Identifying All Ordered Pairs of Positive Integers Using Algebraic Equations

When dealing with algebraic equations, one fundamental task is to find all ordered pairs of positive integers that satisfy a given equation. This article explores a specific problem and its solution, providing a detailed analysis of the process and key steps involved.

Problem Statement

Consider the equation:

4x3y3 4a(b2(a2 - b2

Given that (x geq y), and (a) and (b) are positive integers, we need to find all ordered pairs ((x, y)) that satisfy this equation.

Solution Approach

Let's begin by simplifying and solving the equation step by step.

Rearrange the equation to simplify it:

b  x - y
4x3y3  3ab2a3

Note that:

x242xyy2 11a2 - 10b2

Substitute (b x - y) into the simplified equation:

b23a40  a244 - a

From the equation (3a40 a244 - a), we can determine the possible values for (a). Solving for (a):

44 - a  3a
44  4a
a  11

If (a 11), then (b 0), which results in (x y 22). Check that (xy 22 times 22) is a solution.

Check for other possible values of (a):

(a 2) (b^2 46 a^2 - 42 40): Not a solution because (frac{42}{21}) is not a perfect rational square. (a 3) (b^2 49 a^2 - 41 42): Not a solution because (frac{42}{21}) is not a perfect rational square. (a 4) (b^2 52 a^2 - 40 40): Not a solution because (frac{52}{4}) is not a perfect rational square.

Continue checking for higher values of (a):

(a 8) (b^2 64 a^2 - 36 48): Yes! (x 7), (y 1) (or vice versa).

Alternative Method

Another approach is to use the algebraic identity:

x3y3 x242xyy2

which simplifies to:

(frac{x}{y}(x-1)frac{y}{x}(y-1) 42)

Steps to Solve

Note that each term must be non-negative and an integer.

Let (frac{x}{y}(x-1) 42cos^2 z) and (frac{y}{x}(y-1) 42sin^2 z).

Multiplying both terms:

((x-1)(y-1) 42^2 cos^2 z sin^2 z 441 sin^2 2z 441 frac{1 - cos 4z}{2})

Since ((x-1)(y-1)) is an integer and (cos 4z) can only take values (0, pm 1, pm frac{1}{2}), these values do not allow ((x-1)(y-1)) to be an integer.

If (cos 4z 1), then ((x-1)(y-1) 0), so either (x 1) or (y 1). Neither works because the original equation is not satisfied.

If (cos 4z -1), then ((x-1)(y-1) 441).

Thus, the factor pairs of 441 are:

1 x 441 3 x 147 7 x 63 9 x 49 21 x 21

Since (x eq y), the only valid factor pairs are 21 x 21, which results in the ordered pair ([22, 22]).

Conclusion

The only ordered pairs of positive integers ((x, y)) that satisfy the given equation are ([1, 7]) and ([7, 1]).