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.