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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 7 Apr 2019
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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

    5:27 AM - 7 Apr 2019
    • 1,596 Retweets
    • 5,863 Likes
    • Nathan💥 Andrew McAuliffe Erica Pagan Mostafa Sabiloo Maggie Z✌️ Fatemeh S Vlad Zmitrovich Mnoyi BrightShadow
    45 replies 1,596 retweets 5,863 likes
      1. New conversation
      2. Jonathan Mboyo Esole‏ @JMEsole 7 Apr 2019
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        Replying to @fermatslibrary

        In this case, the algebraic proof is also simple enough: For any integer n, we have 2n+1= 2n+1 + n^2 -n^2 = (n+1)^2-n^2.pic.twitter.com/tQ35a5iFXT

        2 replies 18 retweets 265 likes
      3. NIall Litchfield‏ @nlitchfield 7 Apr 2019
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        Replying to @JMEsole @fermatslibrary

        Indeed. But it isn't half so pretty.

        2 replies 1 retweet 18 likes
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      2. 418: No Coffee 4 U  🤖‏ @ramriot 7 Apr 2019
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        Replying to @fermatslibrary

        But which odd integer is the smallest one that is the sum of two cubes in two different ways?

        4 replies 0 retweets 10 likes
      3. Caly‏ @Calybium 7 Apr 2019
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        Replying to @ramriot @fermatslibrary

        1729 10³+9³ 12³+1³ There's a famous story about GH Hardy going to see Ramanujan in a cab numbered 1729, and he said he wished it'd been more interesting a number, and Ramanujan said, "nonsense. It's the smallest imteger that can be expressed as the sum of two cubes in two ways"

        3 replies 9 retweets 138 likes
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      2. Kaleem‏ @legendu_da 7 Apr 2019
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        Replying to @fermatslibrary

        Also the fact that sum of consecutive odd numbers = squares of numbers ex. 1 + 3 + 5 = 9 can be proved from the last grid.

        2 replies 5 retweets 48 likes
      3. AJ Witt‏ @ITAMRocks 8 Apr 2019
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        Replying to @legendu_da @fermatslibrary

        Can you expand on this one? Don't quite understand it

        1 reply 0 retweets 1 like
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      1. FlyingOctopus0‏ @FlyingOctopus0 7 Apr 2019
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        Replying to @fermatslibrary

        Or equivalently sum of n odd numbers from 1 to 2n-1 is a square of n. n^2 = 1 + 3 + ... + (2n-1) .To get the above clamin you just substract two formulas with n+1 and n. So (n+1)^2-n^2 = 2n+1

        0 replies 1 retweet 16 likes
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      2. Viktor Arvidsson‏ @anotherday____ 7 Apr 2019
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        Replying to @xfyfz @fermatslibrary

        Hydraulics!

        1 reply 0 retweets 34 likes
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      2. M[un]ira‏ @mundanemun 7 Apr 2019
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        Replying to @fermatslibrary

        I need more help

        3 replies 0 retweets 1 like
      3. Karen Rylander‏ @KarenRylander 7 Apr 2019
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        Replying to @mundanemun @fermatslibrary

        Those seem to be single tiles made up of two parts, so the odd number here is 5. Odd numbers are just evens (a number divisible by 2) + 1: 5=2*2+1. The 1 makes a corner position. So 5 is like (2+1)*(2+1)-2*2 (the 2*2 are the 4 tiles that fill the rest of the square).

        2 replies 0 retweets 7 likes
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