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fermatslibrary's profile
Fermat's Library
Fermat's Library
Fermat's Library
@fermatslibrary

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Fermat's Library

@fermatslibrary

A platform for illuminating academic papers. We publish an annotated paper every week. Our chrome extension for arXiv: https://fermatslibrary.com/librarian 

fermatslibrary.com
Joined September 2015

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    Fermat's Library‏ @fermatslibrary 31 Oct 2018
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    Every odd integer is the difference of 2 squares Proof: 1) Pick an odd number ░░ ░░ ░░ ░░ ░░ 2) Bend it in half ░░ ░░ ░░ ░░ ░░ 3) Fill in the rest ░░ ░░ ░░ ░░ ▓▓ ▓▓ ░░ ▓▓ ▓▓ The odd nbr is the area difference of the big and small squares

    5:45 AM - 31 Oct 2018
    • 2,267 Retweets
    • 7,098 Likes
    • Vincent Lu Sharath Chandra Sheripally Logan Corley Phil Arvind Prakash Josh Chua Albert socialmonkey🦍 iPengi
    83 replies 2,267 retweets 7,098 likes
      1. New conversation
      2. Tom Ruen‏ @Tom_Ruen 31 Oct 2018
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        Replying to @fermatslibrary

        Yes, n=[(n+1)/2]^2-[(n-1)/2]^2 A special case of: ab=[(a+b)/2]^2-[(a-b)/2]^2 Like 19x1=100-81=19 18x2=100-64=36 17x3=100-49=51 16x4=100-36=64 15x5=100-25=75 14x6=100-16=84 13x7=100-9=91 12x8=100-4=96 11x9=100-1=99 10x10=100-0=100

        2 replies 9 retweets 74 likes
      3. Steve Howell‏ @showell2020 31 Oct 2018
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        Replying to @Tom_Ruen @fermatslibrary

        331 is a difference between two cubes in any base—octal, decimal, etc

        2 replies 3 retweets 29 likes
      4. End of conversation
      1. Tamás Görbe‏ @TamasGorbe 31 Oct 2018
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        Replying to @fermatslibrary

        1+2+3+...+n=n×(n+1)/2 Proof: 1) Draw 1,2,...,n ░░ ░░ ░░ ░░ ░░ ░░ 2) Duplicate and rotate it by 180° ▓▓ ▓▓ ▓▓ ▓▓ ▓▓ ▓▓ 3) Put them together to form a rectangle of size n×(n+1) ░░ ▓▓ ▓▓ ▓▓ ░░ ░░ ▓▓ ▓▓ ░░ ░░ ░░ ▓▓

        1 reply 7 retweets 34 likes
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      1. New conversation
      2. Sinan Синан سنان‏ @DrSinanovic 31 Oct 2018
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        Replying to @fermatslibrary

        two CONSECUTIVE squares

        2 replies 0 retweets 43 likes
      3. 018‏ @CER0DIECI0CH0 31 Oct 2018
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        Replying to @DrSinanovic @fermatslibrary

        Not necessarily, I mean, "6^2 - 3^2 = 36 - 9 = 27"

        2 replies 0 retweets 1 like
      4. 4 more replies
      1. Pierre Popineau‏ @PierrePopineau 31 Oct 2018
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        Replying to @fermatslibrary

        Or you could use : (k+1)^2-k^2 = 2k+1 for every integer 🙂

        1 reply 2 retweets 58 likes
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      2. Luis Batalha  🇵🇹 🇺🇸‏ @luismbat 31 Oct 2018
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        Replying to @fermatslibrary

        Alternatively, since every odd can be written as 2k+1=1(2k+1)=(k+1-k)(k+1+k)=(k+1)²-k² 😄

        2 replies 6 retweets 60 likes
      3. CW‏ @pigeonwhole 31 Oct 2018
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        Replying to @luismbat @fermatslibrary

        (k+2)^2 =k^2 +4(k+1) shows that multiples of 4 are also differences of 2 squares. These and the odd integers are the only integers that are the difference of 2 squares.

        1 reply 0 retweets 2 likes
      4. 1 more reply
      1. Gustav Mahler‏ @GustavMahlerJr 31 Oct 2018
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        Replying to @fermatslibrary

        Wow so for any 2 consecutive numbers, add them together and that is the difference of their squares?

        3 replies 0 retweets 11 likes
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      1. New conversation
      2. k8‏ @katgleason 31 Oct 2018
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        Replying to @fermatslibrary

        I went to public schools but our town had an amazing “gifted & talented program” for the 2 kids with the highest iqs from each elementary school. in that program all we did was learn in this exact manner. I never realized how much it shaped my learning until your tweets! Ty ❤️

        1 reply 0 retweets 4 likes
      3. Espen‏ @WZot 31 Oct 2018
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        Replying to @katgleason @fermatslibrary

        The sad thing is that if those who weren’t as gifted learned it this way they would probably “get it”...:-)

        2 replies 0 retweets 2 likes
      4. 1 more reply

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