Diophantine Equations (the factor method)

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 factorise left hand expression and compare it with the factorisation 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 number 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
  • \((xy-7)^2 = x^2 + y^2 \)

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