General Diophantine Equations

How many ordered pairs $(a,b)$ of positive integers satisfy $a^{b} = 64$ ?

Find the sum of all positive integers $b$ such that $b^2=a^3+1$ , where $a$ is a prime number.

What is the sum of integers $x$ that satisfy $\left(x^2-3x+1\right)^{24-x}=1?$

Find the sum of all numbers $N$ that are perfect squares, and can also be written as a product of 4 consecutive odd integers.

Find the sum of all positive integers $x$ such that $\sqrt{\sqrt{x^2+1000x}-x}$ is an integer.