Diophantine Equations | The Factor Method

Let's understand the factor method of Diophantine equations step-by-step. Aso, try the question related to it.

Diophantine Equations

Consider an equation for which we seek only integer solutions. There is no standard technique of solving such a problem, though there are some common heuristics that you may apply. A simple example is $ x^2 - y^2 = 31$. Suppose we wish to find out the integer solutions to this equation.

First notice that if 'x' and 'y' are solutions, so are '-x' and '-y' (and vice versa). So it is sufficient to investigate positive solutions.

The factor method relies on the following steps:

1. First, bring all variables to one side of the equality sign and constants to the other side.
2. Try to factorize left-hand expression and compare it with the factorization of the right hand constant

Illustration

$ x^2 - y^2 = 31 \newline (x-y)(x+y) = 31 $

But 31 is a prime. So the only way 31 can be written as a product of two positive numbers is 1 times 31.

Since x-y is smaller, the only possibility is x-y=1, x+y=31, giving solutions x=16, y=15

Problems

• $ \frac{1}{x} + \frac{1}{y} = \frac{1}{6} $; find all integer x, y that satisfies the solutions
• $ (xy-7)^2 = x^2 + y^2 $

Some Useful Links:

Math Olympiad Program

PRMO Problems and Solutions

AM - GM Inequality - Video