I found new primality criteria for pairs (p,2p+n) with n odd.

Let A be the square product of the odd numbers from 1 to n

A = (1 * 3 * ... * n-2 * n)^2

Let B(n) be the square product of those prime factors of the

odd numbers up to n-2 which are relatively prime to n:

for example B(13)=(3*5*7*9*11)^2

B(15)=(7*11*13)^2

When p > B(n)

(p,2p+n) is a prime pair iff

An[(p-1)!^2-1] = [2A +/-2^(n+2)]p [mod p(2p+n)]

the sign being "+" when n=4k-1, "-" when n=4k+1

When p <= B(n)

the above formula still holds except for a finite number of cases,

namely when simultaneously:

(1) p is a composite number whose prime factors are <n

(2) p and n are relatively prime

(3) 2p+n is prime

The explicit criteria for the smallest values of n are listed below:

(p,2p+1) is a prime pair iff

(p-1)!^2-1 = -6p [mod p(2p+1)]

without exceptions

(p,2p+3) is a prime pair iff

27[(p-1)!^2-1]= 50p [mod p(2p+3)]

without exceptions

(p,2p+5) is a prime pair iff

1125[(p-1)!^2-1]= 322p [mod p(2p+5)]

except for p=9

(p,2p+7) is a prime pair iff

77175[(p-1)!^2-1]= 22562p [mod p(2p+7)]

except for p=15,45,75,225

(p,2p+9) is a prime pair iff

8037225[(p-1)!^2-1]= 1784002p [mod p(2p+9)]

except for p=25,35,49,175,245,1225

(p,2p+11) is a prime pair iff

1188616275[(p-1)!^2-1]= 216120242p [mod p(2p+11)]

except for p=9,15,21,25,45,49,63,81,135,189,225,315,405,525,675,

735,945,1215,1323,1701,2205,3969,4725,6075,6615,8505,

33075,35721,42525,99225

(p,2p+13) is a prime pair iff

237399086925[(p-1)!^2-1]= 36522903682p [mod p(2p+13)]

except for p=9,15,27,33,35,45,63,75,77,99,105,135,147,225,243,245,

275,297,315,363,405,525,539,605,693,729,735,825,1323,

1575,1815,1925,2205,2835,3267,3969,4455,4725,5445,

6075,6237,7623,8085,11025,11907,12705,19845,22275,

22869,24255,25515,31185,38115,40425,51975,56133,59535,

72765,81675,88209,88935,93555,99225,147015,160083,

200475,280665,297675,441045,444675,1029105,1334025,

1403325,1715175,1964655,3087315,4002075,12006225

(p,2p+15) is a prime pair iff

61632455259375[(p-1)!^2-1]= 8217660832322p [mod p(2p+15)]

except for p=49,91,121,169,539,637,847,1001,1183,1859,7007,11011,

13013,143143,1002001

Thanks for any comments

Kind regards

Flavio Torasso