CRITERIA OF PRIMALITY DUE TO SMARANDCHE


1)  Let S(n) be the Smarandache Function:  S(n) is the smallest number such
    that S(n)! is divisible by n.  Let p be an integer > 4.  Then:

                   p is prime if and only if S(p) = p.


    References:
      [1] Dumitrescu, C., "A Brief History of the Smarandache Function",
          <Mathematical Spectrum>, ??
      [2] Smarandache, Florentin, "A Function in the Number Theory", <Analele 
          Univ. Timisoara>,  Fascicle 1, Vol. XVIII, 1980, pp. 79-88;
          reviewed in Mathematical Reviews: 83c:10008.


The following four statements are derived from the Wilson theorem
(p is prime iff (p-1)! is congruent to -1 (mod p)), but improve it
because the factorial is reduced:

2)  Let p be an integer >= 3.  Then:
                                                         p-1
        p is prime if and only if (p-3)! is congruent to ---- (mod p).
                                                           2

      References:
        [1] Smarandache, Florentin, "Criteria for a Positive Integer to be 
            Prime", <Gazeta Matematica>, Bucharest, No. 2, 1981, pp. 49-52;
            reviewed in Mathematical Reviews: 83a:10007.
        [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, 
            Bucharest, 1996, pp. 94-98.


                                     
3)  Let p be an integer > 4.  Then:            
                                                |  p  |
                                                | --- |+1        
                                                |_ 3 _|   |  p+1 |
     p is prime iff (p-4)! is congruent to  (-1)          | ---- | (mod p),
                                                          |_  6 _|
                                                          
           |   |
    where  | x | means the inferior integer part of x, i.e. the smallest 
           |_ _|
    integer greater than or equal to x.

      References:
        [1] Smarandache, Florentin, "Criteria for a Positive Integer to be 
            Prime", <Gazeta Matematica>, Bucharest, No. 2, 1981, pp. 49-52;
            reviewed in Mathematical Reviews: 83a:10007.
        [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, 
            Bucharest, 1996, pp. 94-98.



                                     
4)  Let p be an integer >= 5.  Then:            
                                                 2
                                                r  - 1
    p is prime iff (p-5)! is congruent to rh + -------- (mod p),
                                                  24
              |   p  |
     with h = | ---- | and r = p - 24h,
              |_ 24 _|
           
           |   |
    where  | x | means the inferior integer part of x, i.e. the smallest 
           |_ _|
    integer greater than or equal to x.


      References:
        [1] Smarandache, Florentin, "Criteria for a Positive Integer to be 
            Prime", <Gazeta Matematica>, Bucharest, No. 2, 1981, pp. 49-52;
            reviewed in Mathematical Reviews: 83a:10007.
        [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, 
            Bucharest, 1996, pp. 94-98.


                                     
5)  Let p = (k-1)!h + 1 be a positive integer, k > 2, h natural number.  
    Then:            
                                              t
    p is prime iff (p-k)! is congruent to (-1) h (mod p),

                  |   p  |
     with t = h + | ---- | + 1,
                  |_  h _|
           
           |   |
    where  | x | means the inferior integer part of x, i.e. the smallest 
           |_ _|
    integer greater than or equal to x.


      References:
        [1] Smarandache, Florentin, "Criteria for a Positive Integer to be 
            Prime", <Gazeta Matematica>, Bucharest, No. 2, 1981, pp. 49-52;
            reviewed in Mathematical Reviews: 83a:10007.
        [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, 
            Bucharest, 1996, pp. 94-98.


6)  Let 1 <= k <= p be integers.  Then:  
 p is prime if and only if  (p-k)!(k-1)! is congruent to (-1)^k.


      References:
        [1] Smarandache, Florentin, "Criteria for a Positive Integer to be 
            Prime", <Gazeta Matematica>, Bucharest, No. 2, 1981, pp. 49-52;
            reviewed in Mathematical Reviews: 83a:10007.
        [2] Smarandache, Florentin, "Collected Papers", Vol. I, Ed. Tempus, 
            Bucharest, 1996, pp. 94-98.