I have a proof of a result in number theory, shorter than the existing proofs. Can such results be published and how?
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.