How to Find All Integer Solutions of a Diophantine Equation

Introduction to Diophantine Equations

A Diophantine equation is an equation in which only integer solutions are sought. These equations often arise in number theory and have applications in various fields, including cryptography and algebraic geometry. The solution process revolves around systematically finding and verifying potential integer solutions. This article provides a step-by-step guide on how to find all integer solutions for a given Diophantine equation.

Problem Statement

The problem at hand involves finding all integer solutions to the Diophantine equation:

[ frac{xy}{x^2y^2 - xy} frac{5}{19} ]

To solve this, we will follow a methodical approach involving cross-multiplication, rearranging terms into a standard quadratic form, and then checking the discriminant to see if it is a perfect square. If it is, we can proceed to find the integer solutions for the variables.

Step 1: Cross-Multiplication and Simplification

We start by cross-multiplying to eliminate the fraction:

[ 19xy 5x^2y^2 - 5xy ]

Expanding and rearranging both sides gives us:

[ 19xy 5x^2y^2 - 5xy - 19x - 19y ]

Further simplification leads to:

[ 5x^2y^2 - 5xy - 19x - 19y 0 ]

Rewriting this equation in standard quadratic form:

[ 5x^2 - 5y - 19x - 5y^2 - 19y 0 ]

Step 2: Discriminant and Perfect Squares

To ensure that (x) is an integer, the discriminant must be a perfect square. The discriminant (D) for this quadratic equation in (x) is given by:

[ D 5y - 19^2 - 4 cdot 5 cdot (5y^2 - 19y) ]

Expanding and simplifying:

[ D 5y - 361 - 100y^2 380y ] [ D -100y^2 385y - 361 ]

For (D) to be a perfect square, we need:

[ -100y^2 385y - 361 k^2 ]

Step 3: Testing Integer Values

We then analyze this equation for various integer values of (y). Here are a few examples:

Case (y 1)

[ D -100(1)^2 385(1) - 361 -100 385 - 361 ] [ D -100 385 - 361 -86 ]

Since (-86) is not a perfect square, (y 1) is not a solution.

Case (y 2)

[ D -100(2)^2 385(2) - 361 -400 770 - 361 ] [ D 441 ]

Since (441 21^2) is a perfect square, (y 2) is a solution.

Step 4: Solving for (x)

Substituting (y 2) back into the original quadratic equation:

[ 5x^2 - 5(2) - 19x - 5(2^2) - 19(2) 0 ] [ 5x^2 - 10 - 19x - 20 - 38 0 ] [ 5x^2 - 19x - 68 0 ]

We solve this quadratic equation using the quadratic formula:

[ x frac{-(-19) pm sqrt{(-19)^2 - 4 cdot 5 cdot (-68)}}{2 cdot 5} ] [ x frac{19 pm sqrt{361 1360}}{10} ] [ x frac{19 pm sqrt{1721}}{10} ]

Since (sqrt{1721}) is not an integer, (x) is not an integer for (y 2).

Step 5: Further Testing

We continue to test further integer values of (y) until we find a perfect square or exhaust reasonable values. For instance, testing (y 3, 4, ldots) can provide more insight.

Conclusion

The process involves:

Cross-multiplying to clear the fraction. Rearranging into a standard quadratic form. Calculating the discriminant and checking for perfect squares. Solving the quadratic for (x) when the discriminant is a perfect square.

The integer solutions can be found through systematic checking and solving the resulting equations.

Summary of the Process

Cross-multiply to eliminate the fraction. Rearrange into a standard quadratic form. Calculate the discriminant and check for perfect squares. Solve the quadratic for (x) when the discriminant is a perfect square.

The method of systematically checking various integer values and solving for (x) provides a reliable approach to finding all integer solutions for Diophantine equations.