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 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.