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The conjecture states that if $A$ is a set of natural numbers and $$\sum_{n\in A}\frac1n=\infty,$$ then $A$ contains arbitratily long arithmetic progressions.

I wonder has it been proved?

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    $\begingroup$ I edited it so curious people don't need to visit wikipedia if they don't know the conjecture. Anyhow, it is a safe bet that if it had been proved, or even if a credible proof claim had been announced, the wikipedia article would have said so. $\endgroup$ – Harald Hanche-Olsen Nov 7 '12 at 8:59
  • $\begingroup$ @HaraldHanche-Olsen, ok. thank you. $\endgroup$ – hxhxhx88 Nov 7 '12 at 9:00
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    $\begingroup$ I've deleted my answer to this post (that the resolution of this conjecture is likely years away), as it is unhelpful. $\endgroup$ – JavaMan Nov 9 '12 at 5:07
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This is open even in the simplest case of arithmetic progressions of lengths three. The best result in this direction, for three-terms, is due to T. Bloom building on work of Sanders, and the paper reviews earlier contributions.

You can also see a recent related MO question for further references.

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