Prove the equation 2x^2 + y^2 = 14 has no positive integer solutions.

Question

anonymous · Answer

Solving for y, $$y=\pm \sqrt{14-2x^2}$$
for y to be a positive integer, (14-2x^2) must be a perfect square.
Any positive integer values chosen for x will not yield a perfect square.
There's probably a more rigorous way, but that's my "but just look it!" proof.  ;-)

anonymous · Answer

It's easy to check because the only positive integers in the domain of x are 1 and 2.

anonymous · Answer

Yea, ... I agree with your "just look at it" proof, haha.  I agree... I'm trying to think of a 'numbered list' with facts approach.  Not really sure yet what the teacher expects... may just do an exhaustive proof case by case of the surrounding perfect squares to show that neither x nor y can be integers.  : /    Thanks!

anonymous · Answer

Yeah, it's essentially a proof by example, or proof by contradiction or maybe it's called reductio ad absurdum.. I forget, but basically it means assume that there are positive integer solutions and show that these leads to an absurd result.

anonymous · Answer

Cool!  Appreciate the help.