Hesitation about writing a paper

Question

The situation maybe a little bit unusual, and I am asking for your opinion.

I have taught an undergraduate course on differential geometry of curves and surfaces last Fall semester. One of the students in my class, whom I think is really bright, told me that he found the proof of a certain well known theorem is a little bit hard to understand. We had some discussion on it, but I was quite busy so I wasn't paying too much attention on it, so we didn't get any good conclusion.

Recently I was revising my lecture notes and I was reminded of his question. Then he and I found that no fewer than three books (two textbooks and a research monograph) give wrong proofs of that theorem. We are quite satisfied with what we found, and his question was settled. Then he asked whether we should write a paper on this issue and submit it to some undergraduate research journal, like Involve. At first I didn't think it was a good idea because finding a mistake of a proof of a theorem in three books isn't really a big deal, and I told him we could send an email to the authors of those books about the mistakes. But then I think his idea might not be as bad as it seems to be because, essentially, when an undergrad can find some mistakes in well-known textbooks that (it seems to me that) nobody can find, it means something to me. My concern is I don't know even if we are talking about an undergraduate research journal, do the editors think it's certainly not worth a try or the opposite?

On the math side of that theorem, those three books use the same techniques but with different assumptions (or you may want to call simplifications). However, the proof in each of those books have different mathematical mistakes (i.e., not just some simple typos). So far we only found one textbook which gives a correct proof. The other books which contain a correct proof of this theorem are either too sketchy/wordy or those books are research monographs.

Since I have no experience on writing a paper based on the "discovery" of this nature, any comment is welcome. On one hand I think it's something to an undergrad student and on the other hand, I am not sure about what the editors think.

Thank you.

Answer 1

Given that a correct and satisfactory proof exists in the literature, you would not be providing "research results" per se, so a full research article probably is not worth writing at this stage, unless you can devise a more elegant or straightforward or otherwise "improved" proof.

If you are unable to provide that, it would certainly be appropriate to comment on the difficulty of correctly proving the theorem, and do a "comparative" study in the form of a letter to indicate the challenges and make them known to a wider audience.

But, as you said, it's certainly a matter worth talking about—it really is a question about the appropriate "form" of writing your results up.

Answer 2

Generally, writing a paper about some kind of a discovery is not a bad idea.

However, are you sure (as in: really, really sure) that:

• the proofs in question are wrong;
• you have a correct proof?

Pointing "they are wrong" with a finger is not the best behaviour, even less so if you have little credibility (i.e., you are a student). If you can produce a fail-safe proof of that theorem, then why not. I have seen some papers with titles "A bla bli blup proof of foo biz baz theorem".

It might be that your course professor is the best person to approach first. (And I also highly recommend to do so.) It takes some experience to judge if an idea is valid and fruitful or not. Said professor might also be able to discern that you all are wrong and book authors are correct with little to no image damage to you.

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Update: If you think your discovery is fail-proof, then aim for the publication by any means (with or without) the professor.

Don't fear being an undergrad.

Kolmogorov (you'd heard this name, I am sure) had at some time a hypothesis that multiplication of two n-sized inputs in on Big-Omega of n^2, i.e. that it is impossible to multiply faster than O(n^2). He mentioned it in a seminar. He was a world-known name then already. A week later an undergrad student showed him a proof that it is possible to do so faster.

While the actual story of the publication is also quite fun and breath-taking, you can research it yourself. I'd just mention, that we all know this student by this method, the Karatsuba multiplication.