Number Theory
# 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 positive integers \(x\) such that \(\sqrt{\sqrt{x^2+1000x}-x}\) is an integer.

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