*At @GerryMyerson's suggestion, I have changed the title to "Iterated Twin Prime conjecture". 

The sum of a twin prime pair

Egy ikerprím pár összege

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.

Sejtés. Egy 24-nél nagyobb egyenlő ikerprím pár összeg kifejezhető két ikerprím pár összegeként.

Example.

Példa.

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:

Precízebben:

(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)

...

All sum of twin prime pairs less than 19.999.944 are verified by the following R program.

#
# BHAX-stpp-c
# Copyright (C) 2019 Norbert Batfai, batfai.norbert@inf.unideb.hu
#
#  This program is free software: you can redistribute it and/or modify
#  it under the terms of the GNU General Public License as published by
#  the Free Software Foundation, either version 3 of the License, or
#  (at your option) any later version.
#
#  This program is distributed in the hope that it will be useful,
#  but WITHOUT ANY WARRANTY; without even the implied warranty of
#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
#
#  You should have received a copy of the GNU General Public License
#  along with this program.  If not, see <https://www.gnu.org/licenses/>.
#
# Version history
#
# See also https://bhaxor.blog.hu/2018/12/31/100_eves_a_brun_tetel
#

library(matlab)
library(R.utils)

stp <- function(x)
{
    primes = primes(x)
    diff = primes[2:length(primes)]-primes[1:length(primes)-1]
    idx = which(diff==2)
    t1primes = primes[idx]
    t2primes = primes[idx]+2
    t1plust2 = t1primes+t2primes
    return(t1plust2)
}

spp=stp(10000000)

failures = 0
for (i in 3:length(spp)) 
{
    printf("%d-th %d\n", i, spp[i])
    partitions = 0
    for (k in 1:i) 
    {
        for (j in k:i) 
        {
            if(spp[i]==spp[k]+spp[j])
            {
                printf(" %d=%d+%d\n", spp[i], spp[k], spp[j])
                partitions = partitions + 1
            }
        }
    }
    printf("partitions: %d\n", partitions)
    if(partitions == 0)
    {
        failures = failures + 1
        printf("%d, conjecture failed...\n", i)
    }
}
printf("failures: %d\n", failures)