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PRIME CHAIN OF 3237, 1671 distinct primes

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  • aldrich617
    This prime chain was not produced with the melded equations method that was used in most of the other prime chain postings on this website, but was rather
    Message 1 of 1 , Aug 19, 2008
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      This prime chain was not produced with the melded equations method
      that was used in most of the other "prime chain" postings on this
      website, but was rather found by means of a prime-producing
      generator closely akin to Eric Rowland's formula for an infinite
      chain: a(1) = 7, n>1, a(n) = a(n-1) + gcd(n,a(n-1) .

      It could be argued that neither one is a true prime chain since
      there is a vast sea
      of 1's in-between each prime, and that mine is not really even
      a 'formula' since it
      does not achieve an infinite consecutive list, but I see these
      objections as
      being more semantic than substantial in nature. I also believe that
      the entire set of
      primes occurring in the sequence represented by x^2 - x + 1 will
      eventually appear in the
      formula below,(about 1/2 of all primes) but this is yet to be
      proved, and it is unclear due
      to equipment limitations if the extreme density of primes at the
      beginning of the sequence
      continues indefinitely or breaks down at some point.

      Let f(x) := x^2 - x + 41, x = 1, k = 2, a = 3 .
      Repeat indefinitely a two-step process:
      x := x + 1,
      If GCD( x^2 - x + 41, (x-1)^2 - (x - 1) + 41 - a* k ) >1,
      then k := k + 1;

      The first 20 values of the sequence that do not equal '1' are:
      47, 227, 71, 359, 113, 563, 173, 839, 251,1187,347, 1607,
      461,2099,593, 2663, 743,
      3299, 911,4007

      Many other values for the 'a' variable above, perhaps infinitely
      many, (but not all)
      may be substituted in the formula and produce long initial prime
      chains.
      for a = 3 : 3237 consecutive primes,
      a = 2 : 736
      a = 4 : 817
      a = 5 : 858
      a = 6 : 161
      a = 7 : 1159
      a = 8 : 221
      a = 9 : 284
      a = 10 : 1276 etc.

      Many other equations, perhaps infinitely many, (but not all) may be
      substituted
      for x^2 - x + 1 in the given formula, or a more complicated
      development of that formula,
      with good results. Example: f(x) := 5*x^2 + 5*x + 1, x = 1,
      k = 2, a = 10 : 553 consecutive primes, 276 distinct primes.

      Aldrich Stevens
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