Prime Number Alignment – Helices or Spirals

Prime numbers can be written in the form p(i)+np(i-1)#, often in multiple forms, thus taking a random selection of prime numbers (6947 onwards):

3 | 5 | 7 | 11 | 13 | |

Prime Number | 2# | 3# | 5# | 7# | 11# |

6947 | 3+3472×2 | 5+1157×6 | 17+231×30 | 17+33×210 | 17+3×2310 |

6949 | 3+3473×2 | 7+1157×6 | 19+231×30 | 19+33×210 | 19+3×2310 |

6953 | 3+3475×2 | 5+1158×6 | 23+231×30 | 23+33×210 | 23+3×2310 |

6959 | 3+3478×2 | 5+1159×6 | 29+231×30 | 29+33×210 | 29+3×2310 |

6961 | 3+3479×2 | 7+1159×6 | 31+231×30 | 31+33×210 | 31+3×2310 |

6967 | 3+3482×2 | 7+1160×6 | 37+231×30 | 37+33×210 | 37+3×2310 |

6971 | 3+3484×2 | 5+1161×6 | 11+232×30 | 41+33×210 | 41+3×2310 |

6973 | 3+3485×2 | 7+1161×6 | 13+232×30 | 43+33×210 | 43+3×2310 |

Above 11 each prime numbers can thus be expressed in terms of two smaller prime numbers; the non appearance of smaller primes in this pattern is not a “problem” in that the overriding patterns explains their presence, but in a simpler manner. Larger prime numbers appear on more than one set of patterns. The larger the prime number the greater the number of helices it will lie on. This structure appears at first sight to be as remarkable and surprising as the double helix of DNA – perhaps visualistion will assist in understanding this. For each helix there is a line of symmetry. However due to the stepped nature of prime number distribution, each successive helix starts at the next highest prime number – thus smaller prime numbers are not included in subsequent helices

Hello, an internet search on patterns and symmetry in prime numbers brought me to your site. My son and I have been playing around with this subject a little over Christmas holiday, and I’m interested to explore the current state of the art on the subject, i.e. where would be a good authoritative source of information on patterns and symmetry in the sequence of prime numbers.

All the best for a happy new year!

Pete Quinn

Victoria, BC, Canada

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Terence Tao, UCLA: Clay/Mahler lecture series

I managed to download a 3.7MB powerpoint titled “structure-and-randomness-in-the-prime-numbers” that may assist. If you look for Terence Tao at UCLA you should be able to find more.

Alternatively I have been told that Dirichlet’s 1837 paper “There are infinitely many prime numbers in all arithmetic progressions with first term and difference

coprime” (By Mr. Lejeune-Dirichlet) covers some if not all of what is shown on this website…..it’s just buried in obscure mathematical language.

As far as I can tell what is shown on this website is not available elsewhere, but am happy to be proved wrong!

Jon

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Thanks for the reply, Jon. Sorry, I don’t actually know who you are – I may have missed this info on the site?

I’ve been searching quite a bit, and have found very few references working in the direction I sense you are, which is quite similar to what we’ve come up with (presumably well after you). I did find one interesting post in this discussion thread:

http://www.mymathforum.com/viewtopic.php?p=8996

Hinting that at least one poster there has the same angle, at least in a general way.

None of the peer reviewed work I’ve found seems to cover the same ground, although I would expect most of the good work in this field remains unpublished for obvious reasons. Mind you, I’m not a mathematician, so a lot of the formal math language is unintelligible to me, and the same result may be hiding in plain sight. 🙂

Feel free to drop me an email if you’d like to chat further.

Cheers,

Pete

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Hi Pete, I’m Jon based in Kent near London, England.

There are a number of papers both published and on the internet (mostly the latter) that note that the composite numbers removed by each Prime Factor p in the Eratosthenes sieve have a repeatable pattern/intervals with periodicity p#.

I don’t think I have seen one that has taken this observation any further; however if you divide these composite numbers by each p, you end up with an infinite series of sets, each of which contain ALL the prime numbers >p. Clearly however they also contain non primes or pseduo primes, which are predictable (p x p etc).

Based on the repeatability of the Eratosthenes composities, the observation that there were repeatable patterns in the primes was then obvious……although not so obvious for the numerous texts on primes to make this basic observation. The observation that there were lines of symmetry in these patterns was mine as far as I know, although I expect Tao has this wrapped up in obscure mathematical language.

I would like to visualise the prime distribution and I can see a series of helices built upon each other, but I don’t know whether this would be useful!?

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Thanks Jon, I’ll try to elaborate a little more when I get a chance.

I was reading one of his Christmas presents over the holidays (The Math Book by Clifford Pickover – a really fun read!), and one of the short articles (about Ulam’s spiral) got me interested in looking for patterns in prime numbers. I wanted to try to generate Ulam’s spiral for myself but didn’t want to spend the time to program it, and took an easier way that led to probably a more interesting path, mostly by accident I think.

When I started to see patterns and symmetry, I wanted to know if they had been described before. I assumed they must have (there are almost never “new” ideas) but nevertheless began my google search.

I still don’t think I’ve found anything that includes all of what we’ve come up with (at least not in an easily understood way – there are probably peer reviewed math papers that describe what we’re seeing quite neatly, but my formal math training is buried deeply in the past). You seem to be working largely along a similar path, but with some key differences.

Your idea of helixes is one we have not explored. But we have used a lot of visualization and brute force mathematical experimentation, as opposed to formal mathematical analysis.

I’ll try over the next short while to compile what we’ve come up with as succinctly and clearly as possible to put out there for people to look at.

Can I ask why you are working on this, out of curiosity? For me it’s just an interesting holiday diversion that I dearly need to let go of. 🙂 For my son it’s of some relevance to his undergraduate studies, but more a hobby than a vocation.

Cheers,

Pete

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Thanks Pete. I have found describing the patterns in English quite difficult – that’s why mathematical terminology works, if you can understand it!

Why am I looking? – well it’s a long story starting with a quiz book I had as a child that listed the prime numbers up to 100 – a “random” pattern that defied explanation. It’s only a pastime, but I was spurred on when I came across a book Born on a Blue Day and perceived there’s a bit of Asbergers in me…..maybe!

Now I am 54, and I started thinking about the problem again a few years ago. I thought that the prime distribution might be explained by the interference of waves – each wave caused by separate integers. A little playing showed I need only consider the waves caused by prime numbers themselves and with a little playing on Excel I found Erathosthenes sieve – yes I started playing before I had read anything!

It then became clear that the combination of these “waves” caused the prime distribution. I believe mathematicians may have got caught in a furrow of thinking that primes are more fundamental than composites that the primes should be able to explain themselves. However the primes are the left over numbers!

Another thing I didn’t yet explain relates to the series of composites that arise from Erathosthenes sieve. I believe these are truely special as for any prime p they contain multiples of all primes higher than p, but with a rhythm of all primes less than p. Thus for example for a sieve of 7, the integers removed are 49, 77, 91…..or 7×11, 7×13, 7×17 etc:

11 13 17 19 23 29 31 37

77 91 119 133 161 203 217 259

At 287 (7*41) the pattern starts to repeat itself i.e. at an interval of 210 or 7# = 7*5*3*2.

The only annoying thing is that composites work their way in higher up as multiples of the primes (I have to check but somehthing like 121 and 11*13 etc)

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Thanks Jon, it seems there is considerable similarity between your thinking and what we have produced. I’ll try to provide more detail when I can, but for the moment I need to focus on work.

Have you checked out either of these:

http://terrytao.wordpress.com/2008/02/07/structure-and-randomness-in-the-prime-numbers/

http://projecteuler.net/index.php?section=problems

Cheers,

Pete

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Jon, let me elaborate. I’ve also posted this a few hours ago at PrimeGrid here:

http://www.primegrid.com/forum_thread.php?id=2991

It seems there is a vast collection of people there working on similar problems, who would know wehther any aspect of this is original or not.

For any primorial Pn#, there is a pattern of possible primes that repeats forward to infinity. For example, P2# (i.e. 3#, or 6), the repeating pattern is 6m +/-1.

We can find the possible prime repetition pattern for P(n+1)# by first repeating the Pn# pattern P(n+1) times, and then removing all prime products of Pn, using only other prime numbers greater than P(n+1) but less than Pn#.

For example, start with 1,0,0,0,1,0 as a series of possible primes for P2#. This is the P2# sieve. To get the P3# sieve, repeat the series 5 times:

1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0 (here 1 indicates a number in this position “may” be prime, and a 0 indicates the number is not prime)

Now remove all multiples of five from this series. The only ones that remain involve a product of 5 with 1 and other primes greater than or equal to 5. In this case that means only 5 (since 5 x 7 is outside P3# and is therefore of no interest for this step), so we remove 5 and 25. The following represents the possible primes for every subsequent group of 30 integers:

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

Note, the values removed through this first sweep are symmetric about the mid-point of P3#. This seems to be the case for all primorial sets for the first pass of removal. Therefore, we only need to remove the prime multiples of 5 up to P3#/2, and then we can simply reflect the result about P3#/2 (e.g. reflecting 5 about 15 would give us 25 as expected).

This leaves the P3# sieve, which can be repeated forever as an indication of all possible future primes, but refined again at the next level (i.e. converted to a 7# sieve by repeating 7 times and then removing all products of 7 with 1, then primes from 7 to 30, including any products involving more than one other prime) to help us get the primes at subsequent levels.

The next step within P3#, to leave only prime numbers standing (the whole point, right?), is to remove all products of primes using only prime numbers greater than P3 (5 in this case). For Pn=5 there are no other products of primes greater than 5 less than 30, so this step makes no difference.

Note that for 1 to 30, this pattern is correct except for the primes already established up to P2#, which need to be retained in the list of known primes. In this stage we’re ony interested in finding the primes between 6 and 30, as primes up to 6 were previously decided and recorded.

Note that as Pn increases, and we have more primes as factors in Pn#, the process attracts a little more complexity. Recall the sieve for the next higher primordial uses the following sequence:

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

but will be repeated 7 times:

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

To generate the sieve for the next primorial, remove all prime products of 7 as previously described, e.g. 7, 49, 77, 91, 119, 133, 161, 203. The distribution of these prime products involving 7 and one other number will be symmetric, so we could only find the first four, then reflect then about 105.

1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0

0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

1,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,0,0

1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,1,0,0,0,1,0,0,0,0,0,1,0

0,0,0,0,0,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0

We can duplicate this pattern 11 times to get the possible primes for every 2310 (11#) numbers.

However, we first want to identify all the prime numbers between 5# and 7# (30 and 210). To do this, we now need to remove all products of primes GREATER than 7 yielding a product of interest. This means if we delete 121 (11 x 11), 143 (11 x 13), 169 (13 x 13) and 209 (11 x 19), it will leave us with only prime numbers up to 210.

The process is the same for larger Pn#, except we need to consider products of a larger number of primes in the sieving within a given range from P(n-1)# to Pn#.

At first glance this might seem to boil down to an equivalent of the sieve of Erathosthenes, but I think it reduces to an exponentially smaller number of calculations as you are able to take advantage of the patterns, periodicity and symmetry embedded in the sequence of prime numbers.

This would be easier for me to explain with some of the graphs I used to visualize this.

I’ve only explained the general process of generating the sieves and focussing the search to eliminate the rest of the non-pimes, but it should be obvious that those other steps can also be described coherently.

Please note I tend to be a little careless with details, so it’s quite possible (likely even!) there are some minor errors embedded in this post. I don’t think they detract from what I’ve presented, but perhaps they do, and feel free to let me know.

Cheers,

Pete

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I’ve read your postings on primegrid and feel we may have covered some of the same ground, reflecting the same issue – the non primes or composites repeat themselves within the p# frameworks (i.e. one for each p). As a result there are locations within the integers where we can confidently predict where there are no primes. Similarly we can predict where some large gaps in the primes are likely to be. Predicting the primes however remain elusive in part because the pattern is stepped and becomes more complex with higher integers, and thus the frequency of primes goes down……….

I don’t think I understand the precise details of what you have done, although I can re read. As previously indicated, using English for this is exhausting, long winded & ambiguous!

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Thanks Jon,

while I agree determining the primes becomes more complex as Pn# grows, I think it remains a straightforward process. I’ll try to add more to explain it better, but I’m in deep trouble with some real work deadlines so I’d better leave this alone for a bit.

I also posed the question to Dr. Terry Tao here:

http://terrytao.wordpress.com/2008/02/07/structure-and-randomness-in-the-prime-numbers/

The post is similar but I’ve mentioned your blog there as well.

Cheers,

Pete

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Hi Jon,

I saw a link to this site from Pete’s comment on the Tao blog.

I’ve been looking at the symmetrical primorial patterns for a while and have posted an attempted twin prime proof the linked blog.

If it’s correct it would also probably solve Polignac’s conjecture. I’m trying to get a few people to look at it to see if they can spot any mistakes.

Hugh

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Hello.This post was really motivating, particularly since I was searching for thoughts on this issue last Sunday.

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Utterly composed content material, Really enjoyed reading through.

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