**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*Q-6 or 4*Q-12, 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 pre-dates)

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 = 39-5=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 ...

sps.gp

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

- well, messing with GC a bit but my latest observation ties Twin Primes to the P/PP (Primorials/Primorial Products) as follows:

on surface inspection, look for TP in the vicinity of (N2-N1)/2 where the N are adjacent entries of OEIS A129912

(12-6)/2=3 and we see (3,5)

similarly, (30-12)/2 gives 9 and we see (5,7),(11,13) both (60-30)/2 and (210-180)/2=15 and we find (11,13) and (17,19)

(360-210)/2=75 and one has (71,73) (420-360)/2=30 and we have (29,31)

skipping, (2310-1260)/2 yields (521,523) (4620-2520)/2 yields (1049,1051) (12600-6300)/2 yields (3119,3121)

skipping, (5405400-3063060)/2 yields (117241,117243)

using largest I have, (4162158000-4073869800)/2=44144100 and we see (44144141,44144143)

It would seem to be holding ...

Twin Primes: I finally plotted the data, here it is thru 32-bit 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 (N2-N1)/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+0.76995X-1.623 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*30-53 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 < 2Q-1 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 < (24-1)

Though generally the primes "fill-in" 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)

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