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Original Problem: Counterexample given below by user francis-jamet.

Let $A\subset \mathbb Z_n$ for some $n\in \mathbb{N}$.

If $A-A=\mathbb Z_n$, then $0\in A+A+A$


New Problem: Is the following statement true? If not, please give a counterexample.

If $A-A=\mathbb Z_n$ and $0\not\in A+A$, then $0\in A+A+A$.

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    I've restored the original problem to go side by side with new problem. I think it is less confusing for readers this way.2012-06-28

1 Answers 1

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For the original problem, there is a counterexample for $n=24$ and $A=\{3,9,11,15,20,21,23\}$.

There are no counterexamples for $n \leq 23$.

For the new problem, there is a counterexample:

$n=29$ and $A=\{4,5,6,9,13,22,28\}$.

There are no counterexamples for $n \leq 28$.

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    @sdcvvc Interesting, as it turns out there are only two other counterexamples for n=29. [2, 3, 11, 14, 17, 19, 21] and it's negation along with the other two francis and I have already listed.2012-06-29