# Iterated Twin Prime conjecture

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.)

• How is "$120=36+84$" a sum of twin-prime pairs ? – Peter Mar 4 at 8:45
• @Peter I believe this is due to $120 = 59 + 61$, $36 = 17 + 19$ and $84 = 41 + 43$. – John Omielan Mar 4 at 9:06
• Now posted to MO, mathoverflow.net/questions/324673/… – Gerry Myerson Mar 5 at 12:06
• At @GerryMyerson's suggestion, mathoverflow.net/questions/324673/… , I have changed the title to "Iterated Twin Prime conjecture". – Norbert Bátfai Mar 6 at 13:39
• Since all sums $\ge 12$ of twin primes are divisible by 12, it's convenient to divide by 12. The resulting sequence is oeis.org/A002822: 1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58, 70, 72, 77, 87, 95, 100, 103,... – YCor Mar 6 at 14:04