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23042[PrimeNumbers] Re: Pythagorean Sets of Consecutive Primes

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  • Jens Kruse Andersen
    Aug 31, 2011
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      David wrote:
      > Here is the first set of 9:
      >
      > {v=[
      > 206710003, 206710013, 206710019,
      > 206710027, 206710037, 206710043,
      > 206710051, 206710061, 206710067];
      > print(vector(#v-1,k,v[k+1]-v[k]));}
      >
      > [10, 6, 8, 10, 6, 8, 10, 6]

      A gap cycle of 6, 8, 10 in some order can only get one longer than
      this before becoming inadmissible modulo 5.
      If we want more than 10 primes then we need other gaps.

      Puzzle:
      If (a, b, c) and (b, c, d) are Pythagorean triples then is a=d?
      I assume so in the following.
      This means all solutions must have a gap cycle of 3 numbers.
      If 5 divides the sum of the 3 numbers then the cycle never
      becomes inadmissible modulo 5.

      (10, 24, 26) is the smallest even Pythagorean triple with sum
      divisible by 5.
      The first case of 12 consecutive primes for that cycle in some order:
      490537270893409 + d,
      for d = 0, 10, 34, 60, 70, 94, 120, 130, 154, 180, 190, 214.
      The gaps are 10, 24, 26, 10, 24, 26, 10, 24, 26, 10, 24.
      A few hundred cases of 12 non-consecutive primes were found
      during the search.

      I suspect the above is the first case of 12 consecutive primes
      with any gaps.
      A cycle with (10, 24, 26) cannot give more than 12 primes because
      it becomes inadmissible modulo 7.
      In order to get an admissible pattern with 13 primes I think we
      would need gaps of a size which makes it very hard to find
      consecutive primes.

      If the k-tuple conjecture (*) is true then there are arbitrarily
      long solutions.
      For example, (3*n#, 4*n#, 5*n#) is a Pythagorean triple for all n,
      and the gaps never become inadmissible modulo primes up to n.

      (*) There are variations of the k-tuple conjecture.
      We only need one saying that all admissible patterns have at least
      one occurrence.
      Then it follows that they also have occurrences with consecutive primes.

      --
      Jens Kruse Andersen
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