• ## A goldbach conjecture for twin primes

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• There has been so much discussion about the Goldbach conjecture and also about twin primes that I can t resist adding a conjecture which sort of combines them.
Message 1 of 8 , Nov 29, 2001
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There has been so much discussion about the Goldbach conjecture and also
about twin primes that I can't resist adding a conjecture which sort of
combines them.

Define a t-prime to be a prime which has a twin.

Conjecture: Every sufficiently large even number is the sum of two
t-primes.

More exact: This is true for all even numbers greater than 4208. There are
only a few even numbers less than or equal to 4208 for which this is not
true.

Harvey Dubner
• To: primenumbers From: Harvey Dubner Date sent: Thu, 29 Nov 2001
Message 2 of 8 , Nov 29, 2001
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From: "Harvey Dubner" <hdubner1@...>
Date sent: Thu, 29 Nov 2001 23:15:40 -0500
Send reply to: "Harvey Dubner" <hdubner1@...>
Subject: [PrimeNumbers] A goldbach conjecture for twin primes

> There has been so much discussion about the Goldbach conjecture and also
> about twin primes that I can't resist adding a conjecture which sort of
> combines them.
>
> Define a t-prime to be a prime which has a twin.
>
> Conjecture: Every sufficiently large even number is the sum of two
> t-primes.

1)
Couldn't this conjecture be extended to any constellation?

2)
It seems your conjecture implies both the twin prime conjecture
and the goldbach conjecture.

Michael Hartley : Michael.Hartley@...
Sepang Institute of Technology
+---Q-u-o-t-a-b-l-e---Q-u-o-t-e----------------------------------
"Research without development is impossible - because research inspires development.
Development without research is impossible - because research inspires development."
• Harvey Dubner conjectured that ... which seems to have a good heuristic. As Michael Hartley observed, a comparable heuristic should work, in practice, for any
Message 3 of 8 , Nov 30, 2001
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Harvey Dubner conjectured that

> Every sufficiently large even number is the sum of two t-primes.

which seems to have a good heuristic.

As Michael Hartley observed, a comparable heuristic should
work, in practice, for any constellation with presumed
O(1/log(N)^k) density, be it k=1 for the primes in Goldbach,
k=2 for the twins in Harvey's conjecture, or what you will.

For reference, these (I think!)
are the k=2 exceptions that Harvey had in mind:

[ 2, 4,
94, 96, 98,
400, 402, 404,
514, 516, 518,
784, 786, 788,
904, 906, 908,
1114, 1116, 1118,
1144, 1146, 1148,
1264, 1266, 1268,
1354, 1356, 1358,
3244, 3246, 3248,
4204, 4206, 4208]

I like it that Harvey and Michael have combined
two of the great unprovens into one unproven.

David
• ... So did nobody read my mail from the day before either of the above? Phil Don t be fooled, CRC Press are _not_ the good guys. They ve taken Wolfram s money
Message 4 of 8 , Nov 30, 2001
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On Fri, 30 November 2001, d.broadhurst@... wrote:
> I like it that Harvey and Michael have combined
> two of the great unprovens into one unproven.

So did nobody read my mail from the day before either of the above?

Phil

Don't be fooled, CRC Press are _not_ the good guys.
They've taken Wolfram's money - _don't_ give them yours.
http://mathworld.wolfram.com/erics_commentary.html

Find the best deals on the web at AltaVista Shopping!
http://www.shopping.altavista.com
• ... This conjecture goes back a way. The sequence is Sloane s A007534. It is listed as finite , which would imply the conjecture. Normally he doesn t put
Message 5 of 8 , Nov 30, 2001
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At 02:22 PM 11/30/2001 +0000, d.broadhurst@... wrote:
>Harvey Dubner conjectured that
>
> > Every sufficiently large even number is the sum of two t-primes.

>[ 2, 4, ...
>4204, 4206, 4208]

This conjecture goes back a way. The sequence is Sloane's A007534. It is
listed as "finite", which would imply the conjecture. Normally he doesn't
put "finite" in there unless it is known to be finite, but that doesn't
seem to be the case here. Also see the end of:

http://mathworld.wolfram.com/TwinPrimes.html

%I A007534 M1306
%S A007534 2,4,94,96,98,400,402,404,514,516,518,784,786,788,904,906,908,1114,
%T A007534
1116,1118,1144,1146,1148,1264,1266,1268,1354,1356,1358,3244,3246,3248,
%U A007534 4204,4206,4208
%N A007534 Even numbers not the sum of a pair of twin primes.
%D A007534 D. Wells, The Penguin Dictionary of Curious and Interesting
Numbers. Penguin Books, NY, 1986, 132.
%H A007534 <a
href="http://www.research.att.com/~njas/sequences/Sindx_G.html#Goldbach">Index
entries for sequences related to Goldbach conjecture</a>
%H A007534 E. W. Weisstein, <a
The World of Mathematics.</a> (Currently unavailable)
%e A007534 The twin primes < 100 are 3, 5, 7, 11, 13, 17, 19, 29, 31, 41,
43, 59, 61, 71, 73. 94 is in the sequence because no combination of any two
numbers from the set just enumerated can be summed to make 94.
%t A007534 p = Select[ Range[ 4250 ], PrimeQ[ # ] && PrimeQ[ # + 2 ] & ]; q
= Union[ Join[ p, p + 2 ] ]; Complement[ Table[ n, {n, 2, 4250, 2} ],
Union[ Flatten[ Table[ q[ [ i ] ] + q[ [ j ] ], {i, 1, 223}, {j, 1, 223} ]
] ] ]
%Y A007534 Cf. A051345.
%K A007534 nonn,nice,fini
%O A007534 1,1
%A A007534 njas, Robert G. Wilson v (rgwv@...)
%E A007534 Conjectured to be complete.

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• Jud, Thanks for the information and references about the sum of twin primes. I was not aware of the previous work. The conjecture that every even number
Message 6 of 8 , Nov 30, 2001
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Jud,

Thanks for the information and references about the sum of twin primes. I
was not aware of the previous work. The conjecture that every even number
greater than 4208 is the sum of two t-primes is actually a consequence of
the main thrust of my paper.

Define a "middle number" to be the even number sandwiched between a pair of
twin primes. The main thrust of my paper is the conjecture that every
middle number greater than 6 is the sum of two middle numbers.

Harvey
• Hello all Harvey s conjecture ... than 4208. could be stated thusly: Every even number over 4208 can be written at least once as the sum of two primes, each
Message 7 of 8 , Nov 30, 2001
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Hello all

Harvey's conjecture

> Define a t-prime to be a prime which has a twin.
> Conjecture: Every sufficiently large even number is the sum of two
> t-primes. More exact: This is true for all even numbers greater
than 4208.

could be stated thusly:

Every even number over 4208 can be written at least once as the sum
of two primes, each prime of which is exactly two units from another
prime.

I can't resist another conjecture, and it is this: Every even number
over 8 can be written at least once as the sum of two primes, each
prime of which is exactly six units from another prime.

I have used my proprietary, massively parallel computer system (which
very much resembles a human's neural network) to confirm or refute
this. Up to 2n = 80 the conjecture holds.

An almost identical but stronger conjecture: Every even number over 8
can be written at least once as the sum of two primes, each of the
two primes being exactly six units more than another prime, or each
of the two primes being exactly six units less than another prime.

For instance, 68 = 7 + 61 = (13-6) + (67-6).

40 = 11+ 29 = (5+6) + (23+6).

The conjecture is very likely true since from from 5 to 113 the
number 71 is the only prime that is not 6 units away from another
prime. Also, primes separated by 6 (not necessarily consecutive
primes)should be on average twice as common as primes separated by
two.

Mark

--- In primenumbers@y..., "Harvey Dubner" <hdubner1@c...> wrote:
> There has been so much discussion about the Goldbach conjecture and
also
> about twin primes that I can't resist adding a conjecture which
sort of
> combines them.
>
> Define a t-prime to be a prime which has a twin.
>
> Conjecture: Every sufficiently large even number is the sum of two
> t-primes.
>
> More exact: This is true for all even numbers greater than 4208.
There are
> only a few even numbers less than or equal to 4208 for which this
is not
> true.
>
> Harvey Dubner
• ... also ... sort of ... There are ... is not ... I think primes satisfy GC not because of their primality but because of their random enough
Message 8 of 8 , Dec 2, 2001
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--- In primenumbers@y..., "Harvey Dubner" <hdubner1@c...> wrote:
> There has been so much discussion about the Goldbach conjecture and
also
> about twin primes that I can't resist adding a conjecture which
sort of
> combines them.
>
> Define a t-prime to be a prime which has a twin.
>
> Conjecture: Every sufficiently large even number is the sum of two
> t-primes.
>
> More exact: This is true for all even numbers greater than 4208.
There are
> only a few even numbers less than or equal to 4208 for which this
is not
> true.
>
> Harvey Dubner

I think "primes" satisfy GC not because of their "primality" but
because of their "random enough" distribution. In fact, with ever
larger even numbers, ever less percent of primes is needed to satisfy
GC. With my computer power, less than 1% of the primes would suffice.

Defining g(i) as the product of (1-2/p) for all prime p<=sqrt(n), the
number of Golbach pairs for the even n is almost (n/4)*g(i).
Similarly, the number of t-primes between p^2 and (p+2)^2 is almost
2p*g(i), which means t-prime distribution is far denser than what is
needed to satisfy GC. In fact, every set of random odd numbers with a
spacing of the order n^1/3 is enough to make all evens with their
paired sums, while t-prime spacing is of the order log(n)^2.

For the same reason, the GC-like conjecture for the triplet-primes is
also eventually true.

Kaveh
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