Prev: Another game called X-it
Next: Are space and time separate issues that can be treated in a parallel way.
From: raycb on 12 Jul 2010 09:08
Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1 are
prime.

Conjecture: 2n can be written as the sum of two primes in a way that
both primes are members of twin prime pairs.

For example:

6 = 3 + 3

12 = 5 + 7

18 = 7 + 11

30 = 11 + 19

42 = 11 + 31

I've gone up to 2n = 1290 without finding a counterexample.
From: hagman on 12 Jul 2010 17:07
On 12 Jul., 15:08, raycb <ra...(a)live.com> wrote:
> Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1 are
> prime.
>
> Conjecture: 2n can be written as the sum of two primes in a way that
> both primes are members of twin prime pairs.
>
> For example:
>
> 6 = 3 + 3
>
> 12 = 5 + 7
>
> 18 = 7 + 11
>
> 30 = 11 + 19
>
> 42 = 11 + 31
>
> I've gone up to 2n = 1290 without finding a counterexample.

Since twin primes are of the fomr 6 k +- 1 (except for the pair
involving 3),
any such decomposition (if 2n>6, i.e. when 3 cannotbe involved) must
be of the form
2n = 6 m = (6 a + 1) + (6 b - 1)
and has a companion decomposition
2n = 6 m = (6 a - 1) + (6 b + 1).
It is more convenient to look for m = a + b with m, a, b in the set
of
numbers k such that 6 k +- 1 are twin primes
If it comforts you: There are no counterexamples up to 2n = 14999460

hagman
From: Tim Little on 13 Jul 2010 02:46
On 2010-07-12, raycb <raycb(a)live.com> wrote:
> Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1 are
> prime.
>
> Conjecture: 2n can be written as the sum of two primes in a way that
> both primes are members of twin prime pairs.

This appears likely to be true, given empirical patterns of density of
twin primes. Even if Goldbach's conjecture is false and also that
there are not infinitely many twin primes, it could still be true.


> I've gone up to 2n = 1290 without finding a counterexample.

I've covered n <= 10^8, and the average number of sums of twinned
primes appears to increase with larger n.


- Tim
From: Dan Cass on 13 Jul 2010 08:14
> On 2010-07-12, raycb <raycb(a)live.com> wrote:
> > Let 2n be an even number larger than 4 such that 2n
> - 1 and 2n + 1 are
> > prime.
> >
> > Conjecture: 2n can be written as the sum of two
> primes in a way that
> > both primes are members of twin prime pairs.
>
> This appears likely to be true, given empirical
> patterns of density of
> twin primes. Even if Goldbach's conjecture is false
> and also that
> there are not infinitely many twin primes, it could
> still be true.
>

This last statement seems problematic. First if Goldbach's conjecture is false,
then there are n so that 2n is not the sum of two primes,
much less the sum of two primes each in a twin prime pair.
On the other hand, if Goldbach's conjecture is true,
and each 2n is a sum of primes 2n = p + q,
then as n--> oo the larger of the primes
p,q --> oo. So if there weren't
infinitely many twin primes, then the OP's statement
....Conjecture: 2n can be written as the sum of two
....primes in a way that
....both primes are members of twin prime pairs.
would be false for large enough n.

> > I've gone up to 2n = 1290 without finding a
> counterexample.
>
> I've covered n <= 10^8, and the average number of
> sums of twinned
> primes appears to increase with larger n.
>
>
> - Tim
From: Tim Little on 13 Jul 2010 19:53
>> On 2010-07-12, raycb <raycb(a)live.com> wrote:
>>> Let 2n be an even number larger than 4 such that 2n - 1 and 2n + 1
>>> are prime. [etc]

[I wrote:]
>> Even if Goldbach's conjecture is false and also that there are not
>> infinitely many twin primes, it could still be true.

On 2010-07-13, Dan Cass <dcass(a)sjfc.edu> wrote:
> This last statement seems problematic. First if Goldbach's
> conjecture is false, then there are n so that 2n is not the sum of
> two primes, much less the sum of two primes each in a twin prime
> pair.

The conjecture applies only to even numbers that are midpoints of a
twin prime pair. There might be no counterexamples to GC of that
form.


- Tim
 |  Next  |  Last
Pages: 1 2
Prev: Another game called X-it
Next: Are space and time separate issues that can be treated in a parallel way.