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16383Re: Dat Dastardly 788 : (Now a stronger GC)

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  • Mark Underwood
    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
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