Fermat's theorem on sums of two squares | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares


00:04:58 1 Proofs of Fermat's theorem on sums of two squares
00:06:30 2 Related results
00:10:44 3 See also



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SUMMARY
=======
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime p can be expressed as:

  
    
      
        p
        =
        
          x
          
            2
          
        
        +
        
          y
          
            2
          
        
        ,
      
    
    {\displaystyle p=x^{2}+y^{2},}
  
with x and y integers, if and only if

  
    
      
        p
        ≡
        1
        
          
          (
          mod
          
          4
          )
        
        .
      
    
    {\displaystyle p\equiv 1{\pmod {4}}.}
  The prime numbers for which this is true are called Pythagorean primes.
For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways:

  
    
      
        5
        =
        
          1
          
            2
          
        
        +
        
          2
          
            2
          
        
        ,
        
        13
        =
        
          2
          
            2
          
        
        +
        
          3
          
            2
          
        
        ,
        
        17
        =
        
          1
          
            2
          
        
        +
        
          4
          
            2
          
        
        ,
        
        29
        =
        
          2
          
            2
          
        
        +
        
          5
          
            2
          
        
        ,
        
        37
        =
        
          1
          
            2
          
        
        +
        
          6
          
            2
          
        
        ,
        
        41
        =
        
          4
          
            2
          
        
        +
        
          5
          
            2
          
        
        .
      
    
    {\displaystyle 5=1^{2}+2^{2},\quad 13=2^{2}+3^{2},\quad 17=1^{2}+4^{2},\quad 29=2^{2}+5^{2},\quad 37=1^{2}+6^{2},\quad 41=4^{2}+5^{2}.}
  On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4.
Albert Girard was the first to make the observation, describing all positive integral numbers (not necessarily primes) expressible as the sum of two squares of positive integers; this was published in 1625. The statement that every prime p of the form 4n+1 is the sum of two squares is sometimes called Girard's theorem. For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares)  in a letter to Marin Mersenne dated December 25, 1640: for this reason this version of the theorem is sometimes called Fermat's Christmas theorem. 
Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as the sum of two squares, by applying Fermat's theorem to the prime factorization of any positive integer n, we see that if all  the prime factors of n congruent to 3 modulo 4 occur to an even exponent, then n is expressible as a sum of two squares. The converse also holds. This equivalence provides the characterization Girard guessed.