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This is my question

https://mathoverflow.net/questions/262039/central-properties-of-the-integers-mathbbz-over-other-structures-with-sum

I think the question is clear enough and it is easy to understand what I'm asking. The question is not easy, it is difficult, but this is not a good excuse to closed it or put in on-hold. It should be open because someone could answer it. I could be agreed on put it as community-wiki, but why have they closed it?

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    $\begingroup$ It was closed for the reasons stated, mainly insufficient clarity. I have studied logic and some model theory, and I know what wording I would use for a question like this. However that would reflect my experience and understanding, not yours, so it would not be the same question. There is the chance of reopening the question after you revised it. I suggest not using anabelian geometry as an example until you can make its relevance clear. You might also respond to the comments already made: do they answer your intent? Gerhard "This Needs More Community Understanding" Paseman, 2017.02.12. $\endgroup$ – Gerhard Paseman Feb 12 '17 at 21:17
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    $\begingroup$ @GerhardPaseman Excuse me, what do you think it is bad in the redaction exactly? What is not clear about what Z is and what Z is not? $\endgroup$ – Chronos Feb 12 '17 at 21:26
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    $\begingroup$ For starters, your sentences run on, making them hard to parse. At this writing, I give up on sentence 2 paragraph 1, sentence 1 paragraph 2. I am unsure that paragraph 3 is relevant to your intent; an explanation for those ignorant of anabelian geometry might help. Finally, the basic answer you seek is simple but impossible to justify using equational logic: The sum of two like powers of high potency are also not powers of the same potency. But this is just a restatement. I think just looking at structures is not enough. Gerhard "FLT Is Not An Equation" Paseman, 2017.02.12. $\endgroup$ – Gerhard Paseman Feb 12 '17 at 21:35
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I will do my best to give an honest interpretation of what is happening here, with the caveat that cutting-edge research into FLT is not within my range of expertise. (Disclaimer: I didn't vote up or down on the question, and (obviously) wasn't one of the closers; I am responding as a site moderator with fairly broad acquaintance with how the MathOverflow site works.)

You say the question is "clearly research level". If you mean this question is posed with a view to doing research to simplify the proof of the FLT (as suggested by the question title), then I think the message that is coming from commenters such as Olivier is that the question as posed is really not the right type of question to ask. Please bear in mind that the reason for closure given, "unclear what you're asking", is taken from a very restricted menu of message templates accompanying question closures, and might not capture too accurately what the actual misgivings of the closers are. I have a feeling that it's more like "what you're asking is not useful" (if you hover your mouse over the vote up/down buttons, you'll see that's one of the criteria).

It would be good if closers see this thread and could amplify further, but my guess is that the truth of the specific FLT statement is, according to current understanding, more like an epiphenomenon of deeper truths such as the modularity conjecture. Meaning that asking what ring-theoretic properties of $\mathbb{Z}$ are responsible probably doesn't reflect, to the minds of the closers, a research-level acquaintance with what is truly at stake; to put it another way, it's very likely unclear what sort of response to the question as asked could be really satisfactory to people who really understand this stuff.

Don't be too discouraged. The bar at MathOverflow is set pretty high, and it can take a while to figure out what types of questions are likely to fly. The bar is set even higher for highly notorious propositions like FLT and RH, so that any question touching on them had better be coming from someone who shows a pretty deep engagement with the underlying issues, and it looks like the question as posed doesn't quite cut it in this respect. My advice would be to aim at a level closer to where you've really done serious mathematical battle. Paraphrasing one form of advice of Polya: if there's a question you can't solve, there's probably a simpler question that you also can't quite figure out. Keep going until you're right at the edge of your understanding.

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    $\begingroup$ That is good advice from Polya. $\endgroup$ – Guido Jorg Feb 26 '17 at 19:21

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