Twin primes sums conjecture

I have found an interesting conjecture between twin primes sums. I don't know if it is already described by someone else. I have checked in internet, but I didn't find any mention of such conjecture.

Could you please tell me if the following conjecture is well known for mathematicians or not?

Let's take twin prime pair 11 and 13 and do the sum:

11+13=24

Now find out sums of two prime numbers which gives the result 24.

5+19=24

7+17=24

You can see that sums 5+19 and 7+17 contain twin primes 5, 7 and 17, 19. Sum of twin primes 11 + 13 can be written by sum of combinations of twin primes 5, 7 and 17, 19.

Another example: Let's take sum of twin prime pair 17, 19:

17+19=36

Its sum can be written with using of other twin primes 5, 7 and 29, 31 as:

5+31=36

7+29=36

We can define twin prime sums conjecture as follow.

Let’s p and q are twin prime numbers; q=p+GAP. For each p+q = a+b = c+d. Where a, b, c, d are prime numbers, where a, c and b, d are prime pairs where c=a+GAP ; d=b-GAP. For twin primes sums conjecture GAP is equal to 2.

Twin prime sums conjecture doesn’t work for prime pairs p, q equals (3, 5), (5, 7), (197, 199), (347, 349).

I have checked the first 100 000 twin primes. The largest checked twin prime pair was 18409199 + 18409201 = 1019 + 36817381 = 1021 + 36817379. It has been found that there are exceptions of p, q for which this conjecture doesn’t work. For sums 3+5; 5+7; 197+199; 347+ 349 it doesn’t exist a, b and c, d which fulfill conjecture.

If twin prime sum conjecture is true then there are unlimited twin prime pairs, because it shows that each twin prime pair sum can be written by sum of combination of smaller and larger twin prime pairs.

For twin primes gap between p,q is equal to 2. I have found that similar conjecture works also for larger even gaps.

Example for GAP equals 6

23+29=52

We can see that GAP between primes 23 and 29 is equal to 6. 29 – 23 = 6. Now we will find sums of other primes which have equal sum to 52.

5+47=52

11+41=52

We have combination of primes 5, 11 and 41, 47. We can see that GAP between those primes is equal to 6, (11 - 5 = 47 - 41 = 6) and is same as the GAP between primes 29 and 23 .

I have also checked this generalized conjecture for primes below 100 000 and even gaps below 100. All of the gaps have few exception pairs p, q for which conjecture doesn’t work, but in major amount of cases conjecture works and for sum of primes with gap we can find another two sums of prime pairs with same gap.

• One thing you might try: take some even numbers (not the sum of twin primes) and see if you can still express them the way you want. There are lots of primes and (for small size) there are lots of twin primes. You might just be seeing some consequences of that. – lulu Nov 24 '15 at 11:06
• Too beautiful to be true but I wish you good luck in your idea, it is not impossible a priori (I think, however, that great mathematicians have failed to demonstrate the infinity of twin primes). Good luck again, your observation is acute. – Piquito Nov 24 '15 at 11:21
• – Lucian Nov 24 '15 at 16:47
• @Lucian Goldbach conjecture says that every even number can be written by sum of two primes. It doesn't say how these primes should looks like. I don't see relation between this conjecture and Goldbach. If this conjecture is true it doesn't proof the Goldbach's and otherwise if Goldbach is true it doesn't proof this one. I see only relation between this conjecture and twin prime conjecture. If this conjecture is true then there are unlimited twin primes. – Ivan Sas Nov 24 '15 at 18:34
• @IvanSas: Goldbach's conjecture stipulates that all numbers $>3$ lie midway between two primes, i.e., $p_{1,2}=n\pm d.$ You set out to investigate the case $d=2.$ Also, for each n, d can have more than one solution. For the cases where $d=2$ is one of the solutions, you have set forth to investigate if such a solution is unique. You have found only a handful of such examples, which is not at all surprising, since the number of solutions for d increases as n increases. – Lucian Nov 24 '15 at 18:44