Exploring Integer Solutions to the Diophantine Equation xy / (x^2 - xy - y^2) 10/111

This article delves into the process of finding positive integer solutions to a specific Diophantine equation. We will employ techniques such as modular arithmetic and algebraic manipulation to derive the solution. This method not only provides an in-depth understanding but also demonstrates a structured approach to solving complex algebraic problems.

Introduction to the Equation

The given Diophantine equation is:

[ frac{xy}{x^2 - xy - y^2} frac{10}{111} ]

This equation can be rewritten as:

[ 111xy 1^2 - 1y - 10y^2 ]

Initial Steps and Divisibility

A key step is to establish the divisibility properties of (x) and (y). From the equation (111xy 1^2 - 1y - 10y^2), it is clear that the left-hand side (LHS) of the equation is divisible by 3:

[ 3 mid 111xy 1^2 - 1y - 10y^2 Rightarrow 3 mid x^2 - xy - y^2 ]

By examining the squares modulo 3, we find that (2x - y^2 equiv 0 pmod{3}). This implies that (2x equiv y^2 pmod{3}), which further leads to (x equiv y pmod{3}) or (x equiv -y pmod{3}). This establishes that both (x) and (y) are divisible by 3.

Substitution and Simplification

Given (3 mid x) and (3 mid y), we can write (x 3m) and (y 3n) for some positive integers (m) and (n). Substituting these into the equation, we get:

[ 111(3m)(3n) 10(3m)^2 - 10(3m)(3n) - 10(3n)^2 ]

Further simplification yields:

[ 10m^2 - 10mn - 10n^2 - 37m - 37n 0 ]

Reformulating the Equation

To solve the above equation, we will reformulate and complete the square. The equation can be rewritten as:

[ 10m^2 - 10mn - 10n^2 37m 37n ]

Carefully completing the square, we obtain:

[ 10m^2 - 10mn - 10n^2 - 3710m - 5n^2 370n - 37m ]

Further simplification gives:

[ 20m - 10n - 37^2 300n^2 - 2220n ]

Finally, completing the square in (n) leads to:

[ 20m - 10n - 37^2 - 310n 37^2 4 cdot 37^2 ]

Solving for (m) and (n)

Letting (a 20m - 10n - 37) and (b 10n - 37), we need to solve:

[ a^2 - 3b^2 4 cdot 37^2 ]

This equation has solutions in all sign combinations:

[ ab pm 73, pm 7, pm 47, pm 33, pm 37, pm 37, pm 26, pm 40, pm 74, 0 ]

Considering the constraints, we find that the valid solutions are:

[ ab -73, -7, 47, 33, -37, -37, -26, -40, -74 ]

With the conditions (b equiv 3 pmod{10}), this reduces us to:

[ ab -73, -7, 47, 33, -37, -37, -26, -40, -74 ]

Solving for (m) and (n), we find that (mn 7, 3, 3, 7).

Conclusion

Finally, since (x 3m) and (y 3n), the integer solutions are:

[ boxed{xy 21, 9, 9, 21} ]

This detailed process not only solves the given Diophantine equation but also illustrates the fundamental techniques of modular arithmetic and algebraic manipulation in handling such problems.