I have a proof of a result in number theory, shorter than the existing proofs. Can such results be published and how? Actually I'm proving that ∀n∃p∈P|n&lt;p&lt;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&lt;p&lt;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?
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.