## Is Goldbachs conjecture a special case of the twin primes problem?

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• Dear all, As far as I can see there is no literature concerning this question, except for some numerical results which show that even numbers are not always a
Message 1 of 2 , May 31, 2013
Dear all,

As far as I can see there is no literature concerning this question, except for some numerical results which show that even numbers are not always a result of the addition of twin primes. However, the amount of primes found which do not conform to this rule is quite small (33 for N = 2 to 20.000.000.000) which indicate that there may be a connection which lies somewhat deeper than the assumption that even numbers can be written as an addition of twin primes.

The twin primes conjecture is a special case of the Polignac's conjecture, which in it's turn can be seen as a special case of the conjecture that there are infinite amount of sets of 3 primes which are separated by the interval p(2)  P(1) and p(3)  p(2). This can be extended further but is not relevant for the current question.

Goldbach's conjecture in the form of the statement that every prime is an addition of 3 primes can be seen as a subset of the extended conjecture of Polignac.

In the Goldbach conjecture of this form there is an interesting subset of solutions of the form 2p(1) + p(2) which is a different way of writing the Goldbach conjecture that every even number is an addition of two primes.

This means that the Goldbach conjecture is a subset of the Poulignac Conjecture. If the Poulignac Conjecture fails so will the Goldbach conjecture fail too.

Besides comments on the question I would also like to know whether anyone knows literature concerning this subject and more specific: Does anyone know the exact reference of the article of Harvey Dubner?

A Goldbach Conjecture Using Twin Primes, MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 147
JULY 1979, PAGE 1071

TWIN PRIME CONJECTURES
HARVEY DUBNER
• ... The first paper is by Dan Zwillinger http://www.ams.org/journals/mcom/1979-33-147/S0025-5718-1979-0528060-5/S0025-5718-1979-0528060-5.pdf The second is
Message 2 of 2 , Jun 1 3:39 AM
"ch.augustijn" <ch.augustijn@...> wrote:

> Besides comments on the question I would also like to know whether anyone knows literature concerning this subject and more specific: Does anyone know the exact reference of the article of Harvey Dubner?
>
>
> A Goldbach Conjecture Using Twin Primes, MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 147
> JULY 1979, PAGE 1071
>
> TWIN PRIME CONJECTURES
> HARVEY DUBNER

The first paper is by Dan Zwillinger"
http://www.ams.org/journals/mcom/1979-33-147/S0025-5718-1979-0528060-5/S0025-5718-1979-0528060-5.pdf

The second is here:
http://oeis.org/A007534/a007534.pdf
and was published as follows:
Harvey Dubner, "Twin Prime Conjectures",
Journal of Recreational Mathematics,
vol. 30, no 3, 2000, p. 199-205

David
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