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Given the vast number of new papers / preprints that hit the internet everyday, one factor that may help papers stand out for a broader, though possibly more casual, audience is their title. This view was my motivation for asking this question almost 7 years ago (wow!), and it remains equally true today (those who subscribe to arXiv feeds, MO feeds, etc., may agree).

I was wondering if the MO-users would be willing to share their wisdom with me on what makes the title of a paper memorable for them; or perhaps just cite an example of title they find memorable?

This advice would be very helpful in helping me (and perhaps others) in designing better, more informative titles (not only for papers, but also for example, for MO questions).

One title that I find memorable is:

The response to this question has been quite huge. So, what have I learned from it? A few things at least. Here is my summary of the obvious: Amongst the various "memorable" titles reported, some of the following are true:

  1. A title can be memorable, attractive, or even both (to oversimplify a bit);
  2. A title becomes truly memorable if the accompanying paper had memorable substance
  3. A title can be attractive even without having memorable material.
  4. To reach the broadest audience, attractive titles are good, though mathematicians might sometimes feel irritated by needlessly cute titles
  5. Titles that are bold, are usually short, have an element of surprise, but do not depart too much from the truth seems to be more attractive in general. 5.101 Mathematical succinctness might appeal to some people---but is perhaps not that memorable for me---so perhaps such titles are attractive, but maybe not memorable.
  6. If you are a bigshot, you can get away with pretty much any title!
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    $\begingroup$ I'd have put in "A Contribution to the Mathematical Theory of Big Game Hunting" as an answer, but that's carrying a joke too far I think. $\endgroup$
    – J. M. isn't a mathematician
    Commented Oct 31, 2010 at 15:19
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    $\begingroup$ Entertaining as this list may be, I seriously doubt that it will be a useful prescriptive guide as to how to title one's papers. Editors' and readers' tastes also change over the years $\endgroup$
    – Yemon Choi
    Commented Oct 31, 2010 at 19:35
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    $\begingroup$ Since this question seems to have turned into a big list of "memorable/amusing paper titles," ignoring the primary question "what makes the title of a paper memorable?", perhaps it might be helpful to re-ask that question but without the loophole "...or perhaps just cite an example of title they find memorable". $\endgroup$
    – Mike Shulman
    Commented Nov 1, 2010 at 0:23
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    $\begingroup$ I have now caught a duplicate answer for the second time in as many days on this thread. To me this casts doubt on the usefulness of this thread, but I acknowledge that I have a long-standing bias against these types of questions, which from previous discussions on meta seems not to be shared by most people $\endgroup$
    – Yemon Choi
    Commented Nov 2, 2010 at 1:19
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    $\begingroup$ For some reason no further answers can be posted, so let me share with you Continuing horrors of topology without choice by C. Good and I.J. Tree, and related to that Horrors of topology without AC: A nonnormal orderable space by E.K. van Douwen, Disasters in topology without the axiom of choice by K. Keremedis, Disasters in metric topology without choice by E. Tachtsis. $\endgroup$
    – Martin Brandenburg
    Commented May 23, 2014 at 14:26

127 Answers 127

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I can't believe no one's mentioned this:

  • Pavol Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one, arXiv:math/0105080
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    $\begingroup$ I tend to remember this as and call it "Some title..." $\endgroup$
    – David Roberts ♦
    Commented Jan 24, 2011 at 3:45
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    $\begingroup$ inching towards the proverbial 100 $\uparrow$ $\endgroup$
    – Suvrit
    Commented Sep 15, 2011 at 9:58
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    $\begingroup$ Yes, it's my greatest claim to fame on MO. $\endgroup$
    – David Roberts ♦
    Commented Sep 15, 2011 at 10:20
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    $\begingroup$ Memorable yes, but I would not consider this a 'good title.' Titles should effectively communicate the content of the work, in addition to whatever good aesthetics, wittiness, etc. (nor though they be memes that encourage readers to not take the work seriously). $\endgroup$
    – user100272
    Commented Dec 18, 2017 at 18:36
  • $\begingroup$ Very nice self reference you did there : I had to click on the arxiv link to see the very title you mentionned! $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented May 4, 2020 at 15:34
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"Hodge's general conjecture is false for trivial reasons."

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    $\begingroup$ I love that title so much. $\endgroup$
    – Gunnar Þór Magnússon
    Commented Oct 31, 2010 at 15:58
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    $\begingroup$ I dissent. This title is arrogant and vulgar ("trivial" is a ugly word), unworthy of Grothendieck who invented a lot of beautiful, decently modest, and often informative title, such as "Sur quelques points d'algèbre homologique" or "Récoltes et semailles", or "à la poursuite des champs". $\endgroup$
    – Joël
    Commented Dec 28, 2010 at 21:44
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    $\begingroup$ It is a modest title (though it might be taken to be offensive) - he doesn't say: "The Hodge Conjecture is false for very deep reasons and only I could have disproved it." $\endgroup$
    – Lennart Meier
    Commented May 2, 2011 at 22:12
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    $\begingroup$ I don't think it's offensive at all -- all one has to do is read a few words of the paper to see that Grothendieck is merely performing a small but useful service. The title is catchy enough that one is easily invited to discover just that. $\endgroup$
    – Todd Trimble
    Commented May 8, 2011 at 14:50
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    $\begingroup$ Grothendieck's paper is 5 pages, and seems to explain itself pretty clearly. $\endgroup$
    – David E Speyer
    Commented Feb 6, 2018 at 19:48
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The flattering lie You Could Have Invented Spectral Sequences by Timothy Y. Chow.

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    $\begingroup$ subtitled "Also, that shirt looks good on you." $\endgroup$
    – Cam McLeman
    Commented Oct 31, 2010 at 17:14
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    $\begingroup$ +1 because more people should learn of the existence of this paper. $\endgroup$
    – Ketil Tveiten
    Commented Nov 1, 2010 at 12:39
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    $\begingroup$ Nice one. I have sometimes wondered if I should have chosen a better title than Prime Simplicity for my joint paper with Catherine Woodgold, setting the record straight about what Euclid did and did not do in a certain well-known but not-well-known proof. $\endgroup$
    – Michael Hardy
    Commented Nov 1, 2010 at 19:56
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    $\begingroup$ There should be a series of "You Could Have Invented" papers. $\endgroup$
    – Akiva Weinberger
    Commented Jun 2, 2016 at 2:19
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One that comes immediately to mind is Can one hear the shape of a drum?

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    $\begingroup$ I've heard of this paper, but not read it. What is the answer ;-) ? $\endgroup$
    – Suvrit
    Commented Oct 31, 2010 at 19:06
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    $\begingroup$ In brief, no; there are "drums" with different shapes but the same "sound". $\endgroup$
    – Gerry Myerson
    Commented Oct 31, 2010 at 20:23
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    $\begingroup$ @Suvrit: Yes, but only if one knows a priori that the drum is convex. $\endgroup$
    – Simon Lyons
    Commented Oct 31, 2010 at 21:43
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    $\begingroup$ @SimonLyons: Sorry to revive this comment after almost eight years, but do you have a reference for the answer being "yes" in the convex case? I remember hearing in several discussions that the problem is still open even for convex two-dimensional domains with analytic boundary. $\endgroup$
    – Jochen Glueck
    Commented May 23, 2018 at 18:19
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    $\begingroup$ It was a long time ago and not my field - I think someone mentioned a result along those lines in a lecture I saw. This might be the right reference: arxiv.org/abs/math/9901005 $\endgroup$
    – Simon Lyons
    Commented May 23, 2018 at 20:09
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A minus sign that used to annoy me but now I know why it is there by Peter Tingley.

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Mark Van Raamsdonk's Princeton PhD thesis in string theory was called "Making the most out of zero branes and a weak background". Priceless.

http://adsabs.harvard.edu/abs/2000PhDT........40V

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My favourite : "My Graph", by H.S.M. Coxeter.

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  • $\begingroup$ My graph brings all the boys to the yard? $\endgroup$
    – Hollis Williams
    Commented Oct 24, 2020 at 16:08
  • 2
    $\begingroup$ Maybe that's the graph that Nickelback wanted to show us all along? $\endgroup$
    – TheOutZ
    Commented Feb 10 at 2:01
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Given the atmosphere of terror and fear in recent years, I did a double take when I first glanced at Bruce Berndt's paper "Ramanujan's association with radicals in India".

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    $\begingroup$ That reminds me of a course description from the Harvard course catalogue, circa 1970: something like, "The theory of blowing up, with special attention to local problems." Fortunately, this was offered by the Department of Mathematics, not Social Relations. $\endgroup$
    – Gerry Myerson
    Commented Oct 31, 2010 at 20:30
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    $\begingroup$ I know this paper, but I'm not entirely sure Berndt was being deliberately provoking... :) $\endgroup$
    – J. M. isn't a mathematician
    Commented Oct 31, 2010 at 23:13
  • $\begingroup$ In my experience with him I'd say he wasn't, but there's a streak in him I wouldn't put it past. $\endgroup$
    – Adam Hughes
    Commented Nov 16, 2010 at 5:09
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    $\begingroup$ @GerryMyerson about blowing up, see also the story about the algebraic geometry people at airport security, at mathoverflow.net/a/53738/5340 (and previously at mathoverflow.net/a/23074/5340 ) $\endgroup$
    – Zsbán Ambrus
    Commented Sep 21, 2016 at 20:15
  • $\begingroup$ The joke went around UChicago in the early 2000s that Ahmed Chalabi might get in trouble with the Coalition Provision Authority because of his own association with radicals (thesis). $\endgroup$
    – LSpice
    Commented Jul 12 at 21:43
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And in the graph theory corner we have the famous Harary/Read paper "Is the null-graph a pointless concept?"

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    $\begingroup$ This epic paper appears in Lecture Notes in Math., vol. 406, Springer, 1974, 37-44. The abstract is the following: The graph with no points and no lines is discussed critically. Arguments for and against its official admittance as a graph are presented. This is accompanied by an extensive survey of the literature. Paradoxical properties of the null-graph are noted. No conclusion is reached. $\endgroup$
    – Richard Stanley
    Commented Nov 2, 2010 at 1:31
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    $\begingroup$ I don't know now whether i like the abstract more or the title; absolutely fantastic. $\endgroup$
    – Suvrit
    Commented Nov 3, 2010 at 17:29
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    $\begingroup$ I like the title more. There are many cases where the degenerate case is hard: 0!, a^0 vs 0^a, 1 is a prime, which we resolve by how many theorems need special cases. $\endgroup$
    – Ross Millikan
    Commented Nov 10, 2010 at 4:57
  • $\begingroup$ @RossMillikan All answers are obvious for natural reasons: $0! = 1$, $a^0 = 1$, $0^a = 0$ for $a \geq 1$, 1 is not prime. $\endgroup$
    – Arshak Aivazian
    Commented Apr 18, 2022 at 12:38
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John Stallings' "How not to prove the Poincare Conjecture" is lovely.

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The book  $A=B$.  

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  • $\begingroup$ Could you give a reference? $\endgroup$
    – Martin Brandenburg
    Commented Oct 31, 2010 at 17:38
  • $\begingroup$ @Martin: I did: the text is a link (but MO+MathJax do not seem to underline or otherwise emphasize link when the linking text is just math) Googling for "A=B" gets you there, too. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Oct 31, 2010 at 17:48
  • $\begingroup$ Just add some other text to the ink, then - e.g. "The book A=B". $\endgroup$
    – Harry Altman
    Commented Oct 31, 2010 at 18:30
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    $\begingroup$ +1. This also brings to mind "Generatingfunctionology", which is itself pretty memorable. $\endgroup$
    – James
    Commented Oct 31, 2010 at 20:34
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"On $O_n$" by D.E. Evans. ($\mathcal{O_n}$ is notation Cuntz gave for the algebras he introduced in "Simple $C^*$-algebras generated by isometries".)

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    $\begingroup$ That resembles Connes/Consani's "Fun with F_1" - 1 in french is "un". $\endgroup$
    – Peter Arndt
    Commented Oct 31, 2010 at 20:49
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    $\begingroup$ There is also the sequel, "On $O_{n+1}$" by Araki, Carey, and Evans: ams.org/mathscinet-getitem?mr=757434 $\endgroup$
    – Jonas Meyer
    Commented Nov 2, 2010 at 17:36
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    $\begingroup$ Have you ever tried to google that paper (without knowing who wrote it, of course)? Hopeless! :-) $\endgroup$
    – Ulrich Pennig
    Commented May 2, 2011 at 21:30
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    $\begingroup$ @Ulrich: The first result that Google Scholar returns when I type On O_n into it is precisely this paper. I wouldn't exactly say that googling this paper is “hopeless”. $\endgroup$
    – Dmitri Pavlov
    Commented May 8, 2011 at 18:50
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    $\begingroup$ @Dmitri: I just tried that in Google Scholar and it didn't work for me. $\endgroup$
    – Timothy Chow
    Commented Nov 17, 2011 at 3:32
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A Midsummer Knot's Dream, by Allison Henrich, Noël MacNaughton, Sneha Narayan, Oliver Pechenik, Robert Silversmith, Jennifer Townsend

It is quite funny to read

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The AKS paper PRIMES is in P is a pretty memorable title for me.

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    $\begingroup$ Certainly it's a memorable title, but I keep having to fight the urge to reply "No, they isn't!" I would have preferred a title like "Deterministic, polynomial-time primality testing," but that would not have been memorable, so perhaps they made the right choice. $\endgroup$
    – Henry Cohn
    Commented Oct 31, 2010 at 16:07
  • $\begingroup$ I guess I should have put in my motivation: while I'm not normally interested in number theory, the title "jumped" at me, so to speak. $\endgroup$
    – J. M. isn't a mathematician
    Commented Oct 31, 2010 at 16:48
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    $\begingroup$ @Henry, to be fair, the title was actually "PRIMES is in P", where 'PRIMES' refers not to the set of primes, but the (hypothetical) (deterministic) algorithm to test for primality. $\endgroup$
    – dorkusmonkey
    Commented May 3, 2011 at 9:48
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    $\begingroup$ Actually, it does refer to the set of primes (see the first page of the article). I agree that the title is syntactically correct; I'm just bothered by how it sounds when you read it out loud. $\endgroup$
    – Henry Cohn
    Commented May 3, 2011 at 13:25
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Finding composite order ordinary elliptic curves using the Cocks-Pinch method, by D. Boneh, K. Rubin and A. Silverberg. (To appear in the Journal of Number Theory.)

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    $\begingroup$ +1 because I'm so shallow. :D $\endgroup$
    – J. M. isn't a mathematician
    Commented Oct 31, 2010 at 16:50
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    $\begingroup$ We have ways of making number theorists talk... $\endgroup$
    – Todd Trimble
    Commented May 8, 2011 at 14:57
  • 1
    $\begingroup$ Journal of Number Theory, Vol. 131 (5), 2011, pp. 832-841. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2020 at 3:15
  • $\begingroup$ This is obviously in direct contrast to the Cox-Zucker machine. $\endgroup$
    – TheOutZ
    Commented Jan 12 at 17:53
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I find it dubious that anyone here will get better at choosing titles for their papers by reading these examples.

Nevertheless, I like the title "The homotopy category is a homotopy category" by Arne Strøm. I also like the very apt title "$\overline{\mathcal{M}}_{22}$ is of general type" by Gavril Farkas. The paper starts like this:

The aim of this paper is to prove the following result:

Theorem: The moduli space of curves of genus 22 is of general type.

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    $\begingroup$ Yes: a title can be effectively eye-catching not just by being humorous or off the wall, but also simply by being very mathematically expressive and succinct. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented Oct 31, 2010 at 18:54
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The paper Division by three by Peter G. Doyle and John Horton Conway

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    $\begingroup$ Reading this paper is a pure moment of happiness! (Except for figure 2, which makes look complicated a proof which is perfectly clear when expressed in words). $\endgroup$
    – Joël
    Commented Nov 15, 2016 at 3:26
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    $\begingroup$ And, one-upping this title: Patrick Lutz, Conway Can Divide by Three, But I Can’t. $\endgroup$
    – darij grinberg
    Commented Jan 20, 2022 at 18:15
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The missing axiom of matroid theory is lost forever

A emotional variation on absolute negative results.
Refs : Vámos, Peter (1978), "The missing axiom of matroid theory is lost forever", Journal of the London Mathematical Society, II. Ser. 18: AT : http://jlms.oxfordjournals.org/content/s2-18/3/403.extract

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A Group of Order 8,315,553,613,086,720,000 by J H Conway, Bull. London Math. Soc. (1969) 1 (1): 79-88, https://doi.org/10.1112/blms/1.1.79

Maybe it's cheating to call this memorable - I remembered there was a Conway paper with a title of this type, but I certainly don't claim to have remembered the exact title!

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    $\begingroup$ I'm told that shortly after the Hall-Wales paper "The Simple Group of Order 604,800" (J. Algebra 9 (1968), 417-450) was published, the editors received an anonymous submission entitled "The Simple Group of Order 604,801". Sure enough, 604801 is prime. $\endgroup$
    – Henry Cohn
    Commented Mar 8, 2011 at 18:12
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    $\begingroup$ There's also a Youtube video called Finite Simple Group (of Order Two), youtube.com/watch?v=UTby_e4-Rhg $\endgroup$
    – Gerry Myerson
    Commented Jun 16, 2018 at 0:38
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An application of Poincaré's recurrence theorem to academic administration by Kenneth Meyer is a title that is hard to resist looking into.

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I always remember the paper entitled "On groups of order one." It turned out the title referred to groups defined by generators and relations, so the problem was to determine when a set of elements (together with its conjugates) generated a free group. I cannot imagine any mathematician who would not look at this paper to see what it was about.

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"I know I should have taken that left turn at Albuquerque" by Gady Kozma and Ariel Yadin (https://arxiv.org/abs/1008.4258)

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    $\begingroup$ You should have informed a certain user on this site about this, and let them post it. $\endgroup$
    – Steven Gubkin
    Commented Mar 4, 2011 at 13:49
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The book Free rings and their relations by P.M. Cohn.

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    $\begingroup$ +1: I've encountered that book many times during trips to the stacks over the past dozen years or so. Every time I stop and scratch my head. One day I suppose I'll actually read it... $\endgroup$
    – Pete L. Clark
    Commented Nov 1, 2010 at 14:37
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Very late addition (August '13) J. J. Sylvester, Thoughts on inverse orthogonal matrices, simultaneous sign-successions and tesselated pavements in two or more colors, with applications to Newton rule, Ornamental tile-work, and the theory of numbers. Phil Mag 34 (1867), 461-475.

This title is unbeatable!

Late addition (March, '13): Long and Wigderson's " How discreet is the discrete log?."

Gale and Shapley's "College Admissions and the Stability of Marriage, " was a great title to a great paper. (link JSTOR)

"Moments in mathematics" Papers from the American Mathematical Society annual meeting held in San Antonio, Tex., January 20–22, 1987. Edited by Henry J. Landau. (Link: Google book)

This is about "moments" in the technical sense but the double meaning of the title is very cute. (There is also a book entitled "great moments in mathematics" with the ordinary meaning of moments.)

About 1-2 decades ago Sylvain Cappell and Shmuel Weinberger planned writing a book called "A piece of the action" about group actions. This is a memorable title but I think the book was not completed.

One obvious: Aigner and Ziegler's Proofs from the book. (Link: WikipediA)

Joel Spencer's title "Six standard deviations suffice." is also memorable. (Link: JSTOR)

Jack Edmonds',(1965) "Paths, Trees and Flowers". (Link: ps file.)

For some reasons I found the title "Defect Sauer results" of a paper by Bollobas and Radcliffe memorable. (Link)

Branko Grunbaum has a paper entiled "The importance of being straight" (I could not find a link), and Irit Dinur and Shmuel Safra have a paper entitled "On the importance of being biased". (A link to a later version with a different title.) (There is a paper by A. Dillof published in Michigan Law Review with very similar name.)

Jorg Wills had a memorable title "decomposable skeleta" for a paper he sent for the 100th birthday of a well known mathematician. But I think at the end he changed the title.

Saharon Shelah has several memorable titles like this one: "On what I do not understand (and have something to say). I" .Although, I forgot the most memorable one.

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    $\begingroup$ If you forgot it (last sentence), it couldn't have been the most memorable! :-) Maybe "The last forcing standing"? $\endgroup$
    – Joseph O'Rourke
    Commented Nov 1, 2010 at 1:03
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    $\begingroup$ You may be thinking of "You can enter Cantor's paradise". It is also worth mentioning that Shelah numbers his publications, and reserves special numbers for significant papers. Paper 666 is the one you mentioned. $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 1, 2010 at 3:19
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    $\begingroup$ I think "Why I am so happy" was the title of an abstract by Shelah, but the paper probably got a more serious title (involving the "main gap"). $\endgroup$
    – Andreas Blass
    Commented Nov 3, 2010 at 19:37
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    $\begingroup$ The Shela very nice title "On what I do not understand (and have something to say)" It is useful for some (including myself) to know that it is a reference to Wittgenstein quote: "Whereof one cannot speak, thereof one must be silent" $\endgroup$
    – Jérôme JEAN-CHARLES
    Commented Nov 5, 2010 at 19:21
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    $\begingroup$ "There is also a book entitled 'great moments in mathematics' with the ordinary meaning of moments." Actually, there are two, by Howard Eves; one is called Great Moments in Mathematics Before 1650, the other, Great Moments in Mathematics After 1650. I'm still waiting for the third volume, Great Moments in Mathematics During 1650. $\endgroup$
    – Gerry Myerson
    Commented Jul 21, 2020 at 2:37
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Here is a list of papers in Theoretical Computer Science with cute titles. Some that I like from the list (aside from "Mick gets some" which is good enough to deserve its own answer anyway).

  • A Smaller Sleeping Bag for a Baby Snake
  • The Art of Pointless Thinking: a Student's Guide to the Category of Locales
  • Scott is not always sober

Also: Mangoes and Blueberries.

And in a similar vein, a quote from "Quotients homophone des groupes libres - Homophonic quotients of free groups," that appears on the first linked page page: "Ah, la recherche! Du temps perdu."

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    $\begingroup$ +1 for mentioning that last dazzling paper $\endgroup$
    – Georges Elencwajg
    Commented Nov 1, 2010 at 12:49
  • 1
    $\begingroup$ The last phrase of "Scott is not always sober": $\underline{\text{Acrostic}}$ apologies, in conclusion, to Dana Scott for the wholly unjustified slur on his good name in the title of this article. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 5, 2021 at 19:30
  • $\begingroup$ In case anyone else wonders, the word "Acrostic" is there because an acrostic runs through the paper spelling out "I'm not being personal, Dana", where the final 'a' is supplied by the word "Acrostic" itself. Perhaps Johnstone enjoyed The Vane sisters? $\endgroup$
    – LSpice
    Commented Jul 13 at 0:01
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"Holey Sheets" - Pfaffians and Subdeterminants as D-brane Operators in Large N Gauge Theories.

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Integrity of ghosts, by Gert Almkvist.

Gert Almkvist's generalization of a mistake by Bourbaki, by Doron Zeilberger. (And a few more by the same author.)

The Point of Pointless Topology, by Peter Johnstone.

The absolute classic: Go To Statement Considered Harmful, by Edsger Dijkstra.

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  • $\begingroup$ Memorable, but that last one is a computer science paper... $\endgroup$
    – J. M. isn't a mathematician
    Commented Nov 1, 2010 at 0:15
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    $\begingroup$ $\mathrm{ComputerScience}\subset\mathrm{Math}$. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Nov 1, 2010 at 3:18
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    $\begingroup$ Mariano, the Library of Congress agrees with you, QA76 $\subset$ QA. $\endgroup$
    – Gerry Myerson
    Commented Nov 1, 2010 at 11:41
  • $\begingroup$ Dang, I was about to go post "The Point of Pointless Topology." Anyway, I'd give you +1 for for the Dijkstra paper alone $\endgroup$
    – David White
    Commented May 4, 2011 at 16:43
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    $\begingroup$ "Disturbing the Dyson Conjecture (in a GOOD Way)" by Andrew V. Sills and Doron Zeilberger (Exp. Math. 15 (2006), 187-191) is especially GOOD. (To whom it concerns, "Good" is for "Jack Good", a combinatorialist.) $\endgroup$
    – Wadim Zudilin
    Commented Jul 11, 2011 at 13:48
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There are not exactly five objects by Andreas Blass

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  • $\begingroup$ OK, now I'm wondering, does anyone know what Morley's solution for the p=5 case was? $\endgroup$
    – Harry Altman
    Commented Nov 4, 2010 at 2:18
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    $\begingroup$ I learned later that "my" proof had actually been published earlier by Kenneth Appel. I can't find any record of what Morley's 1977 proof was, but later, in 1984, Morley sent me a short formula that does the job (for arbitrary primes): $\forall x_0\dots x_{p-1}\,\exists t_0\dots t_{p-1} \bigwedge_{\sigma\in p^p}\big((\bigwedge_{i=0}^{p-1}t_i=x_{\sigma(i)})\to x_{F_0(\sigma)}=x_{F_1(\sigma)}\big)$ where $F_0,F_1:p^p\to p$ are chosen so that for all $\sigma$, $F_0(\sigma)\neq F_1(\sigma)$ and if $\sigma$ is not one-to-one, then $\sigma(F_1(\sigma))=\sigma(F_0(\sigma))$. $\endgroup$
    – Andreas Blass
    Commented Nov 4, 2010 at 17:02
  • $\begingroup$ See also mathoverflow.net/questions/416648/… $\endgroup$
    – Emil Jeřábek
    Commented Jan 3, 2023 at 15:23
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