Fermat's little theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-28

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Fermat%27s_little_theorem


00:00:25 1 History
00:00:36 1.1 Further history
00:01:21 2 Proofs
00:01:33 3 Generalizations
00:02:07 4 Converse
00:05:05 5 Pseudoprimes
00:05:39 6 Miller–Rabin primality test
00:05:45 7 See also
00:05:55 8 Notes
00:06:04 9 References
00:12:33 10 Further reading
00:13:43 11 External links
00:14:56 Miller–Rabin primality test
00:15:28 2s  d + 1, with d odd, an odd integer for which primality has to be tested, choose randomly a such that 1  a  p; then compute b
00:16:27 See also



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SUMMARY
=======
Fermat's little theorem states that if p is a prime number, then for any integer a, the number ap − a is an integer multiple of p.  In the notation of modular arithmetic, this is expressed as

  
    
      
        
          a
          
            p
          
        
        ≡
        a
        
          
          (
          mod
          
          p
          )
        
        .
      
    
    {\displaystyle a^{p}\equiv a{\pmod {p}}.}
  For example, if a = 2 and p = 7, then 27 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7.
If a is not divisible by p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols:

  
    
      
        
          a
          
            p
            −
            1
          
        
        ≡
        1
        
          
          (
          mod
          
          p
          )
        
        .
      
    
    {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.}
  For example, if a = 2 and p = 7, then 26 = 64, and 64 − 1 = 63 = 7 × 9 is thus a multiple of 7.
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem.