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## General Diophantine Equations

We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.

# Factoring

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.

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