106
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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. $\endgroup$ 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. $\endgroup$ 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

89 Answers 89

90
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Every rational r is xyz(x+y+z) for some rational x,y,z.
Proof: Euler (1749) found x(r),y(r),z(r). Nobody knows how.

I have a guess. http://www.math.harvard.edu/~elkies/euler_14t.pdf

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  • 8
    $\begingroup$ And there are some simpler than Euler's x(r), y(r), and z(r). See mathoverflow.net/q/302933/24165 . $\endgroup$ Jul 20 '18 at 4:05
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    $\begingroup$ Thank you for that reference. I already found x,y,z a bit simpler than Euler's, but not nearly that simple! Now I should revisit this to figure out if and how the Klyachko-Mazhuga-Ponfilenko fits into the elliptic-fibration picture . . . $\endgroup$ Jul 20 '18 at 5:01
174
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Four color theorem: any planar graph can be colored with 4 colors. Only proof by computer. SAD.

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5
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    $\begingroup$ Thx for the correction! $\endgroup$ Apr 26 '17 at 13:59
  • 5
    $\begingroup$ Expressing the essence of how it can be reduced to a huge finite problem for computer proof would be nice, though... $\endgroup$ Apr 26 '17 at 16:37
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    $\begingroup$ As years pass by (and more so if we go outside USA) the probability that someone gets the pun tends to zero. SAD. $\endgroup$
    – leonbloy
    Apr 29 '17 at 13:14
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    $\begingroup$ ??????????????? $\endgroup$ Mar 3 '18 at 23:45
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    $\begingroup$ @LutzMattner It's a political joke about Donald Trump, president of America (2016-202?). He likes to say "SAD!" and it's made into jokes. $\endgroup$ May 28 '19 at 5:13
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27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵. Nice try, Euler. link

#Counterexamples

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    $\begingroup$ You can shorten it and make it friendlier to read using Unicode superscript numeral characters. $\endgroup$ Apr 26 '17 at 16:34
  • 2
    $\begingroup$ How do we do it? (without using Tex, say on Twitter) $\endgroup$
    – Gil Kalai
    Apr 26 '17 at 17:34
  • 5
    $\begingroup$ 27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵. $\endgroup$ Apr 26 '17 at 20:13
  • 1
    $\begingroup$ @GilKalai you can express x to the n as x^n with the use of the caret $\endgroup$
    – Restioson
    Apr 29 '17 at 7:31
  • 2
    $\begingroup$ @Restioson: I originally typed it out that way, but R.. commented that it is in fact possible to tweet superscript numerals, so I changed it to make it more readable. I was able to type out the superscript 5's in my answer -- without MathJax -- by copy/pasting from this website: txtn.us/tiny-text. $\endgroup$
    – Will Brian
    May 1 '17 at 13:46
113
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The sentence "The first positive integer that cannot be specified in a 140 character tweet" doesn't specify a well defined integer.

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    $\begingroup$ In other words, "There are non-tweetable positive integers, but there is no smallest non-tweetable positive integer. Contradiction." $\endgroup$ Apr 27 '17 at 14:17
  • 2
    $\begingroup$ Forgive my ignorance, but assuming a base-10 representation, is 10^140 not the smallest non-tweetable positive integer? $\endgroup$
    – ayane_m
    Apr 28 '17 at 5:42
  • 2
    $\begingroup$ We mean tweetability in plain languages and mathematical formulae, not just base-10 representations. $\endgroup$ Apr 28 '17 at 6:23
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    $\begingroup$ You can tweet: "Ten to the power 140" $\endgroup$
    – Gil Kalai
    Apr 28 '17 at 6:24
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    $\begingroup$ @彩音M en.wikipedia.org/wiki/Berry_paradox $\endgroup$ Apr 28 '17 at 6:26
88
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Not deep, but if [0,1]² is cut in N triangles of equal area, N is even. If not, extend 2-adic valuation on Q to R, tricolor plane and apply Sperner.

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    $\begingroup$ This is my favorite! Perhaps it would even benefit from removing the humble "Not deep". $\endgroup$ Apr 26 '17 at 16:44
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    $\begingroup$ I like it too, and it suggests a nice general pattern for tweetable mathematics: (1) Statement, (2) Setup, (3) "Apply [known result]." $\endgroup$ Apr 27 '17 at 2:43
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    $\begingroup$ #MonskysTheorem $\endgroup$
    – Igor Rivin
    May 6 '17 at 3:28
88
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Chebyshev said, and I say it again,
there is always a prime between n and 2n.
#BertrandsPostulate

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    $\begingroup$ Where n = 0? If you have space, maybe include that n > 0 $\endgroup$
    – Restioson
    Apr 29 '17 at 7:32
  • 2
    $\begingroup$ Well, its not really me who invented this tweet... (Also, then I should perhaps also include that $n$ is a natural number and not from some unique factorization domain or whatever) $\endgroup$ Apr 29 '17 at 7:40
  • $\begingroup$ yeah, there just might not be enough chars for that... $\endgroup$
    – Restioson
    May 1 '17 at 13:16
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    $\begingroup$ I wish Paul Erdos had a twitter account. $\endgroup$ Apr 27 '19 at 23:56
84
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Integral of exp(-x²) dx over R = Γ(1/2) = √π. Proof: square the Gaussian integral and use polar coordinates!

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    $\begingroup$ This is a great example! $\endgroup$ Apr 27 '17 at 21:29
64
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Erdos: You can color edges between √2ᵏ vertices red/blue so no monochrome size k subgraphs (Pick a random coloring. It probably works)

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    $\begingroup$ Perfect example! I love this one. $\endgroup$ Apr 27 '17 at 0:04
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    $\begingroup$ It's even a counterexample to Gil Kalai's comment that "To demand that a mathematician in the field can fill the details is too much to ask for." $\endgroup$ Apr 29 '17 at 14:12
57
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The n-th binary digit of π is calculable without calculating all the previous digits.

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47
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There are infinitely many primes p for which there is a prime between p+1 and p+246.

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    $\begingroup$ One can add "Still want to know about p+3". $\endgroup$ May 6 '17 at 15:00
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    $\begingroup$ The reference would fit too: arxiv.org/abs/1407.4897 $\endgroup$
    – Matt F.
    Apr 16 '19 at 13:36
42
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Euclid: there are infinitely many primes, if not, multiply all and add 1. Link

If only they had Twitter back then. They had so many tweetable proofs :)

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    $\begingroup$ There are infinitely many composites: Multiple the first n of them. Don't add 1 (attributed to Hendrik Lenstra) $\endgroup$
    – David
    Apr 27 '17 at 20:43
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    $\begingroup$ @JavaMan: that proves that if there are at least two composites, then there are infinitely many… $\endgroup$ Apr 29 '17 at 23:09
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    $\begingroup$ Also, don't forget xkcd. $\endgroup$
    – Ivo Terek
    Apr 30 '17 at 22:18
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    $\begingroup$ Better: "Euclid: there are arbitrarily many primes, by multiplying those given and adding 1." $\endgroup$
    – Matt F.
    May 1 '17 at 18:56
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    $\begingroup$ This phrasing can support a common misunderstanding of the proof though. I would end with: "...and add 1; then factor." $\endgroup$
    – usul
    Nov 16 '17 at 17:45
38
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33 = 8866128975287528³ + (-8778405442862239)³ + (-2736111468807040)³. Andrew Booker! link.

Update 2019-09-05: $(-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3 = 42$, https://people.maths.bris.ac.uk/~maarb/. This completes all the numbers less than $114$.

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    $\begingroup$ This is fantastic!!!!! (Read the link if you didn't already know why.) Honestly, this looks like a winning answer. $\endgroup$
    – Todd Trimble
    Mar 9 '19 at 14:20
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    $\begingroup$ Tim Browning deserves a tradable digital trophy for this solution. This is a computational problem which is easy to verify, difficult to solve, and of mathematical significance. One should be able to construct a cryptocurrency smart contract that can reward someone for solving this kind of problem. mathoverflow.net/questions/322022/… $\endgroup$ Mar 9 '19 at 14:30
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    $\begingroup$ @JosephVanName I do really like and like to think about your linked question - however, as you kind of hinted at, if Andrew Booker were to be awarded merely for his solution, he would be demotivated to actually publish his methods, lest others copy! $\endgroup$
    – Mark S
    Mar 10 '19 at 23:45
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    $\begingroup$ mathoverflow.net/a/223121/88133 $\endgroup$ Sep 6 '19 at 1:21
37
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The most beautifully useless theorem of mathematics IMO:

"This statement is a theorem (and moreover, I can prove it)" — link

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    $\begingroup$ Can I get you interested in joining the tautology club? $\endgroup$
    – Mindwin
    Apr 26 '17 at 15:36
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    $\begingroup$ @Mindwin Ah, but the amazing thing about these "Henkin sentences" is that they are not so trivial to prove. $\endgroup$
    – Gro-Tsen
    Apr 26 '17 at 15:50
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    $\begingroup$ This made my world that much bigger. Thank you. $\endgroup$ May 2 '17 at 2:52
33
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$AB-BA=I$ has no solution in finite matrices. Pf: Trace it!

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    $\begingroup$ Except void matrices ($n=0$ :) $\endgroup$ May 15 '17 at 0:04
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    $\begingroup$ $AB - BA= I$ not solvable in Banach algebras. Pf: Spectra of AB and BA. $\endgroup$ Apr 16 '19 at 13:04
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    $\begingroup$ Except when the dimension is equal to zero in the coefficient ring (e.g. $M_{2 \times 2}(\mathbb{F}_2)$). $\endgroup$
    – WhatsUp
    Apr 16 '19 at 14:12
31
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$n!$ divides $(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1})$.

Proof: $\mathfrak{S}_n$ embeds in $\mathrm{GL}_n(\mathbf{F}_2)$. $\quad \square$

(I have read this elegant justification on MO someday, yet I do not find it now.)

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    $\begingroup$ What does the fancy $\mathfrak{S}_n$ stand for, symmetric group on $n$ letters? $\endgroup$ Jun 7 '18 at 14:24
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    $\begingroup$ @StefanPerko Absolutely! $\endgroup$ Jun 7 '18 at 14:49
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    $\begingroup$ Nice! But I think it should be $\operatorname{GL}_n(\mathbf{F}_2)$ rather than $\operatorname{GL}_2$. $\endgroup$ Jun 7 '18 at 18:16
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    $\begingroup$ @ZachTeitler Thanks for the correction $\endgroup$ Jun 8 '18 at 7:58
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    $\begingroup$ @M.Winter Good question. It is $2^n-4$ (then $2^n-8$, and so on). The reason is that $|\operatorname{GL}_n(\mathbf{F}_2)| = (2^n-1)(2^n-2)(2^n-4)(2^n-8)\dotsm(2^n-2^{n-1})$. And the reason for that is that if you build an invertible matrix column by column, after choosing the first $k$ columns, the next column to be chosen out of the $2^n$ possible vectors must not lie in the span of the first $k$ columns, which is a $k$-dimensional subspace so it has $2^k$ elements, leaving $2^n-2^k$ choices for the $(k+1)$st column. $\endgroup$ Sep 1 '20 at 21:55
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Halting problem - there is no computer program which can determine if an arbitrary computer program halts on a specified input. Link

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26
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You cannot comb a hairy ball, should you wish to.

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    $\begingroup$ The wind always stands still somewhere. $\endgroup$
    – user1504
    May 6 '17 at 3:15
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    $\begingroup$ Hairy donuts on the other hand... $\endgroup$
    – David
    Feb 24 '19 at 4:16
19
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Sphere packing-The E8 lattice provides the optimal packing in eight-dimensional space. Link

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    $\begingroup$ Still within the twitter limit :) : Sphere packing-The E8 lattice provides the optimal packing in eight-dimensional space, Pf: LP+modular forms arxiv.org/abs/1603.04246 $\endgroup$
    – Gil Kalai
    Apr 26 '17 at 8:58
19
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Mihăilescu's theorem (ex Catalan conjecture) - $8$ and $9$ are the only consecutive proper powers of natural numbers. Link

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1
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    $\begingroup$ first proved in 2002? WOW $\endgroup$ May 6 '17 at 3:28
18
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Borsuk's conjecture - a counterexample for DIM>2000, the tensor product of the unit sphere with itself. Link

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Turing discovers problems that even computers cannot solve. Link

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    $\begingroup$ You won't believe the second, and the fifth blew my mind! $\endgroup$ Apr 26 '17 at 13:38
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    $\begingroup$ @PeterSamuelson I don't get it. Second what, fifth what? $\endgroup$
    – Todd Trimble
    Apr 26 '17 at 16:07
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    $\begingroup$ @Todd, it is an internetism, to get people to click on a link to a list of, oh, money saving incredible mathematical tweets? Gerhard "Using This One Weird Tip" Paseman, 2017.04.26. $\endgroup$ Apr 26 '17 at 19:34
  • $\begingroup$ @GerhardPaseman Oh yes, I've seen that. Makes sense now. $\endgroup$
    – Todd Trimble
    Apr 26 '17 at 19:38
16
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One more:

CFSG - The largest sporadic simple group has order 808017424794512875886459904961710757005754368000000000.

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2
  • $\begingroup$ Oops, that was two links, so it doesn't fit the conditions. Feel free to ignore either - the two Wikipedia pages have direct links to each other. $\endgroup$
    – GNiklasch
    Apr 26 '17 at 10:47
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    $\begingroup$ Is there any intrinsic definition of sporadic group? Out of curiosity, I'd like to know if any other finite simple group was once considered as sporadic (= not abelian, alternating, or in any "list" of Lie-flavor type) before it was extrapolated as Lie type. $\endgroup$
    – YCor
    Mar 9 '19 at 15:55
16
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Wedderburn's theorem-Every finite skew field is a field. Pf: class equation on centralizers and cyclotomic irreducibility. Link

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2
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    $\begingroup$ Edited to give some idea on the "essence" of the proof. $\endgroup$ Apr 26 '17 at 14:54
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    $\begingroup$ Alternative proof (not to be confused with "alt-proof"): apply Chevalley(-Warning) to the norm form. $\endgroup$ Apr 27 '17 at 1:00
16
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Cantor's theorem!

The cardinality of the reals is greater than that of the natural numbers.

Also, various incompleteness results along the lines of the Mathematical T-Rex:

The continuum hypothesis is neither provable nor disprovable from ZFC, unless ZFC is inconsistent to begin with!

(The above can be extended to include a myriad of choice-related statements and ZF, for example.)

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    $\begingroup$ You still have room to mention "diagonalization" and "forcing", respectively, if we are to focus on the "essence of a notable mathematical development" aspect. If you split "CH isn't provable..." and "CH isn't disprovable…" into two tweets, you could accommodate the constructible universe, too. $\endgroup$
    – GNiklasch
    Apr 26 '17 at 12:44
15
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Of course, every finite group of odd order is solvable.

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    $\begingroup$ I'd like to see the essence of this development on a tweet. :) $\endgroup$
    – Gil Kalai
    Apr 26 '17 at 12:33
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    $\begingroup$ I have discovered a truly remarkable proof of this theorem which this <s>margin</s>tweet is too small to contain. $\endgroup$
    – chx
    Apr 30 '17 at 0:35
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    $\begingroup$ Savage one .... $\endgroup$
    – DS R
    May 7 '17 at 14:28
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    $\begingroup$ Proof: Apply CFSG ;-) $\endgroup$ Sep 20 '19 at 20:04
15
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Breaking News: A mathematician blew up 6 points on a plane! You won't believe what happened next...

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    $\begingroup$ BREAKING NEWS: a plane got blown up by a mathematician. At least five cups of coffee are thought to be dead. $\endgroup$ Sep 21 '18 at 19:15
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Lindemann's theorem - You need more than a compass and a ruler to square a circle. Link

Edit Or if we really want the essence of the argument, on which a whole lot of stronger results have been patterned since:

Exploiting the properties of the exponential function, if $\pi$ was algebraic there'd be rational integer strictly between 0 and 1.

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  • $\begingroup$ OK, I'll stop now… :) $\endgroup$
    – GNiklasch
    Apr 26 '17 at 9:37
  • $\begingroup$ Well, Lindemann's theorem is much more than this. "Lindemann's theorem: pi is transcendental" would be better. $\endgroup$ Apr 26 '17 at 9:42
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    $\begingroup$ Sure. I intentionally toned it down so as to mean something to a broader audience, who might not want to expend the extra time to follow a link... $\endgroup$
    – GNiklasch
    Apr 26 '17 at 9:51
  • $\begingroup$ Lindemann's theorem does not say $\pi$ can be approximated too well by rationals to be algebraic. As far as we know, $\pi$ is not particularly well approximated by rationals. $\endgroup$ Apr 26 '17 at 15:23
  • $\begingroup$ Thank you @Robert Israel. Fixed the toned-up version (I hope). $\endgroup$
    – GNiklasch
    Apr 26 '17 at 15:33
14
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Invert 2×2 matrix A? Easy: switch diagonal entries, negate off-diags, & divide by Δ=det(A).
Pf: divide A²-tA+ΔI=0 by A, solve for 1/A !

MO link

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This is actually a real tweet by Ryan O'Donnell on Huang's proof of the sensitivity conjecture.

Hao Huang@Emory:

Ex.1: ∃edge-signing of n-cube with 2^{n-1} eigs each of +/-sqrt(n)

Interlacing=>Any induced subgraph with >2^{n-1} vtcs has max eig >= sqrt(n)

Ex.2: In subgraph, max eig <= max valency, even with signs

Hence [GL92] the Sensitivity Conj, s(f) >= sqrt(deg(f))

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13
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The prime numbers contain arbitrarily long arithmetic progressions.

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