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Archive for July, 2012

So there I was, one minute the owner of a free website from Microsoft called http://www.primorialconjecture.org and the next … nothing. Not one electronic-cyber sausage remains. They pulled the plug on their free web hosting. I suppose were I honest with myself I should have thought at the outset that the word ‘Free’ isn’t one that you would normally associate with Microsoft. Still it was good while it lasted and thanks to them for the last 5 years.

I did try ipage but their sign up page went for a burton so here I am .. Good old WordPress.

You may be wondering why I have called it the primorialconjecture blog. Good question. Well in March 2007 I was playing around with Excel having just re-read Marcus Du Sautoy’s book ‘Music of the primes’. I was thinking to myself .. why so much fuss about something that surely must be obvious.

5 years later I still waste far too much time fiddling with numbers in the hope that one day something will appear from the mists that seem to obscure prime numbers. Any prime nut out there will immediately identify with the feeling you get after your latest attempts at some fiendish prime generating polynomial gives several primes in a row … and then you hit the dreaded semi-prime  or the square. Over time I’ve come to hate the numbers 49,91 and 143. were it not for them I’m sure by now the greatest mathematical achievement (excluding the genius who came up with the recipe for Christmas cake) would be all mine. Ha!

Imagine .. every school child in the world having to learn Potter’s prime number formula … I’d be the most hated man in the universe .. well second only to Pythagoras and maybe the person who dreamed up SOHCAHTOA. (sex on hard concrete always hurts the old arse). If truth were told there isn’t a professional mathematician in the world who hasn’t at sometime lapsed into a pipe dream about proving the Reimann hypothesis and receiving a Fields medal. Too late for me I’m over 40 and don’t understand the first thing about the RH. I remember doing pole zero diagrams in control theory many moons ago and decided at the time that it was witchcraft.

Anyway I digress from explaining just what The primorial conjecture is. A primorial is a portmanteau of the words prime and factorial. Most know that a factorial is when you take each integer (that’s a fancy word for a counting number like 1, 2 3 … etc) and multiply them together. So factorial (3) is simply 1 x 2 x 3 or 6 and factorial (5) is 1x2x3x4x5 or 120. A primorial on the other hand is similar but instead of using the integers you only use the prime numbers.

Primes, for those of you who may need reminding, are numbers divisible only by 1 or themselves. 11 is an example of a prime. Yes I know you can divide 11 by 3 or 4 or 5 but when you do you get messy remainders. It has to divide exactly. So 9 isn’t a prime as that divides exactly by 3. Get the idea? The prime numbers are therefore 2,3,5,7,11,13,17,19,23,29,31 and you can go on adding to the sequence  until the cows come home but the elusive thing is there isn’t anyone we know of who can say what the next prime will be based only on the primes that have gone before. Take it on trust for the moment that it isn’t for the want of trying. many brilliant minds have applied their time and talents to trying to solve this age old conundrum.  So far it remains untamed. Some even resign themselves to the fact they are scattered randomly throughout the integers and that’s an end to it.

I do not believe they are random. Fiendish yes, random no.

So .. where was I .. ah yes the primorial conjecture.

It was March 23 2007. I was playing around with a primorial for 11. which is 2x3x5x7x11 = 2310. I was factorising every integer between 1 and 2310 and looking at what was prime and what was not. This is the job of a prime nut. You’d be horrified at the number of people from all walks of life who think about prime numbers. (I tell myself this so as to try and convince myself I am not a lone nutter).

Factorising is a fancy word for working out what numbers you have to multiply together to get a result. So for example the factors of 15 are 5 and 3

I noticed several things. Firstly that the primes closest to 2310 were separated from 2310 by a distance that was equal to a prime number and that this phenomena  continued until you got quite some distance from the primorial 2310.

Something you should know about the distribution of prime numbers. They seem to thin out the bigger they become. During the last century many mathematicians became intrigued by their distribution throughout the spectrum of integers and managed to work out how many primes you should get up to any number. For example there are 25 primes below the value 100. If they were evenly distributed throughout the integers you’d , not unreasonably, expect there to be about 250 primes below a 1000. Wrong. There are about 166.  They thin out logarithmically. It approximates to x/log(x) but that’s not so important.

So back to the primorial conjecture. By 2310 the primes in that vicinity are starting to thin.

Let’s look at the specific  example I was working on to see if I can explain myself better.

The primes leading up to 2310 are:-

… 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309 …..

Now look at the distance each is from the primorial:-

2309 is 1 away .. 2297 is 13 away … 2293 is 17 away … 2287 is 23 away … 2281 is 29 away … 2273  is 37 away …. 2269 is 41 away …. 2267 is 43 away …. 2251 is 59 away …

so what?

well 1,13,17,23,29,37,41,43, and 59 are all primes themselves …

As I said earlier this effect lasts until you stumble across some number that has a factor of 13 as 2310 is primorial 11 (11#:) and the next prime after 11 is 13.

So my conjecture simply stated there was a contiguous set of primes either side of any primorial that form Goldbach pairs of the primorial.

Goldbach pair? this is simply two numbers that add together to form another number. It’s possible to make every number by adding two or more prime numbers together. So the Goldbach pairs in this case would be (2297,13) and 2293,17) etc ….

After I published this simple Conjecture I received an e-mail from a very irate mathematician saying that this doesn’t help and adds nothing to the corpus of understanding about the primes ….. which may be so but he didn’t get the buzz I got when I spotted it.

When I’ve time I’ll salvage whats left of the old web site and publish it on this blog …
Thanks for bothering to read this.

Bob Potter.

So here’s an extract from the old site that Microsoft used to host before they realised I wasn’t having to pay for it.

THE PRIMORIAL CONJECTURE 
The purpose of this site to to add a tiny fraction to that already understood about one of the most intriguing set of numbers pondered by man.
This page details a conjecture first published on March 23rd 2007 on web site www.partnersinprime.org. The original site is now no longer anything to do with mathematics.

This page was last modified on Thursday, September  2, 2010 21:04 PMConjecture formulated 23 March 2007 by R Potter

 

 


Abstract.
Prime numbers are the building blocks of all other numbers. They are themselves a subset of odd numbers (with the notable exception of 2) and proofs exists which argue they must stretch to infinity. There are many conjectures about the prime numbers, the most famous being the Riemann hypothesis and the Goldbach conjecture. The first, the Riemann hypothesis is probably now the most famous and if it could be proven would open the way to many other conjectures being proven. To even understand the Riemann hypothesis needs a very good grasp of mathematics.
 

The Goldbach conjecture, on the other hand, stating that all even numbers are the sum of two primes is easy to comprehend but has to date proven to be fiendishly difficult to prove. The purpose of this site is to share some observations about the way the primes behave in the vicinity of highly composite even numbers that are created by multiplying the primes together. If all the primes are used in sequence then these numbers are referred to as primorials. (“prime factorials”). The work looks at how the primes combine into Goldbach pairs to add to give the value of the primorial in question.A little history

Goldbach’s unproven conjecture stating that every even number is the sum of two primes (as far as I am aware) doesn’t mention how the primes are distributed when making up a set of Goldbach prime pairs or Goldbach partitions for an even number. This work will show that in the vicinity of primorials there are regions where every prime in sequence will combine with some other prime to make a Goldbach prime pair for the primorial.

The conjecture states that:

1) The conjecture. A contiguous sequence of prime numbers when in the vicinity of a primorial (or primorial multiple) will combine with other (probably non contiguous) primes to make a Goldbach pair for the primorial. The length of sequence for which this effect holds increases as the value of the primorial or primorial multiple increases, in the limit the sequence will tend to an infinite contiguous sequence”.

2) A caveat. In this work Goldbach prime pairs can use both the sum and difference to create the primorial under consideration.3)The definition of the primorial Influence.  From the table below it can be seen that the “primorial influence” of 29p#  for example extends from 6469691981 to 6469694347 so by my definition equals 2396. Within this range there is a contiguous sequence of prime numbers as shown in the table which produce Golbach pairs for the primorial.

4) Twin primes. It is well established that the number of primes less than or equal to n follows a logarithmic law such that as n increases it asymptotically approaches n/log n . It is also conjectured that there are an infinite number of twin primes. It is reasonable to assume that the occurrence of twin primes becomes a less frequent event the bigger the primes become.

 
So it is remarkable that whenever a twin pair occurs within the contiguous sequence of primes  under  the  Primorial Influence then there must be a partner pair somewhere else within the primes to form the Golbach pair of the primorial.
 
For example:
 
 29p# = 6469694057(6n-1 component of twin prime) – 827(also twin prime)
29p# = 6469694059(6n+1 component of twin prime) – 829(also twin prime)
5) The promorial gap:  For every Primorial or multiple there is a gap either side (excluding +/- 1) where no primes will be. In the case of the table below this gap is 102 (+41 –> -61) as neither 29p#+1 or 29p#-1 is prime. The first value that could have been prime after these two values would have been 29p# +/- 31 as any other prime less than 31 (in the case of 29p#) would have simply increased or reduced one of the factors of the primorial itself. This gap will however tend to infinity as the primorial tends to infinity.
——————————————————————————–Example

29p# The group below differs from Fig 1 in that it is a distillation of just the primes that exist either side of primorial 29p# ( 1 x 2 x 3 x 5 x 7 x 11 x 13 x 17 x 19 x 23 x29) = 6469693230

The table  lists the primes and their Goldbach partners that must be added or subtracted to make 29p#.

Every prime in contiguous sequence between the co-primes 6469691959 through to 6469694377 (102 contiguous primes in total ) will make a Goldbach pair for the primorial 29p#.

 6469691959 +(31 x 41)    6469692863 +367               6469694003 -773
6469691981 +1249            6469692877 +353               6469694039 -809
6469692001 +1229            6469692883 +347               6469694041 -811
6469692029 +1201            6469692893 +337               6469694051 -821
6469692037 +1193            6469692917 +313               6469694057 -827
6469692059 +1171            6469692923 +307               6469694059 -829
6469692101 +1129            6469692953 +277               6469694093 -863
6469692107 +1123            6469692973 +257               6469694113 -883
6469692114 +1117            6469692989 +241               6469694137 -907
6469692137 +1093            6469693003 +227               6469694171 -941
6469692181 +1049            6469693037 +193               6469694183 -953
6469692221 +1009            6469693051 +179               6469694213 -983
6469692253 +977              6469693057 +173               6469694251 -1021
6469692263 +967              6469693079 +151               6469694291 -1061
6469692283 +947              6469693081 +149               6469694317 -1087
6469692289 +941              6469693129 +101               6469694333 -1103
6469692353 +877              6469693153 +73                 6469694347 -1117
6469692367 +863              6469693163 +67                 6469694377 – (31 x 37)
6469692371 +859              6469693189 +41
6469692373 +857    29p#(6469693230) 0
6469692401 +829               6469693291 -61
6469692403 +827               6469693319 -89
6469692429 +811               6469693327 -97
6469692431 +809               6469693331 -101
6469692469 +761               6469693333 -103
6469692479 +751               6469693381 -151
6469692497 +733               6469693403 -173
6469692511 +719               6469693457 -227
6469692553 +677               6469693469 -239
6469692571 +659               6469693501 -271
6469692583 +647               6469693511 -281
6469692587 +643               6469693513 -283
6469692599 +631               6469693537 -307
6469692611 +619               6469693543 -313
6469692631 +599               6469693561 -331
6469692673 +557               6469693589 -359
6469692689 +541               6469693661 -431
6469692709 +521               6469693663 -433
6469692721 +509               6469693717 -487
6469692763 +467               6469693753 -523
6469692787 +443               6469693777 -547
6469692799 +431               6469693787 -557
6469692809 +421               6469693907 -677
6469692811 +419               6469693939 -709
The contiguous prime sequence within the PI of 29p#——————————————————————————–I have tested this conjecture against small primorials only as even a primorial of the prime 101 produces a 39 digit number!! Those with the facilities to do so may like to try this with much larger primorials.


So, as I’ve already mentioned it I will use 101p#. It’s a slightly bigger primorial than 29p# above but still extremely modest when compared to today’s gigantic primes.
101p# = 232,862,364,358,497,360,900,063,316,880,507,363,070Unlike 29p# above this list is asymmetric and does not include the primes with a value greater than the value of the primorial.
 

All the values in this table (when evaluated) for the contiguous set of primes within the PI of 101p#.

101#:-131 (closest prime to 101p# by subraction )
101#:-139
101#:-149
101#:-167
101#:-239
101#:-461
101#:-463
101#:-491
101#:-509
101#:-523
101#:-709
101#:-739
101#:-857
101#:-953
101#:-983
101#:-1049
101#:-1297
101#:-1307
101#:-1367
101#:-1427
101#:-1447
101#:-1567
101#:-1597
101#:-1657
101#:-1787
101#:-1801
101#:-1933
101#:-1951
101#:-1999
101#:-2243
101#:-2269
101#:-2281
101#:-2693
101#:-2833
101#:-2843
101#:-2971
101#:-3037
101#:-3067
101#:-3089
101#:-3109
101#:-3181
101#:-3319
101#:-3463
101#:-3559
101#:-3581
101#:-3617
101#:-3623
101#:-3659
101#:-3697
101#:-3709
101#:-3853
101#:-3863
101#:-3889
101#:-4013
101#:-4091
101#:-4241
101#:-4561
101#:-4909
101#:-4933
101#:-5011
101#:-5051
101#:-5107
101#:-5261
101#:-5351
101#:-5717
101#:-5801
101#:-5857
101#:-5981
101#:-6029
101#:-6037
101#:-6329
101#:-6421
101#:-6481
101#:-6779
101#:-6991
101#:-7349
101#:-7393
101#:-7487
101#:-7529
101#:-7673
101#:-7877
101#:-7879
101#:-8123
101#:-8269
101#:-8297
101#:-8353
101#:-8447
101#:-8573
101#:-8737
101#:-8783
101#:-8819
101#:-8963
101#:-9041
101#:-9103
101#:-9227
101#:-9319
101#:-9419
101#:-9689
101#:-9733
101#:-9791
101#:-9871
101#:-10259
101#:-10313
101#:-10343
101#:-10567
101#:-10597
101#:-10607
101#:-10739
101#:-10909
101#:-10937
101#:-10987
101#:-11149
101#:-11261
101#:-11369
101#:-11423
101#:-11587
101#:-11633
101#:-11941
101#:-12091 ( First composite 107 x 113)

bobpotter3999 (skype)

bob.potter@arqiva.com

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