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Q: $n\in\mathbb{Z}$. Prove that $3 | 2n^2 +1$ iff $3\nmid n$.

I began by assuming not true. Then $3|n$ implies $n=3k$ where $k\in\mathbb{Z}$.

$3 | 2(3k)^2 +1$ $\rightarrow$ $3 | 2(9k^2)+1$ $\rightarrow$ $3 | 18k^2 +1$.

$3 | 3(6k^2 + \dfrac{1}{3})$ but $6k^2 + \dfrac{1}{3}$ is not an integer. Contradiction.

Is this proof valid?

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    $\begingroup$ The thing you are trying to prove (at least what you've stated) is not true. $\endgroup$
    – paw88789
    Feb 16 at 15:17
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    $\begingroup$ There is a typo somewhere, we have $3\mid 2n^2+1$ iff $3\nmid n$ instead. $\endgroup$
    – Peter
    Feb 16 at 15:18
  • $\begingroup$ Fixed it @paw88789 $\endgroup$ Feb 16 at 15:21
  • $\begingroup$ The statement you are asked to prove asserts "if and only if". Your hypothetical proof by contradiction will still need a two way argument. Better strategy: there are only three cases modulo $3$. Check each one separately. $\endgroup$ Feb 16 at 15:24
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    $\begingroup$ No. You need to check the cases $n \equiv 0,1,2 \pmod{3}$. You can do a lot less algebra if you do the arithmetic modulo $3$ rather than with the forms $3k+r$. Then you see $2^2 \equiv 1$ immediately. $\endgroup$ Feb 16 at 15:39

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Your proof is valid, but here's an easier way to see and write this- $$3|n\iff 3|n^2\iff 3|2n^2\iff 3\nmid 2n^2+1$$ The idea is just to realise that $n$ cannot divide both $f(n)$ and $f(n)+1$.


Since the OP had already solved the problem, I didn't care much about the details. But, since it was pointed out in the comments, here's the details-

$$n=3k+1\implies 2n^2+1=2(3k^\prime)+2+1\equiv 0 \;(\operatorname{mod }3)$$ $$n=3k+2\implies 2n^2+1=2(3k^\prime)+2.2^2+1\equiv 0\;(\operatorname{mod }3)$$

which completes the argument.

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    $\begingroup$ The last assertion in this answer requires more work. It's not true in general that if $3$ does not divide $A+1$ then it divides $A$. $\endgroup$ Feb 16 at 15:27
  • $\begingroup$ @EthanBolker placed an edit. $\endgroup$ Feb 16 at 22:08

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