28

I found a better proof for a theorem than a proof from a published paper. Is it possible for me to publish my proof with the same theorem citing that paper?

  • 1
    Yes, but watch the venue. There might be different journals/conferences which are interested in new results vs new proofs for old results. Or it might be the same ones. – einpoklum - reinstate Monica Sep 2 '17 at 19:35
  • 5
    You can publish virtually anything, whether you should is another matter. – Peter Green Sep 2 '17 at 20:09
  • 10
    As Google Scholar would put it: about 54,200 results say "yes". – Clement C. Sep 2 '17 at 23:31
  • 3
    I would read the introduction to Gowers' paper "A new proof.." to see a beautiful example of how one justifies publishing the third known proof to an old theorem. – Sasho Nikolov Sep 3 '17 at 4:55
  • 2
    In 1948, Selberg and Erdos found an elementary proof for the prime number theorem (proven using complex analysis by de la Vallée-Poussin in 1896), a discovery definitely worth a publication. – Pertinax Sep 3 '17 at 16:49
38

You can publish anything interesting.

There are instance where a new proof of a new theorem is not interesting. There are also cases where a better exposition of the existing proof is interesting.

  • 16
    Keep in mind that interesting isn't something the author should solely judge for themselves. Seek opinions of others before getting invested. If you are interested it's good. If you can't convince anyone else to be interested you need to work on solving that problem. Better mousetraps only sell themselves once people know they're better. – candied_orange Sep 2 '17 at 15:06
17

Why wouldn't it be? This is somewhat common. To cite just one example, Dilworth's Theorem (about antichains in a partially ordered set) originally had a somewhat involved proof when it was first published in 1950. Since then there have been a number of papers published with titles like "A Proof of Dilworth's Chain Decomposition Theorem" (that one by Fred Galvin), with many of these papers being widely cited in their own right. The one by Galvin is a masterpiece of mathematical elegance.

The crucial question is if you can convince the referees that your proof is both sufficiently new and sufficiently better and/or interesting so as to warrant publication.

  • 2
    To give some more famous examples of theorems with multiple proofs published: the fundamental theorem of algebra, the prime number theorem, quadratic reciprocity, ... – Kimball Sep 2 '17 at 22:16
  • In computer science, Hopcroft's algorithm for DFA minimization has been the subject of many papers, some of which are just new presentations (not even new proofs). – Jeffrey Bosboom Sep 3 '17 at 19:53
  • @Kimball, the Pythagorean Theorem (370+ proofs and counting). – Mark Sep 4 '17 at 6:44
  • @Mark Yes, I thought about mentioning the Pythagorean theorem too, but then I couldn't recall if I'd seen proofs published in research journals, or just books/recreational periodicals. – Kimball Sep 4 '17 at 12:57
10

Sure. People publish new proofs of old results all the time. I've done it twice, myself.

Publishability, as with any paper, depends on how much you can get people to care about your new proof. If the result itself isn't very significant, your proof will need to be a big improvement; for a more significant result, it might be enough to improve one of the steps. People will probably care more if your proof is shorter, simpler, requires less material from outside the field (or, conversely, establishes links with other areas), leads to a stronger result, etc.

Be careful that your new proof isn't circular. If Smith proves that every widget has an even number of facets and Jones extends this to prove that the number of facets is actually a multiple of six, you don't get to publish A Short Proof of Smith's Theorem that simply points out that every multiple of six is even.

1

As several people already mentioned, the prime number theorem is a great example:

No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann’s zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say ‘lie deep’ and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten.” - Hardy, 1921

Then, Erdös and Selberg gave a non-complex-analysis proof in 1948, see history of this problem here.

  • OK, but "Paul Erdős managed to do it" doesn't really help us mere mortals. A simple existence proof doesn't really tell us much. – David Richerby Sep 4 '17 at 7:43
  • @DavidRicherby: Well, in my field, algebraic combinatorics, there is the saying that you can prove EVERYTHING twice - first with representation theory, and then with purely combinatorial means. There are several open problems that are of this form, (usually positivity results), where some quantity is proven to be positive, but lacks a combinatorial proof of this fact. – Per Alexandersson Sep 4 '17 at 11:01
  • 1
    That's a much better example, IMO. – David Richerby Sep 4 '17 at 11:14
1

Apart from the trivial answer, yes, I would like to highlight a few points before submitting new proof for an existing theorem to a publishing venue.

  • It depends on where you are submitting it. Certain peer-reviewed journals might not welcome proofs for established theorems.
  • It mainly depends on the subject or discipline of concern. It is much more common to see alternate proofs in venues of mathematics and physics than in computer science.
  • It depends on whether the new proof is theoretical or practical in nature. Unless the theorem is not practically proved before, referees of reputable technical publications might not readily accept application papers of existing theorems.
  • The new proof ought not to be a trivial simplification of an existing proof. Otherwise, it would become an exposition paper than a research paper. This again comes down to the perception of the reviewer and the publisher.

Having stated above, you could technically publish anything new as long as you believe it benefits a wide scientific community.

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.