- 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:>

prime

> 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

> numbers. The most interesting one is related to the Goldbach

number

> conjecture. Given a set S of positive integers a certain even

> greater than 2 is a Goldbach number with respect to S if it is the

three

> 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

> numbers and a distance of two apart, for example, 94,96,98 or

conjecture

> 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

> > and say that any even number > 4 is the sum of two primes, one a

a

> > prime twin and the other a prime twin or cousin.

> >

> > A lot of primes are now becoming unecessary around here. Is that

> > good thing on a list like this? :)

(32,2)

> >

> > 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)

> > (38,2) (56,2) (68,2) (94,2) (136,2) (164,2) (556,2) (1354,2)

one

> >

> > 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

> > > Goldbach partition pair for an even number and of which one

twin,

> > > cousin, OR sexy prime related to each prime of this pair.

the

> > >

> > > 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

> > > > surprising thing is that this appears to be satified if each

of

> > > the

a

> > > > 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

> > > > member of a prime triplet!

two

> > > >

> > > > The New Goldbach conjecture is that every even number greater

> > than

> > > > four can be written as the sum of two odd primes. If these

> > odd

a

> > > > 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

> > > prime

or

> > > > 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

> > > > triplets. (I've checked up to 50,000.)

> > > >

> > > > Mark - 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

> - 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

where a

> "Twin Prime Conjectures" which included the conjecture,

>

> Every even number greater than 4208 is the sum of two t-primes

> t-prime is a prime which has a twin.

the

>

> This was verified up to 4.10^11 and included strong evidence that

> conjecture would hold for larger numbers. The paper also includes

conjecture or GC.

> 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

> PPS: No, I am not working on such a proof.

> - 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:>

wrong,

> 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

> so let me restate this when divided by 24 the remainder is one of

namely

> 0,1,2, or 3.) There is only one of these with == 0 (mod 24),

> 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

state

> (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

> all primes factors with only two Dirichlet equations, and because

P2

> 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

> = 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).

GC

>

> I do see how this can be used as part of the proof of the t-prime-

> if anyone is up to it.

that

>

> 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

> the

includes

> > conjecture would hold for larger numbers. The paper also

> > interestng comparative Goldbach data.

asks.

> >

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

> >

> > 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.

> >