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 PM**Conjecture 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 -709The 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