Have Fun with sums !!

]]>There’s also lots of videos on some of the more simple proofs in number theory such as why there are an infinite number of primes. They also have nice videos on the latest proof that there are an infinite number of primes separated by 70,000,000 (although they don’t actually go through the proof) and suggest that they may be able to get that down to 16 which is coming ever closer to proving the twin prime conjecture .. i.e. that there are an infinite number of twin primes ..

All and all it’s fun stuff to watch compiled by a group of college lecturers who are obviously passionate about their subject. So if Christmas TV listings don’t appeal then google ‘numberphile’ and enjoy!!

Happy Christmas. 2013

]]>One that does work is 2x^2 + k.

When k=29 then when x= 1 2x^2 = 2*1. so 2×2+29 = 31

when x=2 you get 2x^2 = 8 so 2x^2+29 = 37.

The first few terms of this sequence = 29,31,37,47,61.

Type them into OEIS and you’ll find the sequence.

Maybe the prime sequence isn’t one sequence but a myriad of different sequences. Perhaps the job of a prime nut is to identify where any particular prime belongs.

]]>Take the example of when

x=1 –> x^2+x = 2

when

x = 2 –> x^2+x = 6.

when

x= 3 –> x^2+x = 12 ….

if k = 41 then x^2+x+k between x= 0 and x= 3 gives 41,43,47,53.

the gap grows by 2 each time.

To get the first 2 prime results (41,43) you have to start at the smaller value of a twin prime.

The twin primes in this case are 41,43.

This is the already mentioned famous Euler polynomial.

So what if you start at k = 43 and work the other way? i.e. k- x^2+x.

You can imagine an axis of symmetry at 42

Now the sequence grows back towards the negative numbers.

43, 41, 37, 31, 23, 13, 1, -13, -29, -47, -67, -89,-113,-139 ………

Okay so by ** convention 1 **isn’t a prime and there are lots of clever mathematicians who can offer robust reasons why it shouldn’t be. Personally I think it is a prime, if for no other reason than when you look at sequences such as this it saves the day more often than not.

when k = 17 you also get the same effect

17,19,23,29,37,47,59,73 …… when x^2+x+17 ….

and

19,17,13,7,-1,-11,-23,-37,-53,-71,-91 … when 19-x^2+x) .. I said in an earlier blog that I hate 91 .. here’s just one reason why

I call this the reflection conjecture … just because it has a ring to it … :))

What’s worthy of note is that when x^2+x+29 is used (twin primes now being 29,31) … the sequence runs aground by the second term .. 29,31,35 …

and when you do the 31-x^2+x you get no further

31,29,25 ..

so all twim primes of this type … the next being 59,61 are not good candidates for k values.

the (71,73) twin pair do better as 73-x^2+x than x^2+x+71 …

73,71,67,61,53,43,31,17,1,-17,-37,-59,-83 …..

What I’m getting at in this blog .. is that when seeking sequences work the prime numbers from right to left as well as left to right. You’ll be pleasantly surprised at how often the sequences line up.

BP 05/08/2012

]]>Here’s a thought. A Highly Composite Number (HCN) sits in the middle of most big gaps in the primes. In the case above 120 (2^3*3*5) sits in between 113 and 127. Some primorial or partial primorial do sometimes allow a prime to snuggle right up close (211 – (a prime) is one off from 210 which is 2*3*5*7) but those primorials that don’t have a prime buddy create big holes in the prime sequence. In the primorial conjecture I call this the primorial influence.

I once had a choice of buying a house at 27 or 29. No contest, it had to be 29. I grew up in No 23. My mother still lives there and she’s coming up 89.

Post script: Today at the gym. While sitting on a recumant bike and watching the ladies Olympic triatherlon road cycling stage. I managed 149 calories in 10 minutes. It nearly killed me. A top tip is not to try and keep up the same cadence as an Olympic cyclist.

My epitaph would have read he died while still in his prime.

]]>Here’s how it works.

You dream up a sequence of integers that are related in some way .. which may only be tenuous … or it may be some thing that springs from a well defined formula .. like y=x^2 .. and you create a list of results and you can put the results in the OEIS … then someone else comes along and if they think they’d like to put the sequence in too they can type the first few digits in to see if someone else has already thought of it … Having just re-read that last paragraph .. it all sounds a tad dull. But if you’re a prime nut then it’s the dog’s whatsits.

Sequences in the OEIS start with the letter A .. for example A000040 is a list of prime numbers. This page dedicated to this sequence links to loads of other sequences and gives all the learned papers on prime numbers as reference … a veritable vade mecum of all things integer.

So the reason for mentioning this is because on Saturday .. I entered a sequence that got ‘accepted’ by the editors of the site. They tweaked about with it but in essence they let it be. It came about by playing with Excel and generating several thousand results for a very famous prime generating polynomial. x^2 + x + 41. This is one of the most prolific prime generators. The first 40 values of X for example are all prime!! Okay those who have little interest in primes may be non-plused but believe me .. that’s a corker of a formula. Anything that can tip-toe through the integers in a well defined way and reach 1681 before it comes across its first composite number is doing very very well.

Here’s what I noticed. When you get to 41^2 where the prime sequence starts to fall apart and your lovely stream of contiguous primes comes to an end .. something very interesting happens to the semiprimes. A semi-prime is just two primes multiplied together. Euler who first published this polynomial was more than aware it couldn’t continue to generate primes past 41. What I did was to embrace this fact and start to look at how the semi-primes where made up. I started this work about 3 months ago now as it takes quite a while factorizing numbers to see if they’re prime or not. If numbers get very large then I would suggest you go to Alpertron’s ECM site as that will take the grunt out of doing it.

I noticed that the factors themselves started sequences that appeared regularly through the results …

Let me give you an example so as to explain myself a little better.

37 | 1447 | 1447 | |

38 | 1523 | 1523 | |

39 | 1601 | 1601 | |

40 | 1681 | 41 | 41 |

41 | 1763 | 41 | 43 |

42 | 1847 | 1847 | |

43 | 1933 | 1933 | |

44 | 2021 | 43 | 47 |

45 | 2111 | 2111 | |

46 | 2203 | 2203 | |

47 | 2297 | 2297 | |

48 | 2393 | 2393 | |

49 | 2491 | 47 | 53 |

50 | 2591 | 2591 | |

51 | 2693 | 2693 | |

52 | 2797 | 2797 | |

53 | 2903 | 2903 | |

54 | 3011 | 3011 | |

55 | 3121 | 3121 | |

56 | 3233 | 53 | 61 |

57 | 3347 | 3347 | |

58 | 3463 | 3463 | |

59 | 3581 | 3581 | |

60 | 3701 | 3701 | |

61 | 3823 | 3823 | |

62 | 3947 | 3947 | |

63 | 4073 | 4073 | |

64 | 4201 | 4201 | |

65 | 4331 | 61 | 71 |

The leftmost column is the value of X in the equation x^2+x+41. So, for example, if you take the value 60 you get (60*60) + 60 + 41 = 3701 .. which just happens to be prime.

But now look at the result when X = 41 .. here the result is 1763 .. which isn’t a prime number but is a number with two factors i.e. 41 and 43. This is because 41 * 43 = 1763.

Look down columns 3 and 4 and you get all the factors (41*41), (41*43),(43*47),(47*53)(53*61)….. ( I can feel a ‘so what’ moment coming on .. )

Well these factors are the start of the sequence for x^2+x+41 when x = 1 . (See below)

1 | 43 | 43 |

2 | 47 | 47 |

3 | 53 | 53 |

4 | 61 | 61 |

5 | 71 | 71 |

6 | 83 | 83 |

7 | 97 | 97 |

8 | 113 | 113 |

9 | 131 | 131 |

10 | 151 | 151 |

11 | 173 | 173 |

12 | 197 | 197 |

13 | 223 | 223 |

14 | 251 | 251 |

15 | 281 | 281 |

16 | 313 | 313 |

17 | 347 | 347 |

18 | 383 | 383 |

19 | 421 | 421 |

20 | 461 | 461 |

21 | 503 | 503 |

22 | 547 | 547 |

23 | 593 | 593 |

24 | 641 | 641 |

25 | 691 | 691 |

26 | 743 | 743 |

27 | 797 | 797 |

28 | 853 | 853 |

The other thing to note is that the semi-primes appear at a very regular intervals .. 2,4,6,8,10……

Once x gets to 2 x 41 (remember the 41 was added on ( x^2+x+41) .. another sequence starts …

81 | 6683 | 41 | 163 |

82 | 6847 | 41 | 167 |

83 | 7013 | 7013 | |

84 | 7181 | 43 | 167 |

85 | 7351 | 7351 | |

86 | 7523 | 7523 | |

87 | 7697 | 43 | 179 |

88 | 7873 | 7873 | |

89 | 8051 | 83 | 97 |

90 | 8231 | 8231 | |

91 | 8413 | 47 | 179 |

92 | 8597 | 8597 | |

93 | 8783 | 8783 | |

94 | 8971 | 8971 | |

95 | 9161 | 9161 | |

96 | 9353 | 47 | 199 |

97 | 9547 | 9547 | |

98 | 9743 | 9743 | |

99 | 9941 | 9941 | |

100 | 10141 | 10141 | |

101 | 10343 | 10343 | |

102 | 10547 | 53 | 199 |

103 | 10753 | 10753 | |

104 | 10961 | 97 | 113 |

105 | 11171 | 11171 | |

106 | 11383 | 11383 | |

107 | 11597 | 11597 | |

108 | 11813 | 11813 | |

109 | 12031 | 53 | 227 |

110 | 12251 | 12251 | |

111 | 12473 | 12473 | |

112 | 12697 | 12697 |

This time the factors are made up of those that appeared in the values of x between 1 and 40 and new values that do not occur in the sequence. These can be seen to be 163,167, 179,199 ,227 … and they appear regularly spaced too.

You can try this for yourself at home .. (as they say) and you will note every time X increases by 41 another sequence springs up. I have worked out a formula for the sequence starts and their gaps.

While in the lake district this year I got up early most mornings to work on a general formula for all these sequences and come up with this ..

(n^2*x^2) +/- ((n^2-2n)*x) + (n^2*k – (n-1)) ………………. (k = to 41)

The table below shows how each sequence follows the above formula:

n | x | k | (n^2*x^2)+((n^2-2n)*x)+(n^2*k-(n-1) |

1 | 0 | 41 | 41 |

1 | 1 | 41 | 43 |

1 | 2 | 41 | 47 |

1 | 3 | 41 | 53 |

2 | 0 | 41 | 163 |

2 | 1 | 41 | 167 |

2 | 2 | 41 | 179 |

2 | 3 | 41 | 199 |

3 | 0 | 41 | 367 |

3 | 1 | 41 | 373 |

3 | 2 | 41 | 397 |

3 | 3 | 41 | 439 |

4 | 0 | 41 | 653 |

4 | 1 | 41 | 661 |

4 | 2 | 41 | 701 |

4 | 3 | 41 | 773 |

5 | 0 | 41 | 1021 |

5 | 1 | 41 | 1031 |

5 | 2 | 41 | 1091 |

5 | 3 | 41 | 1201 |

I decided to publish this formula against one specific set of results in the OEIS .. It was allocated A214732. and is when n=5 so the generic formula becomes …. 25x^2 + 15x+1021. It produces around 16 contiguous primes starting at x=0 –> x=16.

So based on the theory that the OEIS will be around long after I’ve shuffled off this mortal coil … this along with the primorial conjecture .. is my bold claim on posterity … ya what I hear you say … get a life …..

]]>0 1 |

1 –> 2 |

2–> 5 |

3–> 10 |

5–> 26 |

7–> 50 |

11–> 122 |

13–> 170 |

17–> 290 |

19–> 362 |

23–> 530 |

29–> 842 |

31–> 962 |

37–> 1370 |

41–> 1682 |

43–> 1850 |

47–> 2210 |

53–> 2810 |

59–> 3482 |

61–> 3722 |

67–> 4490 |

71–> 5042 |

—————- |

30030 |

Also

11^2 + 13^2 + 17^2 + 19^2 +23^2 + 29^2 = **2310 .. which is 2x3x5x7x11 (or primorial(11))**

Here is another contiguous prime sequence – this is note able as 199 is the first prime south side of 210 7(p#). You might expect this result if the primes were centered about 105 (p(7#)/2) but clearly they are not in this case.

199+211+223+227+229+233+239+241+251+257 = **2310 (11p#)**

also ( and this, I agree, is slightly more tenuous) but if you sum all the primes between 1 and 59 in the manner shown below

(1^2 + 2^2) + (2^2+3^2) + (3^2 + 5^2) + ….(47^2+53^2) + (53^2+59^2) =** 30032(p#13 + 2**)

5 |

11 |

17 |

29 |

41 |

59 |

71 |

101 |

107 |

137 |

149 |

179 |

191 |

197 |

227 |

239 |

269 |

281 |

+

________

** 2310**

This time I’ve added the (6n-1) component of the first 18 twin primes and got to 2310 again

Just observations that may help someone along the way .

BP 19/11/2009

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.

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**

** **

** **

**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)

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.

29p# = 6469694059(6n+1 component of twin prime) – 829(also twin prime)

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#.

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

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

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.

**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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