CRITERIA OF SIMULTANEOUS PRIMALITY DUE TO SMARANDACHE




1)  Characterization of twin primes:
    Let p and p+2 be positive odd integers.  Then the following 
    statements are equivalent:
    a) p and p+2 are both prime;
    b) (p-1)!(3p+2) + 2p+2 is congruent to 0 (mod p(p+2));
    c) (p-1)!(p-2)-2 is congruent to 0 (mod p(p+2));
    d) [(p-1)!+1]/p + [(p-1)!2+1]/(p+2) is an integer.



2)  Characterization of a pair of primes:
    Let p and p+k be positive integers, with (p, p+k) = 1.  Then:
   p and p+k are both prime iff
   (p-1)!(p+k) + (p+k-1)!p + 2p+k is congruent to 0 (mod p(p+k)).



3) Characterization of a triplet of primes:
   Let p-2, p, p+4 be positive integers, coprime two by two.  Then:
   p-2, p, p+4 are all prime iff
   (p-1)! + p[(p-3)!+1]/(p-2) + p[(p+3)!+1]/(p+4) is congruent to -1 (mod p).



4)  Characterization of a quadruple of primes:
    Let p, p+2, p+6, p+8 be positive integers, coprime two by two.  Then:
   p, p+2, p+6, p+8 are all prime iff
   [(p-1)!+1]/p + [(p-1)!2!+1]/(p+2) + [(p-1)!6!+1]/(p+6) + [(p-1)!8!+1]/(p+8)
   is an integer.




5) More general:
   Let p , p , ..., p  be positive integers > 1, coprime two by two, and
        1   2        n

   1 <= k  <= p , for all i.  Then the following statements are equivalent:
         i     i

   a) p , p , ..., p  are simultaneously prime;
       1   2        n

                                  k
       n                           i   _________
   b) Sigma [(p - k )!(k -1)!-(-1)   ] |       | p  
      i=1      i   i    i              |       |  j
                                        j different 
                                         from i

      is congruent to 0 (mod p p ...p );
                              1 2    n



                                  k
        n                           i   _________
   c) (Sigma [(p - k )!(k -1)!-(-1)   ] |       | p )/(p   ...p ) 
       i=1      i   i    i              |       |  j    s+1    n
                                        j different 
                                         from i

      is congruent to 0 (mod p ...p );
                              1    s

   


                                  k
        n                           i   
   d)  Sigma [(p - k )!(k -1)!-(-1)   ] p / p  
       i=1      i   i    i               j   i
                                        
                                        
      is congruent to 0 (mod p );
                              j

   

                                  k
        n                           i   
   e)  Sigma [(p - k )!(k -1)!-(-1)   ] / p  
       i=1      i   i    i                 i
                                        
                                        
      is an integer;

   



6) Even more general:
   GENERAL THEOREM OF CHARACTERIZATION OF N PRIME NUMBERS SIMULTANEOUSLY
   DUE TO SMARANDACHE:

    Let p  , 1 <= i <= n, 1 <= j <= m , be coprime integers two by two,
         ij                          i

    and let a , r  be integers such that a  and r  are coprime for all i.
             i   i                        i      i

    The following conditions are considered for all i:
    
    (i)  p   , ..., p    are simultaneously prime iff 
          i1         im
                       i 
    
         c  is congruent to 0 (mod r ).
          i                         i

    Then the following statements are equivalent:

    a) The numbers p  , 1 <= i <= n, 1 <= j <= m , are simultaneously prime;   
                    ij                          i


              n
    b) (R/D) Sigma (a c / r ) is congruent to 0 (mod R/D),
             i=1     i i   i


                   n
                  _____
        where R = |   | r ,  and D is a dividor of R;
                   i=1   i


                           m
                            i
              n           _____
    c) (P/D) Sigma (a c / |   | p  ) is congruent to 0 (mod P/D),
             i=1     i i   j=1   ij   


                   n,m
                      i
                  ________
        where P = |      | p  ,  and D is a dividor of P;
                   i,j=1    ij



                             m
                              i
              n             _____
    d)       Sigma a c (P / |   | p  ) is congruent to 0 (mod P),
             i=1    i i      j=1   ij   



                   n,m
                      i
                  ________
        where P = |      | p  ;
                   i,j=1    ij



                            m
                             i
              n           _____
    e)       Sigma (a c / |   | p  ) is an integer.
             i=1     i i   j=1   ij   





 References:
   [1] Smarandache, Florentin, "Collected Papers", Vol. I, Tempus Publ. Hse.,
       Bucharest, 1996, pp. 13-18.
   [2] Smarandache, Florentin, "Characterization of n Prime Numbers
       Simultaneously", <Libertas Mathematica>, University of Texas
       at Arlington, Vol. XI, 1991, pp. 151-155.
   [3] Smarandache, F., Proposed Problem # 328 ("Prime Pairs ans Wilson's
       Theorem"), <College Mathematics Journal>, USA, March 1988, 
       pp. 191-192.