Here is the beginning of the list of sums of twin prime pairs (OEIS A054735): 8, 12, 24, 36, 60, 84, 120, 144, 204, 216, 276, 300, 360, 384, 396, 456, 480, 540, 564, 624, 696, 840, 864, 924,...
"Conjecture. The sum of a twin prime pair greater than or equal to 24 can be expressed as the sum of two twin prime pairs."
Examples:
- 24 = 12 + 12
- 36 = 12 + 24
- 60 = 24 + 36
- 84 = 24 + 60
- 120 = 36 + 84 = 60 + 60
- 144 = 24 + 120 = 60 + 84
- 204 = 60 + 144 = 84 + 120
- ...
to be more precise:
- (11+13) = 24 = 12 + 12 = (5+7) + (5+7)
- (17+19) = 36 = 12 + 24 = (5+7) + (11+13)
- (29+31) = 60 = 24 + 36 = (11+13) + (17+19)
- (41+43) = 84 = 24 + 60 = (11+13) + (29+31)
- (59+61) = 120 = 36 + 84 = (17+19) + (41+43) = 60 + 60 = (29+31) + (29+31)
- ...
Is it always true or are there counterexamples? Is it a known conjecture?
There are no exceptions it works for all sum of twin prime pairs less than 19.999.944.
For example, for (197,199)
15-th 396
- 396=12+384
- 396=36+360
- 396=120+276
Further details can be found in our post: https://bhaxor.blog.hu/2019/03/03/batf41_haxor_stream_conjecture
I would like to know whether is it a known observation? Is it true for all sum of twin prime pairs greater than or equal to 24? I am curious for your opinion.
(When I was trying to check these I found the post: Twin primes sums conjecture that contains a similar conjecture. My question originally was posted as a comment to this.)
It is a already known observation. It seems is not only true for the sum of a twin pair but any multiple of 12 greater than 24 can be written as the sum of two twin pairs, making it the twin prime equivalent of the Goldbach conjecture. I also verified this up to 100 million.
https://www.researchgate.net/publication/283906626_A_Twin_Prime_Analog_of_Goldbach%27s_Conjecture