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Update: Please restrict your answers to "tweets" that give more than just the statement of the result, and give also the essence (or a useful hint) of the argument/novelty.

I am looking for examples that the essence of a notable mathematical development fits a tweet (140 characters in English, no fancy formulas).

Background and motivation: This question was motivated by some discussion with Lior Pachter and Dave Bacon (over Twitter). Going over my own papers and blog posts I found very few such examples. (I give one answer.) There were also few developments that I do not know how to tweet their essence but I feel that a good tweet can be made by a person with deep understanding of the matter.

I think that a good list of answers can serve as a good educational window for some developments and areas in mathematics and it won't be overly big.

At most 140 characters per answer, single link is permitted. Tweeting one's own result is encouraged.

Update: I learned quite a few things, and Noam's tweet that I accepted is mind-boggling.

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    $\begingroup$ I have discovered a truly remarkable proof of this theorem which this Tweet is too small to contain. $\endgroup$ – Glorfindel Apr 26 '17 at 8:26
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    $\begingroup$ I feel like most of the answers are mis-interpreting the question. This doesn't ask for a result whose statement is in 140 characters; that would be too broad: most paper titles fit in them. It asks for a result whose essence is tweetable: given the tweet alone, a mathematician with good knowledge of the field can fill in the details and complete a proof. So I am going to downvote almost all of them. $\endgroup$ – Federico Poloni Apr 26 '17 at 10:16
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    $\begingroup$ @FedericoPoloni: what is it, a kind of joke? Could you please indicate to me where exactly in the question it is written that the tweet should be such that "a mathematician with good knowledge of the field can fill in the details and complete a proof"? Honestly, I do not think that your personal interpretation of the locution "essence of a notable mathematical development" should be taken as a rule here. $\endgroup$ – Francesco Polizzi Apr 26 '17 at 10:21
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    $\begingroup$ Perhaps the next big MO question should be, "What is the essence of a mathematical result?" I myself lean toward Federico's interpretation- pithifying a theorem's statement does not necessarily clarify or illuminate the ideas at play. $\endgroup$ – Neal Apr 26 '17 at 12:52
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    $\begingroup$ My initial intention was indeed that the "tweet" gives more than just the statement of the result but also the essence of the argument/novelty. To demand that a mathematician in the field can fill the details is too much to ask for. $\endgroup$ – Gil Kalai Apr 26 '17 at 12:58

82 Answers 82

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Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.

Not in English, 268 characters so two tweets, @Glorfindel 's comment, but I couldn't not post it.

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  • $\begingroup$ Dear Ethan, warm regards--Gil $\endgroup$ – Gil Kalai Apr 26 '17 at 16:04
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Elliptic curves produce a key exchange that may be safe against quantum computers. link

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  • $\begingroup$ They do? That is awesome. $\endgroup$ – Asaf Karagila Apr 27 '17 at 19:35
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    $\begingroup$ "May be". That is, nobody knows of a sub-exponential quantum algorithm in the supersingular case, because the known sub-exponential algorithms for ordinary elliptic curves require the endomorphism ring to be commutative. But I see no reason for confidence that a sub-exponential quantum algorithm for this case does not exist. $\endgroup$ – Robert Israel Apr 27 '17 at 20:40
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The smallest positive integer not definable in under sixty letters.

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First, my original tweet, on the essential content of IUT, i.e. how it intends to prove ABC, in 140 characters:

mochizuki: invented big deformation machine-tracks how deformed schemethry needs before HA-thry applies-amount of deformation IS Spziro ineq

Now, Twitter recently raised the character limit for all users to 280 characters, so with my whopping 140 extra characters, I will write a new style tweet of the same flavor:

Mochizuki: Invented deformation machine-IUT-which elim. obstruct'ns from applying fund.thm.of HAtheory to schemethry by deforming schemethry. By measuring distort. needed b4 FTHAT applies, ineq. appears-this is content of Spziro conj. hence ABC, modulo rigor check:IUT black box

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  • $\begingroup$ oh and I should say, HA theory is Hodge-Arakelov theory, Mochizuki's initial framework for solving ABC $\endgroup$ – Samantha Y Nov 17 '17 at 20:21
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Cole 1903: 2^67-1=147,573,952,589,676,412,927=193,707,721 × 761,838,257,287 #MoreThanThreeYearsOfSundays

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"The Magic Words Are Squeamish Ossifrage" - to factor a 129-digit semiprime took way less than 40 quadrillion years when early-90's era computers work together using early-90's era factoring algorithms over the early-90's era Internet!

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Every consistent first-order theory has a model of countable size.

A set of sentences is consistent if and only if every finite subset is consistent.

A set of sentences has a model if and only if every finite subset of it has a model.

The first order logic is the only logic with a finitary syntax to possess the Löwenheim-Skolem property and be complete.

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Only nonbinary nontrivial perfect code is ternary w parity matrix rows 11122010000 11210201000 12101200100 12012100010 10221100001. Link

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Sz(q) has two orbits more than PSL(2,q) under the action of its automorphism group - see http://dx.doi.org/10.1081/AGB-120004501, Thm. 3.4.

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  • $\begingroup$ Just to follow the encouragement to post a result of one's own! $\endgroup$ – Stefan Kohl Apr 26 '17 at 14:17
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You can always find a transversal line meeting a family of parallel line segments on the plane such that any 3 can be transversed.

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f_N = (-d/dx)^N cos(x^½)
= A(y)f_0 + B(y)f_1 for y = 1/(4x); A,B∈Z[y], deg O(N)
~ N!/(2N)! as N → ∞ = o(ε^N)
f_1/f_0∈Q ⇒ y∉Q
x = π^2 ⇒ π^2∉Q

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At a party with $4^n$ people and there are either $n$ people who all know each other, or $n$ who are all mutual strangers.

(I used crappy bounds so it could be easily stated. I know the upper bound for $R(n,n)$ is another answer.)

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    $\begingroup$ The upper bound, rather. $\endgroup$ – Andrés E. Caicedo Apr 30 '17 at 0:41
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    $\begingroup$ See also gilkalai.wordpress.com/2017/03/29/r55-%e2%89%a4-48 straight from OP's blog for the recent result that R(5,5) ≤ 48. The tweet could be "You can't have 48 people on a party such that no 5 are all acquainted and no 5 are all strangers." $\endgroup$ – Zsbán Ambrus Aug 27 '17 at 19:15
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There is no smooth surjection from $S^5$ to $S^6$. #Sard

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Erdös-Faber-Lovasz conjecture: If $n$ copies of $K_n$ have pairwise intersection of $\leq 1$, you can color all points with $n$ colors.

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e^x is sum of x^k/k!, k=0,1,...: Binomial theorem on (1+x/n)^n; coefficient of x^k is binom(n,k)/n^k=(1-1/n)...(1-(k-1)/n)/k!→1/k! as n→∞

(137 characters, uses fancy Unicode characters)

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Lattices with exponential kissing number discovered by Serge Vlăduţ. Another home run for algebraic-geometry codes. Link.

Actually, with the new 260 characters policy we can add:

Lattices with exponential kissing number discovered by Serge Vlăduţ. Another home run for algebraic-geometry codes. Will exponential improvement for Minkowski's 1905 lower bound for sphere packing be the next grand slam?

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A group G≠A_n,S_n with a core-free maximal subgroup of index n∈{266,506,759,1045,1288,1463,3795} is sporadic. Proof by GAP. Any other index?

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One cannot hear the shape of a drum. link

Proof via Sunada's Theorem.

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Graham's number is so big, that its digits contain more information than can be contained within the volume of a human brain

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    $\begingroup$ Understatement of the century... $\endgroup$ – Nate Eldredge Apr 26 '17 at 22:19
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    $\begingroup$ And of the last century. $\endgroup$ – Todd Trimble Apr 27 '17 at 1:15
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    $\begingroup$ But does it have a chance for the understatement of the millennium? $\endgroup$ – Asaf Karagila Apr 27 '17 at 10:45
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    $\begingroup$ It's less of an understatement when you realize the word "volume" plays an important role: "It's a number so insanely, absurdly huge that storing all the digits of Graham's number in the brain could create a black hole, said John Baez, a mathematical physicist at the University of California, Irvine, who is researching big numbers. (Only so much information can be stored in a given amount of space, and trying to squish more matter into that space creates a black hole, he said.)" livescience.com/26870-ginormous-numbers-boggle-the-mind.html $\endgroup$ – Dustin G. Mixon Apr 29 '17 at 23:24
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The fundamental theorem of algebre: every polynomial splits in the field of complex numbers.

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$\bar{\rho}$ irreducible Galois rep has finitely many lifts $\rho$ unramified outside of $S$. Proof: $(\rho_2^{-1}\rho_1\rho_2-\rho_1)/\mathcal{l}^r$ is a cocycle.

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Cogito ergo sum #ReneDescartes

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  • $\begingroup$ I find it hilarious that my "tweet" was too short for MO. Andere Städtchen, andere Mädchen, I suppose? $\endgroup$ – Victor Protsak Apr 26 '17 at 21:14
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    $\begingroup$ Isn't that more in the area of Tweetable Philosophy? $\endgroup$ – Frieder Ladisch Apr 26 '17 at 21:54
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    $\begingroup$ Perhaps. Or metamathematics, as I would interpret it. Arguably, that sentence is also at the root of modern scientific method. To those downvoting this answer, I have another one for you: $$\ $$ You are not fun anymore! #MonthyPython $\endgroup$ – Victor Protsak Apr 27 '17 at 1:48
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    $\begingroup$ @Victor Protsak Wait, so "another roof, another proof" [attrib. Erdős] is a play on "andere Städtchen, andere Mädchen"? $\endgroup$ – Noam D. Elkies May 1 '17 at 18:49

protected by Community Apr 26 '17 at 22:19

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