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Reading What topics can I ask about here?, it's unclear whether this kind of question is on-topic or not.

Here is my draft question:

The Fermat's Last Theorem took 358 years of effort to prove. It's a famous problem, no one doubts it, but I wonder whether it gained a large amount of attention largely because of the popularity of Fermat and his margin story, or because it plays an important role in math per se. Yes, it's hard, and it stimulated the development of algebraic number theory and the proof of the modularity theorem, but this doesn't guarantee its importance. I guess there are many other conjectures out there that are as hard or as important as it, but they don't gain much attention like this one.

So is there a way to know its importance in math? I'm not sure how "importance" should be defined and measured, but I propose that it's the number of problems/papers using it in the proof of a different problem, in contrast with the number of proofs/papers trying to prove it.

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    $\begingroup$ In asking about the "importance" of a mathematical theorem, a Question should reflect the research effort that went before making the post. This is true of all Questions but is especially relevant to conveying what specific meaning importance has to the OP. For example, you cite some history for the proposition "this doesn't guarantee it's importance". Perhaps you should clarify the distinction between "importance in math" and "gain[ing] much attention like this one." $\endgroup$
    – hardmath
    Nov 23 at 16:02
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    $\begingroup$ I'm not sure how "importance" is defined and measured, but I propose that it's the number of problems/papers using it in the proof of a different problem, in contrast with the number of proofs/papers trying to prove it. Is that better? $\endgroup$
    – Ooker
    Nov 23 at 16:10
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    $\begingroup$ In general, questions on Math SE should admit authoritative, objectively correct answers. Questions about the "importance" of a theorem are subjective questions, and are, as a general guideline, off-topic here. There is some guidance on this in the Help Center. I would suggest that your best bet is to try to find a more narrow, more objective question (e.g. "What results follow from FLT?" or "Does FLT have any applications?" or something in that vein. $\endgroup$
    – Xander Henderson Mod
    Nov 23 at 16:24
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    $\begingroup$ For what it's worth, I suspect its fame has much more to do with the fact that it only requires (very) basic school algebra to understand the problem, and also the fact that the problem is closely connected with something many people encounter in school, namely integer possibilities for the Pythagorean theorem -- the 3-4-5 triangle, the 5-12-13 triangle, etc. that frequently shows up in school math and is something that those studying for standardized tests such as SAT, ACT, GMAT, GRE, NMAT, CAT, NCEE (gaokao), etc. become very familiar with. (continued) $\endgroup$ Nov 23 at 16:45
  • $\begingroup$ Of course, Fermat's theorem was very famous long before any of these tests, but integer possibilities for the Pythagorean theorem were still well known and often exploited in lower level math 200+ years ago (and various university "admissions tests" 100+ years ago). $\endgroup$ Nov 23 at 16:51
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    $\begingroup$ I would likely vote to close as opinionated, if posted on math.se. It is a soft question, but the question itself is a conjecture forwarding a particular perspective, without adequate research. I get that you've that hard about a means to objectively operationalize the question, but what would that number mean, in relation to other other theorems you haven't investigated as control groups? $\endgroup$
    – amWhy
    Nov 23 at 18:01
  • $\begingroup$ @XanderHenderson would questions about how to measure the impact/importance of a conjecture objective and on-topic? $\endgroup$
    – Ooker
    Nov 24 at 14:24
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    $\begingroup$ @Ooker I don't know? There is likely a way to ask such a question so that it is on-topic (indeed, I suggested ways, above, that this question could be rephrased in an objective manner so that it would be on-topic). It really depends on how, exactly, the question is asked. $\endgroup$
    – Xander Henderson Mod
    Nov 24 at 14:29
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    $\begingroup$ Of course math.se is not an opinion forum, and not a discussion forum. So keep that in mind when formulating all questions. If you want opinions or discussion, you should find the right forum for that. $\endgroup$
    – GEdgar
    Nov 25 at 12:43

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