

If $p>2$ , then both $p$ and $q$ are odd. In this case, the left-hand side is even and the right-hand side is odd, so there are no solutions.

Hence the only possibility is $p=2$ , and we would need to solve $2^q+q^2=112q^4-448q^2+7$

The right-hand side of the equation is always divisible by $7$ ; but it's easy to check (by case bashing $q \pmod7$ ) that the left-hand side is never divisible by $7$ , for any natural number $q$ , so again there are no solutions.