THE PRIMORIAL CONJECTURE
The purpose of this site to to add a tiny fraction to that already understood about one of the most intriging 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 Saturday, January 12, 2008 10:02:25 AM
Conjecture formulated 23 March 2007 by R Potter
I am not a mathematician but let's not forget one of the greatest once said
"In mathematics there is no ignorahimus"
- David Hilbert (1862 1943)
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 refered 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
Golbach'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 Golbach prime pair or Golbach partition 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 logrithmic 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 it is reasonable to assume that the occurrence of twin primes becomes a less frquent 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 -827
29p# = 6469694059 -829
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.
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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 (a total of 102) 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#
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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# = 232,862,364,358,497,360,900,063,316,880,507,363,070
Unlike 29p# above this list is assymmetric 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 (try this e-mail address is if the contact page is playing up!!)