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Above is an article about a researcher disproving an open conjecture in algebra (Kaplansky's unit conjecture, which I was unfamiliar with). It says:

Gardam declined to tell the audience just how he had found the long-sought-after counterexample (except to confirm that it involved a computer search). He would share more details in a few months, he told Quanta. But for now, he said, “I’m still optimistic that maybe I have enough tricks left to get some more results.”

Is that a usual thing in math? I have seen some cryptography results announced that way, where someone demonstrates an attack on some cryptosystem but temporarily withholds details. The intention there is different though: it's to give people using the broken system some time to fix their stuff before revealing the attack to the wider world.

In the math case, I know something like this happened with solutions for cubic and quartic equations in the 16th(?) century but I had the impression that since then, if you've got a general method to solve a given type of problem, that's a bigger deal than cranking a few more specific solutions from it, so you might as well publish early.

Don't want to go too much into whether it's good or bad, but just wondering if anyone has seen stuff like this before.

none
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    Yes, this is reasonably common. As has been mentioned below, sometimes the method of finding the counter-example is less important than the counter-example itself. Sometimes authors enjoy the mystique of presenting something with hidden methods. Perhaps this is a bit meta, but by and large proofs are supposed to hide the worst of the struggles you might have had in the process of discovery -- after all they're meant to be as enlightening as possible. – Ryan Budney Apr 15 '21 at 05:46
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    The example of Gardam's result is perhaps not a good choice. According to Quanta the talk was on (last) February 22, while Gardam put his preprint on ArXiv on February 23... – abx Apr 15 '21 at 07:33
  • @abx it looks like version 3 of the paper went up more recently, and has more detail. – none Apr 15 '21 at 07:35
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    @RyanBudney can you give examples? I have never heard of anything like this. Of course, people often find counterexamples by a meandering process that cannot easily be summarised. But if some kind of systematic search was used, I would expect people to explain it. – Neil Strickland Apr 15 '21 at 08:19
  • Nowadays my impression is that it is very rare for authors to explicitly hide their methods. In fact, so much of mathematics depends on the application of deep techniques that it would often be hard for them to do so, and still present a rigorous proof. But I'm afraid a lot of methods are implicitly hidden, because they are not amenable to presentation in the usual format of research papers. – Mark Wildon Apr 15 '21 at 11:45
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    @Mark, I would like to see how Zagier proposes to express $3$ as a sum of two squares. – Gerry Myerson Apr 15 '21 at 12:02
  • One possible more recent example is Zagier's proof that every number congruent to 1 modulo 4 (corrected after Gerry's comment) is a sum of two squares. This can be found in 'Proofs from the Book' and was discussed here mathoverflow.net/questions/31113/…; in particular Christian Elshotz has a nice paper explaining how one might discover the proof – Mark Wildon Apr 15 '21 at 14:25
  • @MarkWildon :21? – Anthony Quas Apr 15 '21 at 15:53
  • This is probably not quite what you are asking for, but there are many instances of results that are announced but no proof appears for a long time. See this MO question for examples. But in these examples, it's probably because the authors get distracted by other projects, not because they are intentionally trying to keep their methods secret. – Timothy Chow Apr 15 '21 at 16:15
  • There is Squier's paper where he showed the Burau representation preserves a sesquilinear form. In the paper, the form is given by a formula. I think most readers find the origins of the form mysterious (we even have an MO question about how it was derived). It was created as an intersection form, using a Poincare duality construction. Deriving the form from geometry requires far more work than just writing down the end result and verifying directly the algebraic properties of the form. But there's many more examples like this. – Ryan Budney Apr 15 '21 at 17:51
  • I found a similar form for the Lawrence-Krammer representation, where the Poincare duality construction (although informative) is far more involved than simply presenting the form algebraically and verifying its properties. People have even suggested to me (after presentations) that the more mysterious methods of presenting the result is probably the approach they would have chosen. – Ryan Budney Apr 15 '21 at 17:55

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Apparently, it was told about C. F. Gauss that "He is like the fox, who effaces his tracks in the sand", because of his elliptic style of exposition. In fact, he liked to write down only very polished results, without explaining how he had found them.

An informative discussion about the quote above his here:

https://hsm.stackexchange.com/questions/3610/what-is-the-original-source-for-abels-quote-about-gausshe-is-like-the-fox-wh

Francesco Polizzi
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When you are dealing with disproving a conjecture (like that by Kaplansky) you don’t need to know how the counterexample has been found to assess whether it works. The same applies to the case of solutions of algebraic equations that you mentioned. In either case the authors more or less explicitly said they didn’t want to share their methods.

It seems that you can’t do that when, for instance, proving a conjecture that everyone thinks it’s true. However, a not-so-rare thing is to find papers in which very nice results follow from relatively simple arguments, which you can follow perfectly but still can’t figure out how they were conceived. This can apply to very old pieces of math, like for instance some proofs by Archimedes. A good example is the case in which he determines the direction of the tangent to the (Archimedean) spiral, which follows from a lengthy argument consisting of many technical Lemmas, where I never managed to grasp a general direction of reasoning and still regard as something a bit mysterious.

Other examples are most of the arguments in the wonderful “Proofs from the Book”, by Aigner and Ziegler (available here: https://archive.org/details/proofsfrombook00aign_348/page/n131/mode/2up). In these cases, there is no explicit intention of hiding anything, but still the results look so pretty that you wonder how they were developed in that precise form.

(And of course the quintessential case of this kind is Fermat’s last theorem, in which the alleged solution was not shared for lack of space...but perhaps this better fits in the category of “jokes”).

Zach Teitler
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Alessandro Della Corte
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  • Sure, I understand that a counterexample can be checked without knowing how it was found. I'm asking more specifically, whether withholding the method is a common thing in contemporary math culture. It sounds like it does happen sometimes. Btw another historical example: Newton's Principia gaved classical geometry proofs of a bunch of things he actually discovered using calculus. – none Apr 15 '21 at 07:33
  • @none It does happen more often than some people realize. As you can imagine, part of the reason is that people get credit for writing papers and solving problems (or proving theorems), so if you can generate more papers by withholding some of your secrets, you might well do so. I know people who do this, although they don't advertise that this is what they are doing. Of course it is possible only in certain circumstances, where you are able to publish a complete solution to a problem without giving away the tricks you used to come up with the solution. – Timothy Chow Apr 15 '21 at 16:46
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    Here's a non-example to illustrate the point: Christian Krattenthaler's paper on Advanced determinant calculus. This paper explains a large number of techniques for evaluating certain kinds of determinants. Krattenthaler did a great service to the mathematical community by sharing all his tricks. But I think you can see how someone with a less generous spirit might keep those tricks secret and try to get credit for evaluating specific determinants of interest one by one, rather than publish just one "bag of tricks" paper that might not seem so impressive. – Timothy Chow Apr 15 '21 at 16:52