Number Theory

General Diophantine Equations

Diophantine Equations - Solve by Factoring

         

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

Find the sum of all positive integers bb such that b2=a3+1b^2=a^3+1, where aa is a prime number.

What is the sum of integers xx that satisfy (x23x+1)24x=1?\left(x^2-3x+1\right)^{24-x}=1?

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

Find the sum of all positive integers xx such that x2+1000xx\sqrt{\sqrt{x^2+1000x}-x} is an integer.

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