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Re: Dat Dastardly 788 : (Now a stronger GC)

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  • antonioveloz2
    I remember reading about a similar problem a while ago and I came across this paper by Patson MR1812793 (2001m:11010) Patson, Noel(5-CQLI) Interesting property
    Message 1 of 8 , Apr 6, 2005
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      I remember reading about a similar problem a while ago and I came
      across this paper by Patson

      MR1812793 (2001m:11010)
      Patson, Noel(5-CQLI)
      Interesting property observed in the prime numbers.
      Austral. Math. Soc. Gaz. 27 (2000), no. 5, 232--236.
      11A41 (11P32)

      Abstract:
      Using computer power the author investigates the properties of prime
      numbers. The most interesting one is related to the Goldbach
      conjecture. Given a set S of positive integers a certain even number
      greater than 2 is a Goldbach number with respect to S if it is the
      sum of two numbers from S. Let S be the set of twin primes. The
      author finds that, except for thirty-four numbers, all even numbers
      less than 360,994 are Goldbach numbers with respect to S. The first
      exception is 4 and the other 33 numbers are all in clusters of three
      numbers and a distance of two apart, for example, 94,96,98 or
      400,402,404 and so forth.



      Antonio Veloz


      --- In primenumbers@yahoogroups.com, "Mark Underwood"
      <mark.underwood@s...> wrote:
      >
      >
      > Based on your great idea John, we'll tighten Goldbach's conjecture
      > and say that any even number > 4 is the sum of two primes, one a
      > prime twin and the other a prime twin or cousin.
      >
      > A lot of primes are now becoming unecessary around here. Is that a
      > good thing on a list like this? :)
      >
      > Below are the even numbers up to 10,000 that had less than 3
      > solutions. Format: (Even number, number of solutions)
      >
      > (6,1) (8,1) (10,2) (12,1) (14,2) (16,2) (18,2) (20,2) (28,2) (32,2)
      > (38,2) (56,2) (68,2) (94,2) (136,2) (164,2) (556,2) (1354,2)
      >
      > Mark
      >
      >
      >
      > --- In primenumbers@yahoogroups.com, "John W. Nicholson"
      > <reddwarf2956@y...> wrote:
      > >
      > > Mark,
      > >
      > > What if you did this:
      > >
      > > 788 = 61 + 727
      > > {61,727}
      > > twin p,p+2 {59,61}
      > > cousin p,p+4 {none}
      > > sexy p,p+6 {61,67},{727,733}
      > >
      > > See now you can state it as one twin and 2 pair of sexy primes.
      > >
      > > And with this, one can conjecture: There is at least at least one
      > > Goldbach partition pair for an even number and of which one twin,
      > > cousin, OR sexy prime related to each prime of this pair.
      > >
      > > Has anyone conjecture this before?
      > >
      > > John
      > >
      > >
      > >
      > > --- In primenumbers@yahoogroups.com, "Mark Underwood"
      > > <mark.underwood@s...> wrote:
      > > >
      > > > Hi all
      > > >
      > > > Was just revisiting some of my old Goldbachian observations.
      I've
      > > > added a new one at the end, from which derives the title of
      this
      > > > post.
      > > >
      > > > The Old Odd Goldbach conjecture is that every odd number nine
      and
      > > > greater can be written as the sum of three odd primes. But the
      > > > surprising thing is that this appears to be satified if each of
      > > the
      > > > three primes is a member of twin prime pair! Even more
      surprising
      > > is
      > > > that it appears to be satified if each of the three primes is a
      > > > member of a prime triplet!
      > > >
      > > > The New Goldbach conjecture is that every even number greater
      > than
      > > > four can be written as the sum of two odd primes. If these two
      > odd
      > > > primes are taken only from prime twins, if fails 15 times
      before
      > > > 1,000. If each of the primes are pulled only from prime
      triples,
      > > it
      > > > fails 22 times before 500.
      > > >
      > > >
      > > > New Observation: If each of the two odd primes is taken from a
      > > prime
      > > > twin OR a prime triplet, the conjecture holds! EXCEPT for ONE
      > > number -
      > > > 788. 788 ( = 4*197) appears to be the only even number > 4
      > which
      > > > can't be written as the sum of two primes pulled from twins or
      > > > triplets. (I've checked up to 50,000.)
      > > >
      > > > Mark
    • Mark Underwood
      Thank you Antonia. Those clusters of three, that rings a bell. When I first posted here about this type of thing, (a year ago?) I too noted the cluster of
      Message 2 of 8 , Apr 6, 2005
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        Thank you Antonia. Those clusters of three, that rings a bell. When I
        first posted here about this type of thing, (a year ago?) I too noted
        the cluster of three phenomenon.


        I think this conjecture should be top priority. After all, if one can
        prove that every even number > 4 is the sum of two primes, one a twin
        and the other a twin or a cousin, then one has essentially proven the
        twin prime conjecture. :o

        Mark

        PS Speaking of partitioning, some of you guys may find this news
        article from http://www.news.wisc.edu/10833.html an interesting
        read.


        *


        RESEARCH

        Mathematician untangles legendary problem
        (Posted: 3/18/2005)

        Paroma Basu

        Karl Mahlburg, a young mathematician, has solved a crucial chunk of a
        puzzle that has haunted number theorists since the math legend
        Srinivasa Ramanujan scribbled his revolutionary notions into a
        tattered notebook.

        "In a nutshell, this [work] is the final chapter in one of the most
        famous subjects in the story of Ramanujan," says Ken Ono, Mahlburg's
        graduate advisor and an expert on Ramanujan's work. Ono is a Manasse
        Professor of Letters and Science in mathematics.

        "Mahlburg's achievement is a striking one, " agrees George Andrews, a
        mathematics professor at Penn State University who has also worked
        deeply with Ramanujan's ideas.

        The father of modern number theory, Ramanujan died prematurely in
        1920 at the age of 32. The Indian mathematician's work is vast but he
        is particularly famous for noticing curious patterns in the way whole
        numbers can be broken down into sums of smaller numbers,
        or "partitions." The number 4, for example, has five partitions
        because it can be expressed in five ways, including 4, 3+1, 2+2,
        1+1+2, and 1+1+1+1.

        Ramanujan, who had little formal training in mathematics, made
        partition lists for the first 200 integers and observed a peculiar
        regularity. For any number that ends in 4 or 9, he found, the number
        of partitions is always divisible by 5. Similarly, starting at 5, the
        number of partitions for every seventh integer is a multiple of 7,
        and, starting with 6, the partitions for every 11th integer are a
        multiple of 11.

        The finding was an intriguing one, says Richard Askey a emeritus
        mathematics professor who also works with aspects of Ramanujan's
        work. "There was no reason at all that multiplicative behaviors
        should have anything to do with additive structures involved in
        partitions."

        The strange numerical relationships Ramanujan discovered, now called
        the three Ramanujan "congruences," mystified scores of number
        theorists. During the Second World War, one mathematician and
        physicist named Freeman Dyson began to search for more elementary
        ways to prove Ramanujan's congruences. He developed a tool, called
        a "rank," that allowed him to split partitions of whole numbers into
        numerical groups of equal sizes. The idea worked with 5 and 7 but did
        not extend to 11. Dyson postulated that there must be a mathematical
        tool--what he jokingly called a "crank"--that could apply to all
        three congruences.
        Four decades later, Andrews and fellow mathematician Frank Garvan
        discovered the elusive crank function and for the moment, at least,
        the congruence chapter seemed complete.

        But in a chance turn of events in the late nineties, Ono came upon
        one of Ramanujan's original notebooks. Looking through the illegible
        scrawl, he noticed an obscure numerical formula that seemed to have
        no connection to partitions, but was strangely associated with
        unrelated work Ono was doing at the time.

        "I was floored," recalls Ono.

        Following the lead, Ono quickly made the startling discovery that
        partition congruences not only exist for the prime number 5, 7 and
        11, but can be found for all larger primes. To prove this, Ono found
        a connection between partition numbers and special mathematical
        relationships called modular forms.

        But now that Ono had unveiled infinite numbers of partition
        congruences, the obvious question was whether the crank universally
        applied to all of them. In what Ono calls "a fantastically clever
        argument," Mahlburg has shown that it does.

        A UW-Madison doctoral student, Mahlburg says he spent a year
        manipulating "ugly, horribly complicated" numerical formulae, or
        functions, that emerged when he applied the crank tool to various
        prime numbers. "Though I was working with a large collection of
        functions, under the surface I slowly began to see a uniformity
        between them," says Mahlburg.

        Building on Ono's work with modular forms, Mahlburg found that
        instead of dividing numbers into equal groups, such as putting the
        number 115 into five equal groups of 23 (which are not multiples of
        5), the partition congruence idea still holds if numbers are broken
        down differently. In other words, 115 could also break down as 25,
        25, 25, 10 and 30. Since each part is a multiple of 5, it follows
        that the sum of the parts is also a multiple of 5. Mahlburg shows the
        idea extends to every prime number.

        "This is an incredible result," says Askey.

        Mahlburg's work completes the hunt for the crank function, says Penn
        State's Andrews, but is only a "tidy beginning" to the quest for
        simpler proofs of Ramanujan's findings. "Mahlburg has shown the great
        depth of one particular well that Ramanujan drew interesting things
        out of," Andrews adds, "but there are still plenty of wells we don't
        understand."




        --- In primenumbers@yahoogroups.com, "antonioveloz2"
        <antonioveloz2@y...> wrote:
        >
        > I remember reading about a similar problem a while ago and I came
        > across this paper by Patson
        >
        > MR1812793 (2001m:11010)
        > Patson, Noel(5-CQLI)
        > Interesting property observed in the prime numbers.
        > Austral. Math. Soc. Gaz. 27 (2000), no. 5, 232--236.
        > 11A41 (11P32)
        >
        > Abstract:
        > Using computer power the author investigates the properties of
        prime
        > numbers. The most interesting one is related to the Goldbach
        > conjecture. Given a set S of positive integers a certain even
        number
        > greater than 2 is a Goldbach number with respect to S if it is the
        > sum of two numbers from S. Let S be the set of twin primes. The
        > author finds that, except for thirty-four numbers, all even numbers
        > less than 360,994 are Goldbach numbers with respect to S. The first
        > exception is 4 and the other 33 numbers are all in clusters of
        three
        > numbers and a distance of two apart, for example, 94,96,98 or
        > 400,402,404 and so forth.
        >
        >
        >
        > Antonio Veloz
        >
        >
        > --- In primenumbers@yahoogroups.com, "Mark Underwood"
        > <mark.underwood@s...> wrote:
        > >
        > >
        > > Based on your great idea John, we'll tighten Goldbach's
        conjecture
        > > and say that any even number > 4 is the sum of two primes, one a
        > > prime twin and the other a prime twin or cousin.
        > >
        > > A lot of primes are now becoming unecessary around here. Is that
        a
        > > good thing on a list like this? :)
        > >
        > > Below are the even numbers up to 10,000 that had less than 3
        > > solutions. Format: (Even number, number of solutions)
        > >
        > > (6,1) (8,1) (10,2) (12,1) (14,2) (16,2) (18,2) (20,2) (28,2)
        (32,2)
        > > (38,2) (56,2) (68,2) (94,2) (136,2) (164,2) (556,2) (1354,2)
        > >
        > > Mark
        > >
        > >
        > >
        > > --- In primenumbers@yahoogroups.com, "John W. Nicholson"
        > > <reddwarf2956@y...> wrote:
        > > >
        > > > Mark,
        > > >
        > > > What if you did this:
        > > >
        > > > 788 = 61 + 727
        > > > {61,727}
        > > > twin p,p+2 {59,61}
        > > > cousin p,p+4 {none}
        > > > sexy p,p+6 {61,67},{727,733}
        > > >
        > > > See now you can state it as one twin and 2 pair of sexy primes.
        > > >
        > > > And with this, one can conjecture: There is at least at least
        one
        > > > Goldbach partition pair for an even number and of which one
        twin,
        > > > cousin, OR sexy prime related to each prime of this pair.
        > > >
        > > > Has anyone conjecture this before?
        > > >
        > > > John
        > > >
        > > >
        > > >
        > > > --- In primenumbers@yahoogroups.com, "Mark Underwood"
        > > > <mark.underwood@s...> wrote:
        > > > >
        > > > > Hi all
        > > > >
        > > > > Was just revisiting some of my old Goldbachian observations.
        > I've
        > > > > added a new one at the end, from which derives the title of
        > this
        > > > > post.
        > > > >
        > > > > The Old Odd Goldbach conjecture is that every odd number nine
        > and
        > > > > greater can be written as the sum of three odd primes. But
        the
        > > > > surprising thing is that this appears to be satified if each
        of
        > > > the
        > > > > three primes is a member of twin prime pair! Even more
        > surprising
        > > > is
        > > > > that it appears to be satified if each of the three primes is
        a
        > > > > member of a prime triplet!
        > > > >
        > > > > The New Goldbach conjecture is that every even number greater
        > > than
        > > > > four can be written as the sum of two odd primes. If these
        two
        > > odd
        > > > > primes are taken only from prime twins, if fails 15 times
        > before
        > > > > 1,000. If each of the primes are pulled only from prime
        > triples,
        > > > it
        > > > > fails 22 times before 500.
        > > > >
        > > > >
        > > > > New Observation: If each of the two odd primes is taken from
        a
        > > > prime
        > > > > twin OR a prime triplet, the conjecture holds! EXCEPT for ONE
        > > > number -
        > > > > 788. 788 ( = 4*197) appears to be the only even number > 4
        > > which
        > > > > can't be written as the sum of two primes pulled from twins
        or
        > > > > triplets. (I've checked up to 50,000.)
        > > > >
        > > > > Mark
      • Harvey Dubner
        I had a paper published in Journal of Recreational Mathematics, Vol.30(3), 1999-2000, entitled Twin Prime Conjectures which included the conjecture, Every
        Message 3 of 8 , Apr 6, 2005
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          I had a paper published in Journal of Recreational Mathematics, Vol.30(3),
          1999-2000, entitled
          "Twin Prime Conjectures" which included the conjecture,

          Every even number greater than 4208 is the sum of two t-primes where a
          t-prime is a prime which has a twin.

          This was verified up to 4.10^11 and included strong evidence that the
          conjecture would hold for larger numbers. The paper also includes
          interestng comparative Goldbach data.

          I will be happy to send a pdf copy of the paper to anyone who asks.

          Harvey Dubner

          PS: No, the paper does not include a proof of the above conjecture or GC.
          PPS: No, I am not working on such a proof.

          ----- Original Message -----
          From: "antonioveloz2" <antonioveloz2@...>
          To: <primenumbers@yahoogroups.com>
          Sent: Wednesday, April 06, 2005 2:44 PM
          Subject: [PrimeNumbers] Re: Dat Dastardly 788 : (Now a stronger GC)


          >
          >
          > I remember reading about a similar problem a while ago and I came
          > across this paper by Patson
          >
          > MR1812793 (2001m:11010)
          > Patson, Noel(5-CQLI)
          > Interesting property observed in the prime numbers.
          > Austral. Math. Soc. Gaz. 27 (2000), no. 5, 232--236.
          > 11A41 (11P32)
          >
          > Abstract:
          > Using computer power the author investigates the properties of prime
          > numbers. The most interesting one is related to the Goldbach
          > conjecture. Given a set S of positive integers a certain even number
          > greater than 2 is a Goldbach number with respect to S if it is the
          > sum of two numbers from S. Let S be the set of twin primes. The
          > author finds that, except for thirty-four numbers, all even numbers
          > less than 360,994 are Goldbach numbers with respect to S. The first
          > exception is 4 and the other 33 numbers are all in clusters of three
          > numbers and a distance of two apart, for example, 94,96,98 or
          > 400,402,404 and so forth.
          >
          >
          >
          > Antonio Veloz
          >
        • John W. Nicholson
          I have been looking at this paper of Harvey s. I think I have discovered a few things which are very interesting. First let me list the numbers which he found
          Message 4 of 8 , Apr 14, 2005
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            I have been looking at this paper of Harvey's. I think I have
            discovered a few things which are very interesting. First let me
            list the numbers which he found not to be middle numbers < 2*10^10:

            N = 96, 402, 516, 786, 906, 1116, 1146, 1266, 1356, 3246, and 4206.

            Harvey stated to me the correction of 4208 should be 4206.

            Note that all except the second one, 402, is == 1 (mod 5). Also as
            would be expected all are divisible by 6, but they are a complete
            residue set with (mod 4) == (mod 24) (I might be stating this wrong,
            so let me restate this when divided by 24 the remainder is one of
            0,1,2, or 3.) There is only one of these with == 0 (mod 24), namely
            96. There is also a interesting thing with (mod 11) and N/6 == (mod
            11) most have a factor of 2 except the first three. I did look at
            (mod 7). There are no == 1,4 (mod 7). I wonder how this fits with
            http://primepuzzles.net/problems/prob_003.htm ?

            N, (mod 11), N/6 == (mod 11), (mod 7)
            96, 8, 5, 5
            402, 6, 1, 3
            516, 10, 9, 5
            786, 5, 10, 2
            906, 4, 8, 3
            1116, 5, 10, 3
            1146, 2, 4, 5
            1266, 1, 2, 6
            1356, 3, 6, 5
            3246, 1, 2, 5
            4206, 4, 8, 6

            Add to these statement a look a MR1745569,
            http://primes.utm.edu/references/refs.cgi?author=Suzuki
            I have a scaned it if anyone wants a copy.

            Because of the above, 4*n+/-1 and 6*m+/-1 are the only ways to state
            all primes factors with only two Dirichlet equations, and because
            4206 < 6!*6, I don't think there are any more possible. All other
            numbers > 6!*6 have some way of having numbers > 36 = 6^2 in the P2
            = p1*p2 product (prime factors are of 6*n +/- 0, 1, 2, 3, 4, or 5, n
            < 6) + prime as to Chen's theorem (all even = P2 + p3).

            I do see how this can be used as part of the proof of the t-prime-GC
            if anyone is up to it.

            Can anything more be said?

            John


            --- In primenumbers@yahoogroups.com, Harvey Dubner <harvey@d...>
            wrote:
            > I had a paper published in Journal of Recreational Mathematics,
            Vol.30(3),
            > 1999-2000, entitled
            > "Twin Prime Conjectures" which included the conjecture,
            >
            > Every even number greater than 4208 is the sum of two t-primes
            where a
            > t-prime is a prime which has a twin.
            >
            > This was verified up to 4.10^11 and included strong evidence that
            the
            > conjecture would hold for larger numbers. The paper also includes
            > interestng comparative Goldbach data.
            >
            > I will be happy to send a pdf copy of the paper to anyone who asks.
            >
            > Harvey Dubner
            >
            > PS: No, the paper does not include a proof of the above
            conjecture or GC.
            > PPS: No, I am not working on such a proof.
            >
          • John W. Nicholson
            One more thing, the gap are added or subtracted from the number below until there are to primes at the ends. 89 (-7) ((96)) (+1) 97 (+4) 101 (+2) 103 103-89 =
            Message 5 of 8 , Apr 14, 2005
            • 0 Attachment
              One more thing, the gap are added or subtracted from the number
              below until there are to primes at the ends.

              89 (-7) ((96)) (+1) 97 (+4) 101 (+2) 103
              103-89 = 14
              96 +/- 7

              383 (-6) 389 (-8) 397 (-4) 401 (-1) ((402)) (+7) 409 (+10) 419 (+2)
              421
              421 - 383 = 38
              402 +/- 19

              509 (-7) ((516)) (+5) 521 (+2) 523
              523 - 507 = 14
              516 +/- 7

              761 (-8) 769 (-4) 773 (-13) ((786)) (+1) 787 (+10) 797 (+12) 809
              (+2) 811
              811- 761 = 50
              786 +/- 25

              883 (-4) 887 (-19) ((906)) (+1) 907 (+4) 911 (+8) 919 (+10) 929
              929 - 883 = 46
              906 +/- 23

              1109 (-7) ((1116)) (+1) 1117 (+6) 1123
              1123 – 1109 = 14
              1116 +/- 7

              1129 (-17) ((1146)) (+5) 1151 (+2) 1153 (+10) 1163
              1163 – 1129 = 34
              1146 +/- 17

              1249 (-10)1259 (-7) ((1266)) (+11) 1277 (+2) 1279 (+4) 1283
              1283 -1249 = 34
              1266 +/- 17

              1303 (-4)1307 (-12) 1319 (-2) 1321 (-6) 1327 (-29) ((1356)) (+5)
              1361 (+6) 1367 (+6) 1373 (+8) 1381 (+18) 1399 (+10) 1409
              1406 – 1303 = 106
              1356 +/- 53

              3221 (-8) 3229 (-17) ((3246)) (+5) 3251 (+2) 3253 (+4) 3257 (+2)
              3259 (+12) 3271
              3271 – 3221 = 50
              3246 +/- 25

              4201 (-5) ((4206)) (+5) 4211
              4211 – 4201 = 10
              4206 +/- 5


              John


              --- In primenumbers@yahoogroups.com, "John W. Nicholson"
              <reddwarf2956@y...> wrote:
              >
              > I have been looking at this paper of Harvey's. I think I have
              > discovered a few things which are very interesting. First let me
              > list the numbers which he found not to be middle numbers < 2*10^10:
              >
              > N = 96, 402, 516, 786, 906, 1116, 1146, 1266, 1356, 3246, and 4206.
              >
              > Harvey stated to me the correction of 4208 should be 4206.
              >
              > Note that all except the second one, 402, is == 1 (mod 5). Also as
              > would be expected all are divisible by 6, but they are a complete
              > residue set with (mod 4) == (mod 24) (I might be stating this
              wrong,
              > so let me restate this when divided by 24 the remainder is one of
              > 0,1,2, or 3.) There is only one of these with == 0 (mod 24),
              namely
              > 96. There is also a interesting thing with (mod 11) and N/6 ==
              (mod
              > 11) most have a factor of 2 except the first three. I did look at
              > (mod 7). There are no == 1,4 (mod 7). I wonder how this fits with
              > http://primepuzzles.net/problems/prob_003.htm ?
              >
              > N, (mod 11), N/6 == (mod 11), (mod 7)
              > 96, 8, 5, 5
              > 402, 6, 1, 3
              > 516, 10, 9, 5
              > 786, 5, 10, 2
              > 906, 4, 8, 3
              > 1116, 5, 10, 3
              > 1146, 2, 4, 5
              > 1266, 1, 2, 6
              > 1356, 3, 6, 5
              > 3246, 1, 2, 5
              > 4206, 4, 8, 6
              >
              > Add to these statement a look a MR1745569,
              > http://primes.utm.edu/references/refs.cgi?author=Suzuki
              > I have a scaned it if anyone wants a copy.
              >
              > Because of the above, 4*n+/-1 and 6*m+/-1 are the only ways to
              state
              > all primes factors with only two Dirichlet equations, and because
              > 4206 < 6!*6, I don't think there are any more possible. All other
              > numbers > 6!*6 have some way of having numbers > 36 = 6^2 in the
              P2
              > = p1*p2 product (prime factors are of 6*n +/- 0, 1, 2, 3, 4, or 5,
              n
              > < 6) + prime as to Chen's theorem (all even = P2 + p3).
              >
              > I do see how this can be used as part of the proof of the t-prime-
              GC
              > if anyone is up to it.
              >
              > Can anything more be said?
              >
              > John
              >
              >
              > --- In primenumbers@yahoogroups.com, Harvey Dubner <harvey@d...>
              > wrote:
              > > I had a paper published in Journal of Recreational Mathematics,
              > Vol.30(3),
              > > 1999-2000, entitled
              > > "Twin Prime Conjectures" which included the conjecture,
              > >
              > > Every even number greater than 4208 is the sum of two t-primes
              > where a
              > > t-prime is a prime which has a twin.
              > >
              > > This was verified up to 4.10^11 and included strong evidence
              that
              > the
              > > conjecture would hold for larger numbers. The paper also
              includes
              > > interestng comparative Goldbach data.
              > >
              > > I will be happy to send a pdf copy of the paper to anyone who
              asks.
              > >
              > > Harvey Dubner
              > >
              > > PS: No, the paper does not include a proof of the above
              > conjecture or GC.
              > > PPS: No, I am not working on such a proof.
              > >
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