Question #43eb0

May 27, 2015

Warning: this is only a partial solution!

If ${\left(5 - \sqrt{x}\right)}^{2} = y - 20 \sqrt{2}$
then
$y = x - 10 \sqrt{x} + 25 + 20 \sqrt{2}$

$= \left(x + 25\right) - 10 \left(\sqrt{x} + 2 \sqrt{2}\right)$

If $x$ is an integer (given)
then for $y$ to be an integer
$\sqrt{x} - 2 \sqrt{2}$ must be an integer.

There is, of course, the obvious solution:
$x = 8$ since $\sqrt{8} = 2 \sqrt{2}$
(which implies $y = 33$).

The problem is in demonstrating that this is the only solution (or finding more solutions).

Anyone have further thoughts on this?