# Diophantine Equation? Or something?

• Mar 1st 2010, 02:43 PM
davismj
Diophantine Equation? Or something?
• Mar 1st 2010, 03:07 PM
icemanfan
\$\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)\$.
• Mar 1st 2010, 03:10 PM
davismj
Very detailed. Thank you muchly. :)