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3158Primality criteria for pairs (p,2p+n)

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  • torasso.flavio@enel.it
    Oct 8, 2001
    • 0 Attachment
      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