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Actually I'm proving that ∀n∃p∈P|n<p<en. Which is a result similar to Bertrand's Postulate, but with a shorter proof. Should I publish my result? If so, as I'm not affiliated with any Research Institute how do I do that?

Edit

Since yesterday I've improved my proof: ∀n∃p∈P|n<p<2n. This means I'm proving Bertrand's Postulate with a simpler and shorter proof. So, which is the most appropriate Journal to publish this result?

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    Probably start by posting it on arXiv (to establish priority) with a public key (to establish identity). – Nat Feb 8 '18 at 15:10
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    It sounds like you might not (yet) have a doctorate? In principle shorter proofs can be interesting and, in my field at least, it is possible to submit to journals and conferences without an affiliation. However, your problem is determining the significance of your result and for that, you need another mathematician. I don't think we can answer your question, but you should try and find a number theorist at an institute who can, it may even lead somewhere. – Dr. Thomas C. King Feb 8 '18 at 15:32
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    I noticed you didn't use TeX in the MathOverflow post despite them having it enabled (SE.Academia doesn't, sadly); was that an oversight because you're new to StackExchange? In any case, if you go to post this proof, you'll definitely want to ensure that it's typeset well. Poorly formatted work is a huge red flag that'll drive a lot of readers off at first glance while well-formatted work is far more likely to receive attention and understanding. – Nat Feb 8 '18 at 16:28
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    Any alternate proof is potentially interesting, regardless of length. Well I suppose an extremely long proof of something that can be proved more concisely would be less interesting. But alternate proofs are very helpful in understanding the theorem better as well as connecting it to other areas of Mathematics. – Todd Wilcox Feb 8 '18 at 18:13
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    To those who voted to close: how is this even remotely a duplicate of the cited question? OP does not "believe they have solved a famous open problem", since Bertrand's postulate is not an open problem. OP also is not asking anything at all similar to "how do I convince people in the field that I am not a crank?", in fact there is no indication that OP has any concerns about being perceived as a crank - their question is entirely different. Voted to reopen. – Dan Romik Feb 10 '18 at 8:16
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As @Nat says, do post in on arXiv. Many journals have a "Notes" section for short results like yours, which people would find interesting, especially if they were teaching similar results. The MAA Monthly does this, e.g. After you have the proof up on arXiv, submit it to the "Notes" section of an appropriate journal, following the specific directions for that section. At least one professional mathematician will look at it, so you'll get some vetting. Have a look at the Missouri Journal of Mathematical Sciences.

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  • Thanks for your answer. But what would be the best journal to publish for as a non-academic? – user1582006 Feb 8 '18 at 16:23
  • "Best" is hard to say. I mentioned the MJMS because, while it has a good reputation, it's one of the easier journals to get published in, and it has a "short notes" section. – B. Goddard Feb 8 '18 at 16:32
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    I would hardly say MJMS has a "good" reputation; rather I would say it has a "non-junk" reputation. – Alexander Woo Feb 8 '18 at 16:48
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    It is easy for people with institutional email addresses to say "oh, just post it on arXiv" and incorrectly assume that their interlocutor will be able to do that. To be clear: posting on arXiv can be a significant challenge for people without an institutional affiliation. Unless you're able and willing to personally endorse the OP, the initial advice needs to be tempered quite a bit. – E.P. Feb 8 '18 at 19:08
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    @B.Goddard If that works, then I'd be disappointed with arXiv's filtering scheme (or at least, I'd hope they're ready to turn it off the minute it starts getting abused). The restrictions on submitting to the arXiv are there for a reason (note e.g. that arXiv had been running for 13 years when they instituted them), because there's a lot of content (much of which now makes its way to viXra) that we don't want to end up on arXiv. – E.P. Feb 8 '18 at 20:18
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Can such results be published and how?

“Such results” certainly can be published, if by “such results” you mean genuinely new proofs of well-known theorems, especially if they contain a new and interesting idea (rather than being a trivial modification or variant of an existing proof), and especially if they are shorter than existing proofs, although that is not a necessary condition for a new proof to be publishable.

Such new proofs are published quite frequently. For example, when I was a graduate student I published a new proof of Stirling’s formula. It was published in the American Mathematical Monthly, a respectable journal that often publishes papers in this category.

Should I publish my result?

As others have said, you can at the very least write up your proof as a paper and submit it to arXiv to make it available to the community. And you can try to publish it in a journal - if it’s interesting, well-written and novel, I think you have a good chance of getting it published somewhere.

If so, as I'm not affiliated with any Research Institute how do I do that?

See this question for some suggestions.

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    Thanks for your answer. I just revisited my proof and I realised I can also prove Bertrand's Postulate with a minimal change. Maybe it's worthwhile publishing it. – user1582006 Feb 8 '18 at 17:15

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