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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.**

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2 Answers 2

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[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

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    $\begingroup$ it's perfect comment and very nice perspective. i see it:) $\endgroup$
    – queen28
    Commented Aug 28, 2020 at 14:30
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    $\begingroup$ 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
    Commented Aug 29, 2020 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
    Commented Aug 29, 2020 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
    Commented Aug 29, 2020 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
    Commented Aug 29, 2020 at 19:45
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For counterexamples in analysis, a good start would be "Counterexamples in Analysis" by Gelbaum and Olmsted, there's even a Dover edition.

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  • $\begingroup$ Thank you:) I will research its details. $\endgroup$
    – queen28
    Commented Aug 28, 2020 at 17:35
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    $\begingroup$ Also Theorems and Counterexamples in Mathematics by the same authors, springer.com/gp/book/9780387973425 $\endgroup$
    – Gerry Myerson
    Commented Aug 29, 2020 at 0:25

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