Gold Twins
 

Just a question relating Twin Primes to Goldbach's Conjecture:

Suppose p and q = p+2 are twin primes.

Then we automatically have three consecutive even integers
2p, 2p+2 = p+q, and 2p+4 = 2q
which can easily be written as the sum of two (maximal) primes,
i.e. which satisfy the Goldbach Conjecture.

My question is just: which one "wins" or "loses"?

IOW as p gets large and the lesser part of a TP pair,
if we count the number of prime pairs that sum to 2p, 2p+2, 2p+4,
is there any tendancy for one of these three to have
more or fewer solutions than the others?

(Please note: I haven't even looked at large primes for this.)

My hand-waving solution is a guess in the sense that it is based on no theoretical knowledge whatsoever. But since everyone can participate in this puzzle section I'll risk ridicule by writing it down.

[SPOILER]I think the answer depends on the truth or otherwise of the conjecture that there are infinitely many twin primes.
If the conjecture is false, then the answer will be dependent on long-in-the-future (currently nowhere near feasable) calculations of all the twin prime pairs (p,q) and all the divisions of 2p, 2p+2, 2p+4 into sums of primes. In short: we don't know yet.
If the conjecture is true then I think that none of the three "win" as p and q tend to infinity. The tiny advantage of the highest number of the trio - in general larger even numbers have more ways of being expressed as the sum of two primes - will "tend to no advantage" as p tends to infinity.
[/SPOILER]

I agree with Brian.

[SPOILER]If there are an infinite number of twin primes, then I don't expect that either of the three possibilities is more significant as p and q tend towards infinity.

If there are a finite number of twin primes...then I don't know. My mathematical knowledge is limited. [/SPOILER]

[QUOTE=Primeinator;280169][SPOILER]If there are a finite number of twin primes...then I don't know. My mathematical knowledge is limited. [/SPOILER][/QUOTE]

[SPOILER]If there are only finitely many twin primes, then "as p gets large" in the problem specification doesn't make sense.[/SPOILER]

[QUOTE=CRGreathouse;280175][SPOILER]If there are only finitely many twin primes, then "as p gets large" in the problem specification doesn't make sense.[/SPOILER][/QUOTE]

[SPOILER]This is true. My wording was not very good. However, I must be honest and say that my mathematical muscles are not strong enough to attempt a good explanation at this problem. [/SPOILER]

[QUOTE=davar55;280135]Just a question relating Twin Primes to Goldbach's Conjecture:

Suppose p and q = p+2 are twin primes.

Then we automatically have three consecutive even integers
2p, 2p+2 = p+q, and 2p+4 = 2q
which can easily be written as the sum of two (maximal) primes,
i.e. which satisfy the Goldbach Conjecture.

My question is just: which one "wins" or "loses"?

IOW as p gets large and the lesser part of a TP pair,
if we count the number of prime pairs that sum to 2p, 2p+2, 2p+4,
is there any tendancy for one of these three to have
more or fewer solutions than the others?

(Please note: I haven't even looked at large primes for this.)[/QUOTE]

I'm guessing you have a code like this:

[CODE]b=[];c=0;forprime(x=1,1000,if(isprime(x+2),b=concat(b,[2*x,2*x+2,2*x+4])));for(a=1,#b,forprime(y=1,100,forprime(z=1,y,if(z+y==b[a],c=c+1)));b[a]=c;c=0)[/CODE] already.

[QUOTE=science_man_88;280238]I'm guessing you have a code like this:

[CODE]b=[];c=0;forprime(x=1,1000,if(isprime(x+2),b=concat(b,[2*x,2*x+2,2*x+4])));for(a=1,#b,forprime(y=1,100,forprime(z=1,y,if(z+y==b[a],c=c+1)));b[a]=c;c=0)[/CODE] already.[/QUOTE]

oh and for an easier read append:

[CODE]b=vector(#b/3,n,[b[(n-1)*3+1],b[(n-1)*3+2],b[(n-1)*3+3]])[/CODE]

[QUOTE=Primeinator;280206][SPOILER]This is true. My wording was not very good. However, I must be honest and say that my mathematical muscles are not strong enough to attempt a good explanation at this problem. [/SPOILER][/QUOTE]

[SPOILER]It's not your wording but that of the OP. I'm just saying that you can drop the part of your post about "if the twin prime conjecture is false" because then there is no question in the first place.[/SPOILER]

[QUOTE=Primeinator;280206][SPOILER]This is true. My wording was not very good. However, I must be honest and say that my mathematical muscles are not strong enough to attempt a good explanation at this problem. [/SPOILER][/QUOTE]

[SPOILER] Assume there is a last twin prime, q-final. It is going to be a very large number, with more than a few tens of digits, so we can use the average properties of the number of pairs of primes that sum to the number with (waves hands) little error, and there will be a (very) slight bias towards the larger prime in the pair having more choices of prime addend pairs.

My mathematical nose (admittedly not terribly discerning) says that this might be related to the points in which the distribution of primes crosses its first approximation, the first point once estimated to be under skewe's number, now known to be under 10^400 or so...having to do with the zeroes of Riemann's Zeta function.

Might be worth a bit of numerical experimentation...Goldbach's has been verified to 10^17 or so....
[/SPOILER]

[QUOTE=davar55;280135]
IOW as p gets large and the lesser part of a TP pair,
if we count the number of prime pairs that sum to 2p, 2p+2, 2p+4,
is there any tendancy for one of these three to have
more or fewer solutions than the others?[/QUOTE]

My prediction : [SPOILER]The central form "2p+2" will have twice as many pairs as the other two forms, asymptotically. EDIT:- IOW, the ratio tends to 1:2:1[/SPOILER]

[QUOTE=axn;280353]My prediction : [SPOILER]The central form "2p+2" will have twice as many pairs as the other two forms, asymptotically. EDIT:- IOW, the ratio tends to 1:2:1[/SPOILER][/QUOTE]
This answer really intrigues me as I know you always post from a knowledgeable starting point. I'd love to know what you base your (to me counter-intuitive) prediction on, even if it's just a hint which might point me in the right direction. Either from you, or from the OP or anyone else who sees more in this puzzle than I do.

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