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I have recently come across a (recent) problem in a different sub-area, by a well-established mathematician, that he deems likely to be intractable, but to which I have found a particularly nice proof (at least I believe so) using the techniques well known in my own sub-area. I think the result itself has a particularly illuminating interpretation so I would like to get it published somehow, but since I am not familiar with the sub-area nor particularly interested in working in it (by which I simply mean I am not suitable for the sub-area), would the best recourse be to contact the author and ask him to write a paper together on the result?

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Probably more like: Contact the author, and ask for his reaction. It could be: "Great, let me help you publish it". Or it could be: "You made the following well-nown error."

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    $\begingroup$ I am well aware of that possibility of course. $\endgroup$ – Project Book Sep 21 '17 at 0:49
  • $\begingroup$ Which is why I think it might be best to contact the author in the first place, since he probably can tell whether the proof is sound or not, otherwise I would have asked if it would be best to submit it straight to arXiv or something. So should I try to submit it to arXiv as @David G. Stork suggested, or should I contact the author first? $\endgroup$ – Project Book Sep 21 '17 at 0:57
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If you're confident of your result (even if you might not have extensive knowledge of the literature), I would recommend you submit your work to a journal. If your work is so closely based on this well-established mathematician, he or she will likely be contacted by the journal to perform a review. But as @Gerald Edgar points out, be prepared for a review that states: "You make the following well-known error."

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    $\begingroup$ Should I submit it to arXiv then? $\endgroup$ – Project Book Sep 21 '17 at 0:45
  • $\begingroup$ If you're reasonably confident of your results, then yes... I would submit to arXiv. $\endgroup$ – David G. Stork Sep 21 '17 at 0:46

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