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", , University of Texas at Arlington, Vol. XI, 1991, pp. 151-155. [3] Smarandache, F., Proposed Problem # 328 ("Prime Pairs ans Wilson's Theorem"), , USA, March 1988, pp. 191-192.