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I want to publish a paper in math which contains a simple proof of a already existing theorem.

The existing proof of the theorem is quite long and uses tedious concepts to begin with. I found an elementary proof, and by that I don't mean that the proof is easy, but that it uses a more limited amount of math.

My question

Can I publish my paper (which is around 3-4 pages long) in the Proceedings of American Mathematical Society?

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Math journals regularly publish new proofs of known theorems when the proof is interesting enough and contains novel elements, and when the theorem itself is considered interesting enough. If your proof is shorter than the known proofs, certainly that would help to make the case that it should be published (assuming it's correct of course).

If your proof is not shorter than the known proofs but has other aspects that can be claimed as superior to or simply different than the known proofs, again there is some chance that a journal will find that sufficient reason to publish your paper.

Can I publish my paper (which is around 3-4 pages long) in the Proceedings of American Mathematical Society?

Possibly - this journal does occasionally publish such things. I searched the MathSciNet database for papers published in Proceedings of the American Mathematical Society whose title begins with "a simple proof of" or "a new proof of". The search returned 118 results. Here is one.

To publish your paper in the Proceedings of the American Mathematical Society, you have to submit it for publication there. It will be reviewed by a referee and a decision whether to publish it will be made depending on the referee's recommendation. We cannot say what the decision will be based on the details you gave; it all depends on how interesting your contribution is judged to be.

If you have never published a paper before, it would be good to get advice from an experienced mentor before submitting your paper. This would help you to avoid some common mistakes that inexperienced researchers make, such as believing your proof is correct when it isn't, believing your proof is new when it isn't, believing the theorem you are proving is interesting when it isn't, and various other less obvious types of mistakes.

Good luck!

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You can try to publish your proof anywhere you like! Whether it gets published or not will depend, among other things, on the quality of the paper, the significance of the result, and the match between the purpose of the journal and your paper.

Even in those areas where a result has been established for, perhaps thousands, of years, new proofs are published. Consider for example the number of different proofs that one can find of the prime number theorem. ... or try searching Google for the phrase "a new elementary proof".

You say nothing about the nature of the original theorem, nor anything about how significant it is. However, leaving those issues aside, the fact that you are proving a known theorem is unlikely, of itself, to preclude an A.M.S. journal from considering it. The best place to start is in the specific guidance for authors, and background about the purpose of the journal in the journal to which you are thinking of submitting.

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