Lagrange's four-square theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 29/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem


00:02:15 1 Historical development
00:04:02 2 The classical proof
00:15:03 3 Proof using the Hurwitz integers
00:15:30 4 Generalizations
00:15:41 5 Algorithms
00:15:53 6 Number of representations
00:16:06 7 Uniqueness
00:16:18 8 Further refinements
00:16:35 9 See also
00:16:49 10 Notes
00:46:16 11 References
00:48:19 12 External links
00:48:26 d
00:50:39 Algorithms
00:52:04 Number of representations
00:54:16 8(p + 1).Some values of r4(n) occur infinitely often as r4(n)
00:54:47 Uniqueness
00:57:17 Further refinements
00:59:55 See also



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SUMMARY
=======
Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares.

  
    
      
        p
        =
        
          a
          
            0
          
          
            2
          
        
        +
        
          a
          
            1
          
          
            2
          
        
        +
        
          a
          
            2
          
          
            2
          
        
        +
        
          a
          
            3
          
          
            2
          
        
      
    
    {\displaystyle p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}
  where the four numbers 
  
    
      
        
          a
          
            0
          
        
        ,
        
          a
          
            1
          
        
        ,
        
          a
          
            2
          
        
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            3
          
        
      
    
    {\displaystyle a_{0},a_{1},a_{2},a_{3}}
   are integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows:

  
    
      
        
          
            
              
                3
              
              
                
                =
                
                  1
                  
                    2
                  
                
                +
                
                  1
                  
                    2
                  
                
                +
                
                  1
                  
                    2
                  
                
                +
                
                  0
                  
                    2
                  
                
              
            
            
              
                31
              
              
                
                =
                
                  5
                  
                    2
                  
                
                +
                
                  2
                  
                    2
                  
                
                +
                
                  1
                  
                    2
                  
                
                +
                
                  1
                  
                    2
                  
                
              
            
            
              
                310
              
              
                
                =
                
                  17
                  
                    2
                  
                
                +
                
                  4
                  
                    2
                  
                
                +
                
                  2
                  
                    2
                  
                
                +
                
                  1
                  
                    2
                  
                
                .
      ...