Consider Notorious & Difficult Number Theory Problem X (hereafter referred to as "The Problem"). Let's say The Problem has already been solved using “heavy machinery”, so that we can all agree upon the truth of the proposition. Let's now say I have an idea on a possible elementary method of attacking The Problem, and I've worked through some of it, but it's not anywhere near a complete proof. And let's assume, for argument's sake, that I'm not a total crank — call me "an optimistic amateur" instead, with perhaps even a few articles published in well-respected peer-reviewed number theory journals.

How best to start such a thread — with the intentional of rationally discussing the idea towards either a proof, a partial proof (or new partial result), or proving that the idea is a clear dead-end — without the thread spiralling either into total crank-dom on the one hand, or total crank-lynching-dom on the other?

Furthermore, should it immediately be declared community-wiki?

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    $\begingroup$ For better or worse, this site is not well suited for discussion. Indeed, the help page math.stackexchange.com/help/dont-ask states, If your motivation for asking the question is “I would like to participate in a discussion about ______”, then you should not be asking here. So, if your intention is to discuss the general ideas for a proof, you may need to look elsewhere. One posible model is the Polymath project, en.wikipedia.org/wiki/Polymath_Project $\endgroup$ – Carl Mummert Aug 24 '14 at 15:53
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    $\begingroup$ The site is for Questions and Answers, not for Ideas. It often happens that an Idea you have leads to a good, concrete Question, in which case it can certainly be asked, as any other Question. $\endgroup$ – user147263 Aug 24 '14 at 16:00
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    $\begingroup$ Here is a concrete question you could ask without being viewed as a crank immediatly: "What obstacles are there to obtaining X by method Y". But if you cannot formulate it as a concrete question, this is indeed the wrong place. $\endgroup$ – Michael Greinecker Aug 24 '14 at 16:01
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    $\begingroup$ Even the question that Michael Greinecker proposes, although it is a relatively concrete question, might even attract some closure votes. My advice would be to be as concrete as possible, and not to ask many questions of that sort in a short period of time. $\endgroup$ – Carl Mummert Aug 24 '14 at 16:02
  • $\begingroup$ Reading about "especially taboo questions" in the header of a question... is MSE for question&answers in math or about hypotheses on social issues in the social/scientific community? $\endgroup$ – Gottfried Helms Aug 24 '14 at 20:08
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    $\begingroup$ @GottfriedHelms The former, but questions of the form "Is there anything wrong with my proof of twin primes conjecture?" tend to fare poorly. $\endgroup$ – user147263 Aug 24 '14 at 21:31
  • $\begingroup$ Late comment, but in my opinion anyone who has a solid grasp of the basics in the field and does not put forth incoherent arguments (unlike Cantor deniers or anti-logicers) is very far from a crank. Not to say that you have even published in well-respected mathematics journals. As user7530 pointed out, "[cranks] leave the path of honest scholarship by e.g. ignoring related literature proving their approach is hopeless, publishing "proofs" with serious flaws and ignoring attempts by peers to educate them, etc". $\endgroup$ – user21820 May 13 '18 at 11:48

I think a question along the lines of, "Is there an elementary proof of Theorem X?", with your ideas for the sketch in the question body, should be fine.

In the best case, somebody will provide a reference to an existing elementary proof, or will read your ideas and give advice about how to fill the gaps.

There's nothing at all cranky about seeking an elementary proof of an established result, and while I can't guarantee your question won't be closed, I don't see why it would face open hostility.

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    $\begingroup$ There is plenty cranky about seeking an elementary proof of Fermat's Last Theorem, the Twin Prime Conjecture, the Goldbach Conjecture, etc., etc. $\endgroup$ – Gerry Myerson Aug 27 '14 at 23:34
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    $\begingroup$ Of these only FLT is an established result, and I see no compelling reason an elementary (albeit likely not simple) proof of FLT can't exist. On the contrary as with the prime number theorem etc. I believe it's more likely than not that one will be found with time. $\endgroup$ – user7530 Aug 28 '14 at 2:52
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    $\begingroup$ I suppose it depends on what one means by "elementary". In connection with the Prime Number Theorem, it just means a proof that uses only the reals, and not the complex numbers. I'm not sure what you mean by an elementary-but-not-simple proof of FLT. I see 400 years of failed attempts as a compelling reason a proof using nothing beyond intro Calculus and intro Number Theory can't exist. $\endgroup$ – Gerry Myerson Aug 28 '14 at 3:24
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    $\begingroup$ Stepping back a bit, I don't consider anyone a crank who is working on a serious mathematical problem using techniques with some hope of success, no matter how small. Trisectors, cantor deniers, etc are cranks; amateurs working on FLT or even stuff like Goldbach using elementary methods are not until they leave the path of honest scholarship by e.g. ignoring related literature proving their approach is hopeless, publishing "proofs" with serious flaws and ignoring attempts by peers to educate them, etc. $\endgroup$ – user7530 Aug 28 '14 at 6:31
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    $\begingroup$ Regarding your specific question, I see 400 years of failed attempts as evidence that a short proof using elementary number theory doesn't exist. $\endgroup$ – user7530 Aug 28 '14 at 6:37
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    $\begingroup$ For the record, it's not FLT I'm thinking of right now. As to the idea that 400 years of failed attempts is somehow conclusive of anything, I offer this fact: Terjanian came up with the first elementary proof (the most advanced thing being the Jacobi symbol) of the first case of FLT for even exponents in 1977 [!!]. Ribenboim dryly notes, ”it is surprising that it was not found beforehand”. $\endgroup$ – Kieren MacMillan Aug 28 '14 at 18:50

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