# Thread: Diophantine Equation? Or something?

1. ## Diophantine Equation? Or something?

Lost. Sorry.

2. $\displaystyle 4x^2 - y^2 = (2x + y)(2x - y) = 480$.

Now list all the pairs of numbers that can multiply to 480:

$\displaystyle (480, 1); (240, 2); (160, 3); (120, 4); (96, 5); (80, 6);$

$\displaystyle (60, 8); (48, 10); (40, 12); (32, 15); (30, 16); (24, 20)$

The pairs with one odd number and one even number will result in y not being an integer, so you can throw those out. Also, the pairs such that the sum of the numbers is not divisible by 4 will result in x not being an integer, so you can throw those out.

1. $\displaystyle 2x + y = 120$

$\displaystyle 2x - y = 4$

$\displaystyle 4x = 124$

$\displaystyle x = 31, y = 58$ is a solution.

2. $\displaystyle 2x + y = 60$

$\displaystyle 2x - y = 8$

$\displaystyle 4x = 68$

$\displaystyle x = 17, y = 26$ is a solution.

3. $\displaystyle 2x + y = 40$

$\displaystyle 2x - y = 12$

$\displaystyle 4x = 52$

$\displaystyle x = 13, y = 14$ is a solution.

4. $\displaystyle 2x + y = 24$

$\displaystyle 2x - y = 20$

$\displaystyle 4x = 44$

$\displaystyle x = 11, y = 2$ is a solution.

Hence the solutions are $\displaystyle (31, 58); (17, 26); (13, 14); (11, 2)$.

3. Very detailed. Thank you muchly.