
An equation that takes Pythagoras to a new level - soheilpro
https://medium.com/starts-with-a-bang/this-one-equation-10%C2%B2-11%C2%B2-12%C2%B2-13%C2%B2-14%C2%B2-takes-pythagoras-to-a-whole-new-level-ff588f1d13b6
======
chx
The linked paper [https://fermatslibrary.com/s/proof-without-words-
pythagorean...](https://fermatslibrary.com/s/proof-without-words-pythagorean-
runs) gives you the formula and I am much more a symbols guy than a geometry
guy so I will prove it that way, first rearrange it:

(4T_n)^2=(4T_n+1)^2-(4T_n-1)^2+....(4T_n+n)^2-(4T_n-n)^2

Now, (4T_n+k)^2-(4T_n-k)^2=16T_nk (this, of course, shows up in the geometric
proof as four rectangles where one side is 4T_n and the other is k)

so this is equivalent to

16T_nT_n = 16 T_n (1+...n)

which is equvialent to:

T_n=1+...n

But that is the very definition of T_n. Q.e.d.

Also if you want to take Pythagoras to a new level, Edsger W. Dijkstra who is
much better known for his work in CS has a very interesting formulation
[https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/E...](https://www.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD975.html)
proving sgn(alpha+beta-gamma)=sgn(a^2+b^2-c^2) which includes and extends the
Pythagoras theorem.

~~~
imglorp
Wow, fermatslibrary has gone further downhill. It's almost unusable now with
all the engagement stuff.

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jimhefferon
The caption "The equation 1⁰² + 1¹² + 1²² = 1³² + 1⁴², whose answer is that
both sides equal 365, was immortalized in a different form in this 1895
painting: “Mental Arithmetic. In the Public School of S. Rachinsky.” (NIKOLAY
BOGDANOV-BELSKY)" has funky exponents. Medium requires that I make an account
to say that, so I am saying it here.

~~~
CliffStoll
I think the equation on the board is ( 10^2 + 11^2 + 12^2 +13^2 + 14^2 ) / 365
which solves for 2.

The students are expected solve this with "Mental Arithmetic". Try that in
your head!

~~~
delhanty
Assuming one's memorized a table of squares already.

10^2 = 100

11^2 + 13^2 = 121 + 169 = 290

12^2 + 14^2 = 144 + 196 = 340

100 + 290 + 340 = 730

730 / 365 = 2

~~~
lonelappde
You can also "walk it up" if you haven't memorized them:

    
    
          10^2*5
        + (10+11)*4
        + (11+12)*3
        + (12+13)*2
        + (13+14)
        ------------
    
           500
         +  84
         +  69
         +  50
         +  27
         -----
           730
    

It's a bit of detail to juggle in your head but not too bad if you hang on to
your current subtotal and remember where you are in the sequence to compute
the next addend, and you know it's easier to add multi-digit numbers left-to-
right than-right to-left.

You only carry from the ones place twice.

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Avshalom
Ptolemy's theorem is another Pythagoras on steroids

[https://m.youtube.com/watch?v=bJOuzqu3MUQ](https://m.youtube.com/watch?v=bJOuzqu3MUQ)

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gordaco
I thought this was going to help me with Project Euler 261 [0], but it's not
exactly the same, so no luck so far. Cool article anyway; linking geometry and
algebra is always satisfactory.

[0]
[https://projecteuler.net/problem=261](https://projecteuler.net/problem=261)

~~~
lonelappde
How much CPU time and number theory do problems in that range generally
require?

Brute force search of 10^10 (with some automatic monitoring of how wide a
range of 'm' to scan at each candidate) is a lot but not a lot of a lot, if
you code in a CPU efficient language and maybe use parallel computation and
are willing to wait a while.

~~~
gordaco
The difficulty and the required run time vary a lot starting from problem 150
or so. This one seems particularly difficult (it's rated at 85% difficulty,
which is also the highest I've managed to solve). The problem here is that the
search space is not 10^10, but higher, since if you iterate over k you don't
know which values of m or n need to be tested. For m=1 it's clear that you can
just use Pythagorean triples, but for bigger values I don't know if there is a
reliable procedure to generate solutions.

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RockofStrength
Also 3^3 + 4^3 + 5^3 = 6^3

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dr_dshiv
Beautiful.

Pythagoreans believed that the world was made of math -- and that the
harmonies of mathematics revealed the harmonies of the universe. Pretty good
for 6th century BC.

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rk06
Medium has paywalled me. Any non-paywall links?

~~~
weibing
That is the reason I really don’t like Medium. So many click-bait articles and
paywall

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johnr2
> Medium has paywalled me. Any non-paywall links?

The link works here in Firefox with NoScript, also in Dillo (no JS support).

