Waste less time on Facebook — follow Brilliant.
×

The Beauty of Math - Four Square Theorem

Bob Krueger Shared by Bob Krueger 17, USA · 1 year, 4 months ago

I came across this theorem one time and found it especially striking. I even told my friends who supposedly dislike math about it, and they were amazed, trying to calculate values for various numbers.

Lagrange's Four-Square Theorem states that any positive integer can be expressed as the sum of four perfect squares. Try this for numbers like 7, 31, 326. While we know of course of numbers that can be expressed as one square (perfect squares), we know less about numbers that can be expressed as the sum of two or three squares. (Note that 0 is a square). Try to work out this theorem for some other numbers? Can you find any that are the sum of only two squares or only three squares? Is there any pattern to these numbers?

While the proof of the theorem is beyond the scope of the Cosines Group, you can find the proof on the Wikipedia page with the same name.

As a Computer Science extension, can you create a code that will find the four squares that sum up to any given inputted number? Perhaps you can use this to locate any patterns in the system.

Feel free to post any solutions, ideas, questions, extensions, or code below.

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](http://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \­( ... \­) or \­[ ... \­] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

See full formatting guide

Sort by:
Agnishom Chattopadhyay
Dec 16, 2013
(last modified Dec 16, 2013 )

Log in to reply

Agnishom Chattopadhyay
Dec 20, 2013

This can be done in mathematica with this code: PowersRepresentations[n,4,2] Here is an interesting algorithm on it: http://www.alpertron.com.ar/4SQUARES.HTM

Source: http://www.mathisfunforum.com/viewtopic.php?id=19225

Log in to reply

Ayush Alankar
Dec 18, 2013
Ayush Alankar
18, India

class Four_squares { public static void main(int n) { for(int a=0;a<=n;a++) for(int b=0;b<=n;b++) for(int c=0;c<=n;c++) for(int d=0;d<=n;d++) if(aa+bb+cc+dd==n) System.out.println(""+a+b+c+d); } }

Log in to reply

Bob Krueger
Dec 18, 2013
Bob Krueger
17, USA

Short and simple. I like it. However, it's not efficient and its output is the cleanest. Are there any easy fixes to this?

Log in to reply

Agnishom Chattopadhyay
Dec 20, 2013

Very Ineffecient, sorry!

Log in to reply

Debjyoti Chattopadhyay
Dec 18, 2013

superb

Log in to reply

Bob Krueger
Dec 18, 2013
Bob Krueger
17, USA

Thank you!

Log in to reply

Calvin Lin
Dec 16, 2013
Calvin Lin Staff
30, USA

Sorry, started making a post and then realized that it was out of depth for #CosinesGroup, so I moved it.

(last modified Dec 16, 2013 )

Log in to reply

Elliott Macneil
Mar 2, 2014
Elliott Macneil
16 years old

This is so awesome.

Log in to reply

Agnishom Chattopadhyay
Feb 28, 2014

Also, try the PowerRepresentations command in Mathematica

Log in to reply

Harsh Shrivastava
Dec 18, 2013
Harsh Shrivastava
14, India

can you please tell what 'mod' means ? PLEASE !!!!!!!!!!!!!!!!!!!!!! I THINK YOU R GENIUS !!

Log in to reply

Alexander Borisov
Dec 18, 2013

Considering numbers "mod 4" stands for looking at their remainders when divided by \(4.\) Note that if you know the remainders, when divided by \(4\), of any two numbers, you can calculate the remainder of their sum and product. For example, if \( a\) has reminder \(2\) and \(b\) has remainder \(3,\) their sum will always have remainder \(1.\) You may want to take a look here: Modular Arithmetic

(last modified Dec 20, 2013 )

Log in to reply

Harsh Shrivastava
Dec 19, 2013
Harsh Shrivastava
14, India

THANKS

Log in to reply

Michael Tang
Dec 17, 2013
Michael Tang
15, USA

By the way, the numbers that can be expressed as a sum of two squares have been completely characterized. Specifically, a number \(n\) is a sum of two squares if and only if each prime divisor that is \(3\) mod \(4\) has an even exponent in the prime factorization of \(n.\)

(last modified Dec 17, 2013 )

Log in to reply

Yong See Foo
Dec 18, 2013
Yong See Foo
16, Australia

Just a random guess, you got this from PFTB? I got it from there :D

Well it is nice to prove this, here are some guidelines:

  1. Prove that the product of two numbers that can both be represented as sums of two squares, can be represented as a sum of two squares.

  2. Look at \(\pmod 4\), obviously.

  3. Another obvious fact but is used, the product of two squares is a square, and can be represented as a sum of two squares obviously.

  4. You may use Fermat's Christmas Theorem (as it was in a letter dated on 1640's Christmas)..

This should be enough.

Log in to reply

Bob Krueger
Dec 18, 2013
Bob Krueger
17, USA

Interesting!

Log in to reply

Prashanth Kumar
Dec 18, 2013
Prashanth Kumar
22, India

You can also check Goldbach's theorem which states that every positive even number can be expressed as the sum of two prime numbers

Log in to reply

Alexander Borisov
Dec 18, 2013

This is actually not a theorem, but a conjecture, one of the most famous in Number Theory. There are some very interesting partial results, obtained using highly sophisticated methods.The Goldbach Conjecture Wikipedia page has a pretty accurate (though long) description.

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...