12
$\begingroup$

I am a masters student. I am interested in short articles which have counter examples and very few references. I want to write a short and interesting article.

For example; One of the best known shortest and best academic paper articles I read is Counterexample to Euler's Conjecture on Sums of Like Powers by L. J. Lander and T. R. Parkin (Bull. Amer. Math. Soc. 72 (1966) p 1079, doi:10.1090/S0002-9904-1966-11654-3). It has only one reference. It's really fascinating.

Is there any short articles in Mathematics and especially in Analysis/Complex Analysis? I am also looking at Counterexamples books for learning something and I searched open problems in Wikipedia and looked at undiscovered, newly valued and current topics.

So can you share these type of articles you read? I am curious about a good and interesting short article topic. What do you recommend to me about it? You can also give some references.** Thanks for your ideas and answers.**

$\endgroup$
15
$\begingroup$

[a bit too long for a comment]

I understand from the question that the aim is to find a research project based on the search for a counterexample. By construction, this will mean showing that some existing paper in the literature is mistaken. That is typically not a productive way to start a project in a new field, simply because (a) if the author of that paper is clueless then there is not much gained in showing them wrong by finding a counterexample,$^\ast$ while (b) if the author is an expert you are facing an uphill battle if you are just entering a field.

Typically, a more productive way to enter a field is to try to generalize/extend work of others, basically by exploring corners of the field they left untouched (or didn't bother to explore). You may find that this leads you to uncover an error/oversight in the paper you started from, but that would then be a byproduct of your research and not the primary motivation.

$^\ast$ many questions here on MathOverflow can be readily dismissed by finding a counterexample, but that rarely becomes something worthy of a publication

$\endgroup$
  • 3
    $\begingroup$ it's perfect comment and very nice perspective. i see it:) $\endgroup$ – queen28 Aug 28 at 14:30
  • 1
    $\begingroup$ Finding a counterexample to a conjecture might not be hard. James Davis's counterexample to Conway's climb-to-a-prime conjecture comes to mind (although I don't think he got a formal publication out of it). $\endgroup$ – Timothy Chow Aug 29 at 12:41
  • $\begingroup$ @TimothyChow: As that example shows, there exist significant conjectures for which finding a counterexample isn’t hard. But finding such conjectures is hard. $\endgroup$ – Peter LeFanu Lumsdaine Aug 29 at 17:06
  • $\begingroup$ @PeterLeFanuLumsdaine : I may be misreading Carlo Beenakker's answer, but it sounded like he was talking about counterexamples to a published theorem and I was trying to say that it's easier to find counterexamples to a published conjecture. I would not normally describe a person who conjectures something that turns out to be false as "clueless" or even "mistaken." $\endgroup$ – Timothy Chow Aug 29 at 17:37
  • $\begingroup$ "By construction, this will mean showing that some existing paper in the literature is mistaken." That's not at all what a counterexample is! There are many papers whose main contribution is an ingenious counterexample to a well-known published conjecture or folklore question. These can be quite lengthy and occasionally highly technical, and it is uncommon that they only have 1 or 2 references like the OP suggests. $\endgroup$ – R. van Dobben de Bruyn Aug 29 at 19:45
8
$\begingroup$

For counterexamples in analysis, a good start would be "Counterexamples in Analysis" by Gelbaum and Olmsted, there's even a Dover edition.

| cite | improve this answer | |
$\endgroup$

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.