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Building Twin Primes III

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  • Milton Brown
    All twin primes can be constructed by 30*P +(11,13) or +(17,19) or +(29,31) or 30*P + 30f +(11,13) or +(17,19) or +(29,31) where P is a prime and f is a
    Message 1 of 3 , Jun 9, 2005
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      All twin primes can be constructed by

      30*P +(11,13) or +(17,19) or +(29,31) or

      30*P + 30f +(11,13) or +(17,19) or +(29,31)

      where P is a prime and f is a fudge-factor,
      very small 1, 3 or 4.

      This allows all primes to be associated with a
      higher Twin Prime, and since the sequence

      2, 3, 5, ..., P, ... is infinite,

      the number of Twin Primes is infinite.

      The first few Twin Prime can all be constructed with
      30*k + (11,13) or +(17,19) or +(29,31). After that the
      Twin Primes can be constructed with previously built
      smaller Twin Primes. Many can still be constructed with
      just these 3 additions.

      Attached is a table for k from 1 to 20 and some
      higher primes showing their construction.
      This is followed by the construction
      of more Twin Primes from primes and
      previous Twin Primes.

      The assertion is that for every k which is prime,
      there is at least smaller twin prime (already produced)
      which produces a new twin prime.

      (30)k---+(11,13) -------+(17,19) ------+(29,31)
      ================================================
      1 -------(41,43) --------xxxxxxxx------- (59,61)
      2p ------(71,73) --------xxxxxxxx -------xxxxxxx
      3p ------(101,103) ------(107,109) ------xxxxxxx
      4 -------xxxxxxxxxx -----(137,139) ------(149,151)
      5p ------xxxxxxxxxx -----xxxxxxxxxx -----(179,181)

      6 -------(191,193) ------(197,199) ------xxxxxxxxx
      7p ------xxxxxxxxxx -----(227,229) ------(239,241)
      8 -------xxxxxxxxxx -----xxxxxxxxxx -----(269,271)
      9 -------(281,283) ------xxxxxxxxxxx ----xxxxxxxxx
      10 ------(311,313) ------xxxxxxxxxx -----xxxxxxxxx

      11p -----xxxxxxxxxxx ----(347,349) ------xxxxxxxxx
      12 ------xxxxxxxxxx -----xxxxxxxxxx -----xxxxxxxxx
      13p -----xxxxxxxxxx -----xxxxxxxxxx -----(419,421)
      14 ------(431,433) ------xxxxxxxxxx -----xxxxxxxxx
      15 ------(461,463) ------xxxxxxxxxx -----xxxxxxxxx

      16 ------xxxxxxxxxx -----xxxxxxxxxx -----xxxxxxxxx
      17p -----(521,523) ------xxxxxxxxxx -----xxxxxxxxx
      18 ------xxxxxxxxxxx ----xxxxxxxxxx -----(569,571)
      19p -----xxxxxxxxxxx ----xxxxxxxxxx -----(599,601)
      20 ------xxxxxxxxxx -----(617,619) ------xxxxxxxxx

      23 ------690-----(827,829)------ +(137,139)--{27*30+(17,19)}
      +4

      29 ------870-----(881,883)------ +(11,13)

      31-------930-----(1031,1033)---- +(101,103)--{34*30+(11,13)}
      +3

      37 ------1110 ---(1151,1153) ----+(41,43)----{38*30+(11,13)}
      +1

      41-------1230----(1289,1291) ----+(59,61)----{42*30+(29,31)}
      +1

      43 ------1290----(1301,1303)-----+(11,13)

      47-------1410----(1427,1429) ----+(17,19)

      51-------1530----(1667,1669) ----+(137,139)--{55*30+(17,19)}
      +4

      53-------1590----(1607,1609) ----+(107,109)--{56*30+(17,19)}
      +3

      59-------1770----(1787,1789) ----+(17,19)

      61-------1830----(1871,1873) ----+(41,43)----{62*30+(11,13)}
      +1

      67-------2010----(2027,2029) ----+(17,19)

      71-------2130----(2141,2143) ----+(11,13)

      73-------2190----(2339,2341) ----+(149,151)--{77*30+(29,31)}
      +4

      79-------2370----(2381,2383) ----+(11,13)

      83-------2490----(2549,2551) ----+(59,61)----{84*30+(29,31)}
      +1

      89-------2670----(2687,2689) ----+(17,19)

      97-------2910----(2969,2971) ----+(59,61)----{98*30+(29,31)}
      +1

      101------3030----(3167,3169) ----+(137,139)--{105*30+(17,19)}
      +4

      103------3090----(3119,3121) ----+(29,31)

      107------3210----(3251,3253) ----+(41,43)----{108*30+(11,13)}
      +1

      109------3270----(3299,3301) ----+(29,31)

      113------3390----(3389,3391) ----+(-1,1)


      Milton L. Brown

      [Non-text portions of this message have been removed]
    • Jud McCranie
      ... Wrong, because you don t have any limit on f, nor say what f will work for each P. And if you re going to have that fudge factor in there, basically (P+f)
      Message 2 of 3 , Jun 10, 2005
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        At 12:15 PM 6/9/2005, Milton Brown wrote:

        >All twin primes can be constructed by
        >
        >30*P +(11,13) or +(17,19) or +(29,31) or
        >
        >30*P + 30f +(11,13) or +(17,19) or +(29,31)
        >
        >where P is a prime and f is a fudge-factor,
        >very small 1, 3 or 4.
        >
        >This allows all primes to be associated with a
        >higher Twin Prime, and since the sequence
        >
        >2, 3, 5, ..., P, ... is infinite,
        >
        >the number of Twin Primes is infinite.

        Wrong, because you don't have any limit on f, nor say what f will work for
        each P.
        And if you're going to have that fudge factor in there, basically (P+f)
        where f is whatever you need it to be, then there is really no need for
        P. And you're back to square one.
      • Jud McCranie
        ... Can you outline your construction procedure? Also, take p=2^24036583-1 and show how you find an f that will work.
        Message 3 of 3 , Jun 11, 2005
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          At 12:15 PM 6/9/2005, Milton Brown wrote:

          >All twin primes can be constructed by
          >
          >30*P +(11,13) or +(17,19) or +(29,31) or
          >
          >30*P + 30f +(11,13) or +(17,19) or +(29,31)
          >
          >where P is a prime and f is a fudge-factor,
          >very small 1, 3 or 4.

          Can you outline your construction procedure? Also, take p=2^24036583-1 and
          show how you find an f that will work.
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