Introduction

Exploring the integer solutions to the equation 2x^2 - 1 y^2 - y delves into the realm of number theory and algebra. This article details the process of solving this equation and highlights the steps to identify integer solutions through a combination of algebraic manipulation and quadratic formula application.

Solving the Equation

To begin, the given equation is rearranged:

?y^2 - y - 2x^2 2 0

This is a quadratic equation in the variable y. Using the quadratic formula y frac{-b pm sqrt{b^2 - 4ac}}{2a}, where a 1, b -1, and c -2x^2 2, we substitute these values to solve for y.

The quadratic formula becomes:

y  frac{1 pm sqrt{(-1)^2 - 4 cdot 1 cdot (-2x^2   2)}}{2 cdot 1}

After simplification, this reduces to:

y  frac{1 pm sqrt{1   8x^2 - 8}}{2}  frac{1 pm sqrt{8x^2 - 7}}{2}

For y to be an integer, the expression under the square root 8x^2 - 7 must be a perfect square. Let k^2 8x^2 - 7 for some integer k. Rearranging gives:

8x^2 - k^2  7

This can be factored as:

(2x - k)(2x   k)  7

The integer pairs that multiply to 7 can be listed as:

(1, 7) (7, 1) (-1, -7) (-7, -1)

Now, we will solve each case.

Case 1: 2x - k 1 and 2x k 7

Adding the equations gives: 4x 8, so x 2. Substituting into 2x - k 1 gives k 3. This results in y frac{1 pm 3}{2}, giving y 2 or y -1.

Case 2: 2x - k 7 and 2x k 1

Similar calculations as Case 1 give the same x 2 and y 2 or y -1.

Case 3: 2x - k -1 and 2x k -7

Adding the equations gives: 4x -8, so x -2. Substituting into 2x - k -1 gives k -3. This also gives y frac{1 pm 3}{2}, resulting in y 2 or y -1.

Case 4: 2x - k -7 and 2x k -1

Similar to Case 3, this again yields x -2 and y 2 or y -1.

In summary, the integer solutions for x, y are:

(2, 2) (2, -1) (-2, 2) (-2, -1)

The complete set of integer solutions is: (2, 2), (2, -1), (-2, 2), (-2, -1).

Verification Using J Programming Language

The J programming language provides a way to verify the integer solutions by generating all pairs within a specific range and testing them:

x y.: a. odo 2 10000 - test all x y pairs and list solutions
│1 0│1 1│2 3│4 6│11 16│23 33│64 91│134 190│373 528│781 1105│2174 3075│4552 6438│

Note that there are an infinite number of solutions, but this list includes the first 12 pairs within the range from 1 to 10000 that satisfy the equation.

Conclusion

Identifying integer solutions to such equations is an exercise in algebra and number theory, with applications in various fields including cryptography and problem-solving. The method outlined here can be generalized to solve similar types of quadratic equations, particularly those involving perfect squares.