Let $ k $ belong to the naturals. Prove that $3k^2+2$ is never a perfect square

Question

I'm been struggling with this proof for a couple of hours. I originally thought I could prove it by contradiction and let some $n^2=3k^2+2 $to prove there is a contradiction, but it got me nowhere. Any hints?

As a follow up to @RossMillikan's very good hint: consider what you get from $(3k)^2\pmod 3$, $(3k+1)^2\pmod 3$ and $(3k+2)^2\pmod 3$. — Cameron Williams, Feb 16 '15 at 04:39

score 5 · Accepted Answer · answered Feb 16 '15 at 03:20

5

Consider your equation $\pmod 3$

answered Feb 16 '15 at 03:20

Ross Millikan

374,822

score 3 · Answer 2 · answered Feb 16 '15 at 14:30

3

If exist $n$ integer such as $3k^2=n^2-2$

then $3|(n^2-2)$

that is $n^2\equiv2$ (mod 3).

But the last congruence has no solutions because the only squares (mod 3) are 1 and 0.

answered Feb 16 '15 at 14:30

Kamala

170

Nice first post! – Hans Engler Feb 16 '15 at 14:37

Let $ k $ belong to the naturals. Prove that $3k^2+2$ is never a perfect square

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