Is Goldbachs conjecture a special case of the twin primes problem?
- 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
- --- In firstname.lastname@example.org,
"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?The first paper is by Dan Zwillinger"
> A Goldbach Conjecture Using Twin Primes, MATHEMATICS OF COMPUTATION, VOLUME 33, NUMBER 147
> JULY 1979, PAGE 1071
> TWIN PRIME CONJECTURES
> HARVEY DUBNER
The second is here:
and was published as follows:
Harvey Dubner, "Twin Prime Conjectures",
Journal of Recreational Mathematics,
vol. 30, no 3, 2000, p. 199-205