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caveat: please credit the author for any information used (Bill R McEachen CCCSD/POD/CSE)
last edit Apr 9 it turns out OEIS A007534 predates Enchev's insight, though it covers sums of 2 TP pairs. Looking at OEIS A014574 seems to indicate one can relate any # of TP sets (pairs, triads, etc)
last edit Apr 8 2010 I saw Enchev's post in a math forum, very interesting. It says with Pi each a Twin prime member,the following holds:
P1 = [(P2 + P3 + 2)/ 4]  1
essentially, it says that if one starts from a TP pair, there are 2 other TP pairs that can be connected as follows:
for (17,19), take 4*18=72. 72=12+60, where we see the (2) pairs (11,13) and (59,61). Note that one of these pairs was larger than the one we started with.
A second example would use centers at 6,12 and 42 where 12*4=48 =6+42, thus relating (5,7),(11,13) and (41,43)
But does it work with any combo? We try 6*4=24, which is 12+12 for the special case of redundant pairs.
Try 4*42=168, which = 18+150, relating (41,43) and (17,19) to (149,151)
This implies that one may generate infinite TP pairs should the conjecture hold. Note the largest TP center I am aware of is Q=2003663613 ยท 2^{195000}
This would mean a TP could exist at 4*Q6 or 4*Q12, for someone with resources to compute in that realm.
Feb 6 2009 (Bernoulli numbers encountering TP's)
Feb 6 well, I looked at OEIS A002445, which is the Bernoulli number denominators when >0. I analyzed the frequency of how many BN encounter unique adjacent primes , and in particular Twin Primes. I submitted the sequence of encountered TPs as a new OEIS sequence today (A156053)...If I recall there were 202 TP pairs seen in the 1st 15000 BNs, with a prime encounter rate of 23%. I will post a chart of the encountered TPs A156053.svg
Goldbach's conjecture Reformulated
well, came across page by Bernard Farley which obviously captured my Mar 4th reformulation of GC (date unknown), link is:
http://www.math.vt.edu/people/linnell/Ugresearch/farley.html
here is my Pari script for testing (odd pairs): http://billymac00.pbwiki.com/f/mc2.gp
I formulated a conjecture (essentially an elaborate reworking of Goldbach's), but more revealing (the back part is in question, swi to A002110 3/17/2008)
Conjecture Mar 4 2008 (see note above attributing basic part to Bernard Farley before me)
Considering 2 addends of identical parity (at least one of which is composite) the sum can be expressed using two primes >=5 in at least one way, for addends >=6 (sums>=12). Specifically, it can be expressed at least X ways, where X is the number of A002110 sequence entries (6 or more) that the addend midpoint, is greater than or equal to. proving there is at least one way would yield proof of Goldbach's conjecture, as it allows direct substitution of (6n +/1) with a prime.
[ I then came across Conjecture 22 by Rudolf Knjzek, which overlaps (and I assume predates)
http://www.primepuzzles.net/conjectures/conj_022.htm ]
Recall OEIS A002110 begins {2,6,,30,210,2310,...}
Examples (notation here always has the greater original addend shown first)
orig /midpoint /A129912 terms applic. /X /prime pairs
15+9 /12 /6,12 /2 /(17,7),(19,5)
9+5 /7 /6 /1 /(7,7)
121+119 /120 /6,12,30,60 /4 /(127,113),(151,89),**etc
39+25 /32 /6,12,30 /3 /(41,23),(17,47),(11,53),(5,59)
35+21 /28 /6,12 /2 /(37,19),(43,13),(19,37)
Note the manipulation applies a single offset to each addend, with opposite signs.
The maximum offset is then <= max(N1,N2)5 to yield the minimum prime substitute. Thus for the case 39+25 above, this maximum offset = 395=34, which gave (5,59).
6/17/2008 came across OEIS A047160 after independently creating the sequence
I have Pari code implementation to generate/check; note TD Noe's comment there is incorrect, ALL sequence terms are relevant ...
6/30/2008 saw a fella's restatement of GC as every integer>=4 can be expressed as the average of 2 primes, never thought of the sum that way ...also came across OEIS A121611 related to every integer being the average of 3 distinct primes ....
= = = = = = =
Twin Primes
on surface inspection, look for TP in the vicinity of (N2N1)/2 where the N are adjacent entries of OEIS A129912
(126)/2=3 and we see (3,5)
similarly, (3012)/2 gives 9 and we see (5,7),(11,13) both (6030)/2 and (210180)/2=15 and we find (11,13) and (17,19)
(360210)/2=75 and one has (71,73) (420360)/2=30 and we have (29,31)
skipping, (23101260)/2 yields (521,523) (46202520)/2 yields (1049,1051) (126006300)/2 yields (3119,3121)
skipping, (54054003063060)/2 yields (117241,117243)
using largest I have, (41621580004073869800)/2=44144100 and we see (44144141,44144143)
It would seem to be holding ...
Twin Primes: I finally plotted the data, here it is thru 32bit range:TP_A129912_thru77.xls
the summary is posted at:
http://primepuzzles.redgolpe.com/phpBB2/memberlist.php. Encouraging, prediction of the TPs is bang on thru the 50th term (TP at 3603797) or so of A129912 via the (N2N1)/2 prediction. Note only predicts them at the sequence "intervals" ie not every single one of course....I will try to work on a larger entry manually as time permits (at 100th primorial or so). Of course, the correlation may well NOT hold up just as easily
An alternate simpler approach which I haven't checked too high is to look for TPs near squares of the HCN (OEIS 002182). Take HCN=2520.
Merely look in the vicinity of this number squared, ie 6350400. Using http://primes.utm.edu/lists/small/100ktwins.txt we see the closest pair at (6350291, 6350293) quite close.
= = = = =
Nov 5 2008
OK, a post I saw led me to fiddle with symmetric primes, primes symmetrical around key numbers, in this case the entries of OEIS A129912 (primorials). I include here simple Pari script to find/report the pairs as well as the Gnumeric spreadsheet plotting the result ...the limited dataset fit is: pair count = 0.036778*X +109, where X is the sequence entry value. For example, there are 3 symmetrical pairs centered around 12, (11/13), (7/17),(5/19).
Usefulness check? Well, since 2310 formed from knowing only the 1st **prime factors, one can "predict" there are 194 pairs of other (higher) primes centered around it (actual is 190)...obviously these pairs will be in the range {0+ thru 2*2310}. The highest of these primes are 4603,4597,4591,4583, etc. This has some use in the higher prime region, for assessing prime gaps and primes above a known prime. If anyone extends the dataset fit please let me know.
here is Pari script: mirrors.gp
CR Greathouse provided a much superior script specific to the primorials, I'll post HERE$$ shortly
here is spreadsheet: mirror_primes.gnumeric
For ease, I also looked purely at the primorials, A002110 and with limited data derive the following improved fit (R^2=0.999+)
ln(Y)= 0.12155X^2+0.76995X1.623 where Y=# prime pairs and X=# of prime factors. Data became OEIS A147517
http://www.research.att.com/~njas/sequences/?q=A147517&language=english&go=Search
For example, primorial 30030 has 6 prime factors and 1564 symmetrical prime pairs (fit predicts 1592).
I resubmitted the sequence 5,7,11,17,19,19,29,37,37,37,73,47,59... On Nov 15th. Its entries represent the distance of the highest prime of the outermost SPP from 2*N where N is the relevant primorial. Thus, for N=30, the outermost SPP is (7,53) and 2*3053 gives 7 as the entry. They will all be primes (part of Goldbach partitions) by previous conjecture. SPP denotes symmetric prime pair. I give specific credit to TD Noe for his assistance with improving the sequence description, and providing new terms beyond what I show. The new seq link is: http://www.research.att.com/~njas/sequences/A147853
I MUST give credit to Charles Greathouse once again for assistance with the symmetric prime work. As well as offering a script improvement, he offers xtension of the terms as follows:
orig:
ln(pi(5#)) ~= 2.3 vs SP value 1.79
ln(pi(7#)) ~= 3.83 vs 3.40
ln(pi(11#)) ~= 5.84 vs 5.25
ln(pi(13#)) ~= 8.09 vs 7.36
ln(pi(17#)) ~= 10.65 vs 9.75
ln(pi(19#)) ~= 13.38 vs 12.33
Greathouse adds:
ln(pi(23#)) ~= 16.32 (12283531) vs 15.15
ln(pi(29#)) ~= 19.52 (300369796) vs ??pending
ln(pi(31#)) ~= 22.81 (8028643010)
ln(pi(37#)) ~= 26.28 (259488750744)
Nov 18th 2008 well, I quickly refuted a conjecture that fell out of the above SPP work. Essentially it was
(Given the 1st 4 primes) The primes encountered in SPP will encompass all primes < 2Q1 where Q=P# *2^k, k=0,1,2,3...
For example, let =3#*2^1 = 12. Via SPP we find (11,13),(7,17),(5,19) and with the 1st 4 we have all primes < (241)
Though generally the primes "fillin" as one moves upward through the primorials and their even multiples, again emphasizing their inherent structural distribution, there are exceptions. I am trying to parameterize those exceptions...
links to my other math pages:
Primality, GIMPS, and other prime stuff main Highly Composite Numbers (conjecture)