Prove that if n=2 (mod 3) then it cannot be a square.

No idea where to start with this one. Any pointers please?

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- Jul 20th 2008, 11:35 AMfree_to_flyHelp on proving this statement...
Prove that if n=2 (mod 3) then it cannot be a square.

No idea where to start with this one. Any pointers please? - Jul 20th 2008, 11:44 AMMoo
Hmm a table of congruences would work.

Assume that $\displaystyle n=k^2$

$\displaystyle \text{If } k=0 \bmod 3 \text{ then } n=k^2=0 \bmod 3$

$\displaystyle \text{If } k=1 \bmod 3 \text{ then } n=k^2=1 \bmod 3$

$\displaystyle \text{If } k=2 \bmod 3 \text{ then } n=k^2=4 \bmod 3=1 \bmod 3$

And these are all the possibilities. You can see that none of the results are $\displaystyle 2 \bmod 3$ :p - Jul 20th 2008, 11:47 AMThePerfectHacker
Here is an interesting fact. The only (odd) primes that have 2 as a square are of the form $\displaystyle 8k\pm 1$. (Since $\displaystyle p=3$ does not have this form this form it is not a square of $\displaystyle 2$. This is not meant to be a proof of what you asked, just an interesting fact).