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Consecutive primes on a prime leash

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  • Mark Underwood
    Consider: the prime sequence 29, 31, 37, 41, 43, 47, 53. can be regarded as 24 plus: 5, 7, 13, 17, 19, 23, 29. (The prime eleven is missing but what matters is
    Message 1 of 11 , Mar 12, 2006
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      Consider: the prime sequence 29, 31, 37, 41, 43, 47, 53.
      can be regarded as 24 plus: 5, 7, 13, 17, 19, 23, 29.

      (The prime eleven is missing but what matters is that they are all
      prime.)

      So we are looking for basecamp numbers (like 24), after which there
      is a long sequence of primes, each prime a prime distance from the
      base.

      How does the sequence grow? Below is a list showing (basecamp number,
      length of sequence generated) for increasing sequences. Notice how
      jumpy the sequence length can get:

      (8,2)
      (24,7) (as seen above)
      (90,9)
      (120,14)
      (840,19)
      (4410,21)
      (6930,24)
      (10920,26)
      (20370,27)
      (30030,29)
      (115500,31)
      (159390,40)

      Just look at that jump from 31 to 40!

      There are 40 consecutive primes after 159390 which can be written as
      159390 plus :

      (13, 17, 31, 41, 47, 67, 73, 79, 83, 101, 109, 113, 131, 149, 151,
      163, 173, 179, 181, 199, 227, 233, 239, 241, 277, 281, 283, 293, 307,
      311, 317, 331, 347, 349, 373, 379, 383, 389, 397, 401.)

      Unsurprisingly 159390 is factor rich: 159390 = 2 * 3^2 * 5 * 7 * 11
      * 23.

      (Another option would be to account also for primes on the other side
      of the basecamp which might increase the sequence length, sometimes
      dramatically. For instance 120 has 14 primes above and 10 primes
      below it, making a sequence of 24 consecutive primes, each a prime
      away from 120.)

      Longer sequences, anybody?

      Mark
    • Mark Underwood
      Talk about another jump. After (159390,40) the next increases are (2106720,43) and (3573570,53)! So 3573570 has 53 primes immediately after it, such that each
      Message 2 of 11 , Mar 13, 2006
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        Talk about another jump. After (159390,40) the next increases are
        (2106720,43) and (3573570,53)!

        So 3573570 has 53 primes immediately after it, such that each prime
        is a prime from 3573570. The primes are:

        3573570 + (29, 37, 43, 53, 59, 67, 103, 109, 127, 139, 157, 163,
        179, 181, 191, 199, 229, 233, 251, 257, 269, 307, 317, 347, 367, 383,
        401, 409, 443, 467, 479, 491, 509, 521, 541, 557, 571, 587, 599, 617,
        619, 631, 661, 677, 683, 727, 739, 743, 773, 787, 809, 811, 821).

        It so happens that 3573570 also has 26 primes immediately before it
        such that each prime is a prime away from 3573570. The primes are:

        3573570 - (41, 43, 47, 53, 59, 61, 67, 71, 79, 97, 101, 109, 139,
        157, 167, 179, 197, 229, 233, 239, 307, 311, 317, 331, 347, 349).

        That would make 79 consecutive primes, each a prime away from 3573570.


        As would be expected, 3573570 is very composite!
        3573570 = 2 * 3 * 5 * 7^2 * 11 * 13 * 17

        Mark
      • Mark Underwood
        I can t believe I did this, this is so embarassing. Of *course* the sequences of consecutive primes on a leash will grow fast, and arbitrarily long. I thank
        Message 3 of 11 , Mar 13, 2006
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          I can't believe I did this, this is so embarassing. Of *course* the
          sequences of consecutive primes on a leash will grow fast, and
          arbitrarily long. I thank you all for not calling 911 on my behalf.

          in recovery from a brain bubble,
          contritely,
          Mark


          --- In primenumbers@yahoogroups.com, "Mark Underwood"
          <mark.underwood@...> wrote:
          >
          > Talk about another jump. After (159390,40) the next increases are
          > (2106720,43) and (3573570,53)!
          >

          (snip)
        • Phil Carmody
          ... c.f. Lucky numbers perhaps? It may be clear something grows, but that doesn t mean that finding extremal values isn t interesting. c.f. prime gaps. Phil ()
          Message 4 of 11 , Mar 13, 2006
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            --- Mark Underwood <mark.underwood@...> wrote:
            > I can't believe I did this, this is so embarassing. Of *course* the
            > sequences of consecutive primes on a leash will grow fast, and
            > arbitrarily long. I thank you all for not calling 911 on my behalf.
            >
            > in recovery from a brain bubble,
            > contritely,
            > Mark

            c.f. Lucky numbers perhaps?

            It may be clear something grows, but that doesn't mean that finding extremal
            values isn't interesting.

            c.f. prime gaps.

            Phil

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          • Mark Underwood
            Thank you Phil. I ve recovered enough to offer an addendum: The 9 primes after 1260 are 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47). I ve checked up to six
            Message 5 of 11 , Mar 13, 2006
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              Thank you Phil. I've recovered enough to offer an addendum:

              The 9 primes after 1260 are

              1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).

              I've checked up to six million and thus far the 9 string above is the
              longest cluster of, for lack of a better term, doubly consecutive
              primes.

              Mark
            • Phil Carmody
              ... Jens will find you ones up to length 13 trivially, won t you, Jens ;-) Phil () ASCII ribbon campaign () Hopeless ribbon campaign / against
              Message 6 of 11 , Mar 14, 2006
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                --- Mark Underwood <mark.underwood@...> wrote:
                > Thank you Phil. I've recovered enough to offer an addendum:
                >
                > The 9 primes after 1260 are
                >
                > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
                >
                > I've checked up to six million and thus far the 9 string above is the
                > longest cluster of, for lack of a better term, doubly consecutive
                > primes.

                Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

                Phil

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              • Phil Carmody
                ... Jens will find you ones up to length 13 trivially, won t you, Jens ;-) Phil () ASCII ribbon campaign () Hopeless ribbon campaign / against
                Message 7 of 11 , Mar 14, 2006
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                  --- Mark Underwood <mark.underwood@...> wrote:
                  > Thank you Phil. I've recovered enough to offer an addendum:
                  >
                  > The 9 primes after 1260 are
                  >
                  > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
                  >
                  > I've checked up to six million and thus far the 9 string above is the
                  > longest cluster of, for lack of a better term, doubly consecutive
                  > primes.

                  Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

                  Phil

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                • Jens Kruse Andersen
                  ... Here is 18: 1906230835046648293290030 + (13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83) But I didn t find it:
                  Message 8 of 11 , Mar 14, 2006
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                    Phil wrote:
                    > Jens will find you ones up to length 13 trivially, won't you, Jens ;-)

                    Here is 18:
                    1906230835046648293290030 + (13, 17, 19, 23, 29, 31, 37,
                    41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83)

                    But I didn't find it: http://www.ltkz.demon.co.uk/ktuplets.htm#largest18
                    Joerg Waldvogel & Peter Leikauf report it's minimal for 13-83.

                    There are probably smaller 18's for sequences spanning more than 71 numbers.

                    --
                    Jens Kruse Andersen
                  • Jens Kruse Andersen
                    ... I have found 3 cases of 19 doubly consecutive primes . I think they are the first known above 18. n=66916817613475787 and n=304555583497054847 both give
                    Message 9 of 11 , Mar 17, 2006
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                      Mark Underwood wrote:

                      > The 9 primes after 1260 are
                      >
                      > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
                      >
                      > I've checked up to six million and thus far the 9 string
                      > above is the longest cluster of, for lack of a better
                      > term, doubly consecutive primes.

                      I have found 3 cases of 19 "doubly consecutive primes".
                      I think they are the first known above 18.

                      n=66916817613475787 and n=304555583497054847 both give consecutive primes:
                      n + (0,2,6,12,20,62,66,126,132,140,144,146,150,152,180,192,210,242,276)

                      n=149888370336254401 and n=224561886371432461 both give:
                      n + (0,12,18,36,76,112,126,148,186,196,208,222,246,250,252,256,258,298,316)

                      n=56029782174606241 and n=267322720146562081 both give:
                      n + (0,28,58,66,70,96,100,102,106,108,130,148,156,162,240,276,312,336,378)

                      The same 3 in Mark's notation:
                      237638765883579060 + (66916817613475787, ..., 66916817613476063)
                      74673516035178060 + (149888370336254401, ..., 149888370336254717)
                      211292937971955840 + (56029782174606241, ..., 56029782174606619)

                      q=34837285 prp quintuplets were computed on the form:
                      k*17# + (97, 101, 103, 107, 109), for k<734445220000

                      This ensured q*(q-1)/2 ~= 6*10^14 pairs with at least
                      5 consecutive primes in the same pattern.
                      The use of 17# gave the same divisibility properties
                      for primes<=17 to increase odds of more matches.

                      The previous 14 and next 14 prp's were computed for each quintuplet.
                      Comparison gave 3 cases with 14 combined consecutive matches.

                      The GMP library made prp tests. PARI/GP proved the solution primes.
                      The DOS/Windows command "sort" was used on a 1.6 GB file during comparison.
                      (Has "sort" ever been credited in a prime record?)

                      --
                      Jens Kruse Andersen
                    • Mark Underwood
                      Now this is amazing - not only are the primes doubly consecutive in clusters of 19 (!) but they each contain the same minimal quintuple. Brilliant, thank you
                      Message 10 of 11 , Mar 17, 2006
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                        Now this is amazing - not only are the primes doubly consecutive in
                        clusters of 19 (!) but they each contain the same minimal
                        quintuple. Brilliant, thank you Jens!

                        Mark


                        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
                        <jens.k.a@...> wrote:
                        >
                        > Mark Underwood wrote:
                        >
                        > > The 9 primes after 1260 are
                        > >
                        > > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).
                        > >
                        > > I've checked up to six million and thus far the 9 string
                        > > above is the longest cluster of, for lack of a better
                        > > term, doubly consecutive primes.
                        >
                        > I have found 3 cases of 19 "doubly consecutive primes".
                        > I think they are the first known above 18.
                        >
                        > n=66916817613475787 and n=304555583497054847 both give consecutive
                        primes:
                        > n +
                        (0,2,6,12,20,62,66,126,132,140,144,146,150,152,180,192,210,242,276)
                        >
                        > n=149888370336254401 and n=224561886371432461 both give:
                        > n +
                        (0,12,18,36,76,112,126,148,186,196,208,222,246,250,252,256,258,298,316
                        )
                        >
                        > n=56029782174606241 and n=267322720146562081 both give:
                        > n +
                        (0,28,58,66,70,96,100,102,106,108,130,148,156,162,240,276,312,336,378)
                        >
                        > The same 3 in Mark's notation:
                        > 237638765883579060 + (66916817613475787, ..., 66916817613476063)
                        > 74673516035178060 + (149888370336254401, ..., 149888370336254717)
                        > 211292937971955840 + (56029782174606241, ..., 56029782174606619)
                        >
                        > q=34837285 prp quintuplets were computed on the form:
                        > k*17# + (97, 101, 103, 107, 109), for k<734445220000
                        >
                        > This ensured q*(q-1)/2 ~= 6*10^14 pairs with at least
                        > 5 consecutive primes in the same pattern.
                        > The use of 17# gave the same divisibility properties
                        > for primes<=17 to increase odds of more matches.
                        >
                        > The previous 14 and next 14 prp's were computed for each quintuplet.
                        > Comparison gave 3 cases with 14 combined consecutive matches.
                        >
                        > The GMP library made prp tests. PARI/GP proved the solution primes.
                        > The DOS/Windows command "sort" was used on a 1.6 GB file during
                        comparison.
                        > (Has "sort" ever been credited in a prime record?)
                        >
                        > --
                        > Jens Kruse Andersen
                        >
                      • Jens Kruse Andersen
                        ... I computed 47315761 more quintuplets to search for 20 doubly consecutive primes . There wasn t any - but there was 21! n=446863043340173267 and
                        Message 11 of 11 , Mar 21, 2006
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                          Mark Underwood wrote:
                          > The 9 primes after 1260 are
                          > 1260 + (17, 19, 23, 29, 31, 37, 41, 43, 47).

                          I wrote:
                          > I have found 3 cases of 19 "doubly consecutive primes".
                          > 34837285 prp quintuplets were computed

                          I computed 47315761 more quintuplets to search for 20 "doubly
                          consecutive primes". There wasn't any - but there was 21!

                          n=446863043340173267 and n=534402442999154537 both give consecutive primes:
                          n + (0,2,56,62,80,110,146,150,152,170,230,234,252,264,276,
                          290,294,296,300,302,344)

                          The same in Mark's notation:
                          87539399658981270 + (446863043340173267, ..., 446863043340173611)

                          GMP prp'ed and PARI/GP proved. This time "sort" worked on a 3.4 GB file.
                          I had to delete things to get room. The search has stopped.

                          --
                          Jens Kruse Andersen
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