Hey, which of you will give us numeric results for a reasonable size set of numbers, say, twin primes up to 10^6?

This would be a good "project euler" problem....

We have a pair of primes 6n+/-1. So the three numbers are: 12n-2, 12n, and 12n+2. Working mod 12, we see that the following are the valid pairs that can sum up to these numbers.

12n-2: { [B](1,9)[/B], [B](3,7)[/B], (5,5), [B](7,3)[/B], [B](9,1)[/B], (11,11) }
12n : { (1,11), [B](3,9)[/B], (5,7), (7,5), [B](9,3)[/B], (11,1) }
12n+2: { (1,1), [B](3,11)[/B], [B](5,9)[/B], (7,7), [B](9,5)[/B], [B](11,3)[/B] }

However, the bolded pairs are also not valid, because at least one of the pairs is divisible by 3, and hence cannot be prime. So we see that the forms have 2:4:2 valid pairs. The result follows.

We can extend this to higher primes than 3. For example, if we include 5, we note that all twins must be of the form 30n+6k+/-1 where k=0,2,3. Then the three cases are:
k=0: twins are 30n+/-1: the forms are 60n-2, 60n, 60n+2: ratio 6:[B]16[/B]:6
k=2: twins are 30n+12+/-1: the forms are 60n+22, 60n+24, 60n+26: ratio 6:12:6
k=3: twins are 30n+18+/-1: the forms are 60n+34, 60n+36, 60n+38: ratio 6:12:6

[QUOTE=Christenson;280598]Hey, which of you will give us numeric results for a reasonable size set of numbers, say, twin primes up to 10^6?[/QUOTE]

8169. A reasonable estimate on the number up to x is 1.32x/(log x)^2, though for such small numbers it's not all that accurate.* See
[url]https://oeis.org/A114907[/url]
and
[url]https://oeis.org/A007508[/url] .

Note in particular that Tomás Oliveira e Silva calculated the number of twin primes up to 10^18 as
808675888577436
and is extending the calculation to 4 * 10^18.

* For better estimates with small numbers use the integral of 1/log^2 x rather than x/log^2 x.

[QUOTE=CRGreathouse;280636]8169. A reasonable estimate on the number up to x is 1.32x/(log x)^2, though for such small numbers it's not all that accurate.* See
[url]https://oeis.org/A114907[/url]
and
[url]https://oeis.org/A007508[/url] .
[/QUOTE]

I suspect he wanted to compute the statistics as per the OP _for_ the twin primes below 10^6. i.e. compute how many Goldbach representations are there for 2p, 2p+2, 2p+4 for all twin primes (p, p+2) < 10^6.

For your viewing pleasure: :smile:
[CODE]? default(primelimit, 10^6)
? goldbach(e)={my(c=0); forprime(p=3, e/2, c += isprime(e-p)); c}
? p1=5; forprime(p=7, default(primelimit), if(p-p1==2, write("D:\\a.txt", p1 " " goldbach(p1*2) " " goldbach(p1+p) " " goldbach(p*2))); p1=p);
? ##
*** last result computed in 16min, 50,060 ms.
[/CODE]

Yes, it was incredibly naive of me to think that pairs of primes which sum to 2p, 2p+2, 2p+4 would be uniformly distributed. As axn explains, modular arithmetic shows the uneven occurrence of candidate pairs of which one number is divisible by 3 (and 5, and higher primes) with the 2p+2 getting twice as many of these. Thanks.

I guess anyone working on the Goldbach verification project will be well aware of this phenomenon.

The OP did imply the assumption of the TP Conjecture, but there was
another motivation to this "puzzle". In "a.zip", the "Goldbach numbers"
seem to increase "smoothly" over the 2p and 2q, but more raggedly
over the 2p+2. Looking for patterns is a useful pasttime, and the OPer
is interested in the possibility of finding a connection between the
TP Conjecture and the GC, in the sense of mathematical equivalence.
Empirically, for the small numbers (uo to like 10^18 ?) both conjectures
appear to be true, and (admittedly brashly) perhaps by linking the two
via statistical or analytic methods we might help solve both.

[QUOTE=davar55;280790]The OP did imply the assumption of the TP Conjecture, but there was
another motivation to this "puzzle". In "a.zip", the "Goldbach numbers"
seem to increase "smoothly" over the 2p and 2q, but more raggedly
over the 2p+2. Looking for patterns is a useful pasttime, and the OPer
is interested in the possibility of finding a connection between the
TP Conjecture and the GC, in the sense of mathematical equivalence.
Empirically, for the small numbers (uo to like 10^18 ?) both conjectures
appear to be true, and (admittedly brashly) perhaps by linking the two
via statistical or analytic methods we might help solve both.[/QUOTE]

well I could tell you again that GC means every number is equidistant from 2 primes but , it doesn't say if a distance of 1 comes up infinitely often.

[QUOTE=science_man_88;280827]well I could tell you again that GC means every number is equidistant from 2 primes but , it doesn't say if a distance of 1 comes up infinitely often.[/QUOTE]

every integer >3