Primorial bases
Primorial numbers:Regularities in the distribution of the prime numbers are apparent modulo primorials. Prime gaps, prime constellations, etc are perhaps best described in the primorial numerial system.
1, 2, 6, 30, 210, 2310, ...The importance of the primorials in the distribution of the prime numbers is readily apparent upon inspection of the smallest prime numbers. Although we don't typically work with the primorial number system, we can recover something of its character by considering modulo classes of primorials.
Mod two:
The primorial of one is two. Although, the first case is so small as to be trivial, it is responsible for our first result about the primes: all primes other then two are odd (and so equal to one mod 2).
1As consecutive odd numbers are distance two from one another, the smallest prime gaps this allows are of size two, which are between twin primes. The only exception is 2 and 3 which are the only consecutive primes. This is only possible because 2 is prime.
Mod six:
The primes mod six must be in only two classes modulo six: 1 and 5. An immediate consequence of this is that every twin prime $p,p+2$ has a sandwhich number $p+1$ that is a multiple of six.
1,5The only exception to this is the initial system: 2,3,5,7 consisting of a consecutive primes as well as the only consecutive twin primes $p,p+2,p+4$. Whenever you consider primes modulo a primorial, the larger the primorial the fewer the locations that a prime number can be in.
The numbers less the primorial always consitute the only exception. So prime gaps grow as prime numbers get larger then each primorial and from this we can already intuit that prime gaps grow without bound. Indeed, this is the case as determined by the prime number theorem.
Mod thirty:
The mod thirty case is the genuinely first interesting case as it allows us to describe the types of admissable prime patterns: twin primes, cousins, triplets, quadruples, decades, quintuplets, sextuplets, etc.
1,7,11,13,17,19,23,29The classification of max distance four prime systems proceeds in the following manner:
- Lone primes: primes like 53 that have no other primes near that are less distance six
- Tight Cousins: pairs like $379,383$ without any twin primes around them.
- Tight primes: these are primes without any cousins around them like 29,31, 59,61, etc. They are the ind of max distance four clusters that can occur around multiples of thirty.
- Triplets: numbers $p,p+2,p+6$ or $p,p+4,p+6$ such as $67,71,73$ and $277,281,283$.
- Single twin quadruples: by these I mean $p,p+4,p+6,p+10$ systems like $223,227,229,233$ and $307,311,313,317$.
- Decades: as these are all at 11,13,17,19 modulo 30 they are called decades. An obvious example is 191,193,197,199. It doesn't have any cousins so it is merely a quintuplet and not anything more.
- Quintuplets: these are either at 7,11,13,17,19 or 11,13,17,19,23 modulo 30. An example exists at the start of the seventh modulo 210 region: 1481, 1483, 1487, 1489, 1493.
- Sextuplets: a full system modulo 30 the most obvious occurence of this is 97,101,103,107,109,113.
There are no triples of primes $p,p+4,p+8$ such that the three primes are each cousins one after another. This can be seen by the modulo thirty distribution of primes. So basically for prime clusters with a maximum distance of four besides tight cousins we have they are one or twin primes with a number of cousins around them.
Mod 210:
The prime numbers modulo 210 avoid the classes 7,49,77,91,119,161,203. Which means that they must belong to one of the following residue classes:
1 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 121 127 131 137 139 143 149 151 157 163 167 169 173 179 181 187 191 193 197 199 209The usefulness of the mod 210 region is that it lets us determine the locations of max gap six prime clusters, which are much more interesting then the max distance four clusters which previously received a complete classification.
Definition. the separator region is the area between 90 and 120 mod 210. The numbers at the ends of the separator region like 97 and 113 must be distance at least eight away from primes outside the separator region.
The separator region is in fact responsible many of the first instances of larger prime gaps, because it always has prime gaps of at least eight around it from both sides. An example is the separator single twin quadruple 307,311,313,317 which has distances of fourteen around it in both directions.
Definition. a twin prime whose sandwhich is a multiple of 210 is a doubly tight twin prime, because its primes cannot be sexy primes with any other number. Then the minimum distance from any such pair to another prime is at least ten.
Thus, we have two areas of separation inherent to the mod 210 region: the separator region between 90 and 120 and the areas around the multiples of 210 themselves. With these two separation regions, we can places bounds on the min distance six prime clusters:
Proposition. let $C$ be a non-initial interval of primes, then if it has max distance six it has size no more then 78. The largest possible cases are from 11 to 89 and 121 to 199 modulo 210.
As the separator region which is between 90 and 120 modulo 210 occurs in the middle of the modulo 210 region, the most tightly packed systems of primes necessarily occur together at the start or the end of the modulo 210 region while excluding the middle.
Prime clusters:
Prime directions:The prime gaps for any pair of prime numbers are always positive integers, but another thing I have thought about a lot is the gaps between prime gaps. As the prime gaps are not monotone increasing, these are not necessarily positive integers. The gap between gaps can be positive, negative, or zero.
This basically assigns a direction to each prime number, which determines which prime number it is closer to. A positive gap between gaps means that a prime is pointing forwards and a negative gap between gaps means that it is pointing backwards. A balanced prime is one for which its gap in both directions is the same. A balanced prime has no direction, so it is not pointing towards anything.
Definition. let $p_n$ be a prime number other then two, then its direction is the sign of $p_{n+1} - p_{n-1}$. So it can be either $-1$, $0$, or $1$. A balanced prime has a direction of zero.
The directions of the first few primes are described below:
3 -, 5 0, 7 -, 11 +, 13 -, 17 +, 19 -, 23 -, 29 +, 31 -, 37 +, 41 +, 43 -, 47 -, 53 0, 59 +, 61 -, 67 +, 71 +, 73 -, 79 +, 83 -, 89 -The directions give us an idea about how we can cluster different prime numbers together: each prime number is most readily associated to the primes in the direction it is pointing. Prime clusters can be formed starting with a prime pointing in a positive direction and ending with a prime pointing in a negative direction.
That way the ends of the cluster of primes are both pointing inward. An example is twin primes, in that case because all non-trivial twin primes are closer to each other then any other number the two of the form a small cluster together. But numbers may form clusters in a number of different ways, depending upon how far up you want to go.
An example is the single twin separator quadruple 307,311,313,317 then these have prime directions $\rightarrow \rightarrow \leftarrow \leftarrow$. We can therefore get an inward pointing prime cluster in two different ways: 311,313 and 307,311,313,317. The first is at a lower depth level then the second. The grouping of prime numbers at increasingly high levels of depth essentially produces the structure of a tree.
Parse tree of the prime numbers:
The prime numbers are no different then any other unstructured data stream with a lot of inner structure to them, in that we can parse them into a tree structure. The theory of prime directions and prime gaps already developed gives us the means we need to do this.
Towards that end, I will demonstrate by describing the parse tree of the initial cluster. Starting with $2$ and $3$ we see that three points backwards because it is closer to $2$ then it is to five so the first cluster is $2$ and $3$. Then since $5$ is a balanced prime at the next depth level we get $2,3,5,7$ grouped together.
All twins form a natural pair so $11$ and $13$ go together just as $17$ and $19$ do, but $11,13$ taken as a pair are balanced because their distances to nearby clusters are equal and the same is true of $17,19$ so the only result is that $2,3,5,7,11,13,17,19,23$ form a single family of prime numbers. The largest lower set of primes without a prime gap of six in them.
The same principle applies to $29,31$ and $59,61$ as twin primes. For both the single prime quadruple $37,41,43,47$ and the prime triplet $67,71,73$ we see that they form two levels of structure. Finally, the cousins $79$ and $83$ can be grouped together.
Of course, this is only the parse tree of the initial cluster. There are infinitely more primes, but the basic principle is the same. The fact that prime clusters have directions which make them grouped more readily with certain prime clusters instead of others, means that all prime numbers form a tree.
As the primes are infinite, we can never view this entire tree structure. However, given any finite sampling of prime numbers we can parse them into a tree by studying their gaps to one another and then forming clusters from the most tightly packed sets of primes. This can be continued to an arbitrary depth level, so the entire initial cluster itself can be grouped with the sextuplet separator.
A look at the small primes
Initial cluster:The initial cluster 2,3,5,7,11,13,17,19,23, 29,31,37,41,43,47,53,59,61,67,71,73, 79,83,89 is unique in that it is the largest min distance six collection of primes. After the first mod 210 region, these cluster tend to get smaller and rarer.
Sextuple separator:
The region between 90 and 120 mod 210 is always associated with prime gaps of size eight of greater, which is why this is is called the separator region. In this case 97,101,103,107,109,113 forms a sextuple. As this sextuple separator is distance eight from the initial cluster and fourteen from the Mersenne starter system, it has a negative direction so it can be grouped with the initial cluster.
97, 101, 103, 107, 109, 113Mersenne starter system:
The prime collection 127,131,137,139 consists of a cousin prime followed by a twin prime. A notable feature of this collection of primes is that it starts with 127 which is a Mersenne prime. It has a positive direction, because it has a smaller distance to the cousin in the middle then to the sextuple separator.
127, 131, 137, 139Cousin in the middle:
The cousin in the middle system is essentially symmetric in terms of prime gaps. It starts and ends with twin primes. In between them is a cousin surounded by two balanced sexy primes. The location of the cousin pair $163,167$ is why I identify this cluster of primes as the cousin in the middle system.
149, 151, 157, 163, 167, 173, 179, 181Prime quintuplet
After the cousin in the middle system there is a prime quintuplet, consisting of a pair of twin primes. As far as prime constellations, these are comparatively rare and another one doesn't occur until the eight hundreds region.
191, 193, 197, 199Primorial prime:
The prime number $211$ is a primorial prime because it is one after the primorial $210$. It is also a relatively lonely prime having no twins, cousins, sexy prime pairs, etc most of which is a consequence of its location near a primorial.
211Single twin quadruple starter system:
The quadruple starter system comes next. It consists of a single-twin quadruple $223,227,229,233$ and after that it has a twin prime $239,241$.
223,237,239,233, 239, 241Long system:
I call this the long system for lack of any better word for it. It starts with a sexy triplet then a twin prime and a prime triplet.
251, 257, 263, 269, 271, 277, 281, 283Straggler:
The prime number 293 is not a balanced prime, so by no means can be say that it is equidistant to other primes. It is closer to the long system then the great separator quadruple. This negative prime direction makes it a straggler for the long system.
293Great separator quadruple
The 90 to 120 region mod 210 is often associated with larger prime gaps as previously demonstrated. The first region is associated with the first size eight and fourteen gaps. The second, the great separator quadruple, is responsible for the first appearance of size fourteen prime gaps so close to each other. The great separator quadruple is balanced because it is distance fourteen from any other primes.
307,311,313,317Sexy pair:
The prime numbers 331 and 337 are the first sexy prime pair such that both numbers are pointed towards each other.
331,337Backwards system:
This is called the backward system because of the prime directions $\rightarrow \leftarrow \leftarrow \leftarrow$, so that most of its primes are pointing backwards. This is a consequence of the fact that the prime gaps tend to grow as you go further along in this cluster.
347,349,353,359Twinless system:
This is also a cousin in the middle system in the sense that it contains an enclosed pair of cousin primes, but it is distinguished by the fact that it has no twins. The lack of twins is of course something that becomes increasingly common, as the prime gaps tend to grow logarithmically.
367, 373, 379, 383, 389Cousins:
The prime numbers $397$ and $401$ form cousins. Together they are balanced cousins as the nearest primes are distance eight in either direction.
397, 401Another straggler:
The prime number 409 is also a straggler with a negative pointing direction because it has a distance of eight from 401 and a distance of ten to 419 so it is closer to the former rather then the later. But it is not nearly as interesting of a straggler I'm afraid, because it doesn't have anything like the long system around it.
409Twin primes around a multiple of 210
The second multiple of 210 is 420. It is a sandwhich of the twin prime 419,421. As we saw in the theory of the mod 210 region each such pair necessarily has a distance of at least ten in each direction so although these are twin primes they cannot have cousins, triplets, quintuplets, etc. So we are going to have just settle for a twin prime.
419,421Separate twin and cousins
This has a twin and a cousin, but the cousin primes are not cousin to either of the twins. Finally, there is a sexy prime pair at the end.
431, 433, 439, 443, 449Another single twin prime quadruple:
So this is a lot like the single twin prime quadruple 307,311,313,317 that occurred in the separator region. Prime clusters like these of course occur all the time
457, 461, 463, 467Forward pointing straggler:
The number 479 is notable in that it is has a positive direction instead of a negative one like 293 and 409.
479Another cousin:
This is just another pair of cousin primes much like 397, 401. Like in that case, it has a distance of eight between it and any other primes.
487,491Cousin with a sexy prime partner
Before the eighteen gap separator there is a cousin with a sexy prime at the end of it. The eighteen gap comes after the separator, so the separator twin is closer to this cluster then the subsequente sexy primes.
499, 503, 509Eighteen gap separator:
It has been mentioned a couple of times that the separator region is often responsible for some of the first instances of prime gaps. The third separator region is responsible for the first size eighteen prime gap. Instead of having a lot of structure like the first and second separator regions it has a larger prime gap.
521, 523Sexy primes:
Recall that the great separator quadruple is followed by a sexy prime pair: 331 and 337. In this case, the next separator gap region is associated with a subsequent sexy prime pair as well: 541, 547. This is a regularity common to both modulo 210 regions.
541, 547Cousinless twin prime system:
This cluster of primes includes a twin but instead of having cousins like a single-twin quadruple it the twin prime 569,571 has a bunch of sexy primes associated to it.
557,563,569,571,577Second long system:
This is another long system much like the one that occurs toward the end of the 200s. It has the same number of prime numbers, except one of the sexy primes is moved between the first prime and the final triplet.
587,593,599,601,607,613,617,619Prime around a multiple of 210:
As always, primes around multiples of 210 that are not twin primes stand alone. They must necessarily have gaps of at least ten around them.
631Triplet starter system This is a spectacular collection of primes because the amount of the structure it has: it has a twin prime at both the start and the end. This is in stark contrast to the subsequent twinless void that doesn't have any such tightly coupled systems of primes.
641, 643, 647, 653, 659, 661Twinless void:
The twinless void is the first large gap without any twin primes. As the twinless void doesn't contain nearly as much interesting structure as the earlier primes, it will be addressed all at once. Notable attributes including the factorial prime 719 and the 722,733,739,743 cluster in the separator region. This is the first separator region is that isn't asociated with any significantly larger prime gaps.
673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797The first new twins:
The end of the long twinless void is marked by the formation of the tight twin pair 809,811.
809, 811A prime decade
The prime quintuplet 821,823,827,829 is the first such structure to occur after 191,193,197,199. Although there was a long twin void after 661, clearly in the 800s region there is no shortage of twin primes.
821,823,827,829A prime before a multiple of 210
The two main separator regions are between 90 and 120 mod 210 and those numbers that straddle a multiple 210 itself either at modulo 209 like with this number or at modulo 1 like with 211. This number is of the later sort, which means it necessarily has large prime gaps to any other prime numbers.
839Two consecutive single twin quadruples:
Two single prime quadruples then occur one after another in the eight hundreds region. This makes it far easier to memorize them as they are merely repetitions of the same pattern.
853 857 859 863 877 881 883 887The nine hundreds:
The nine hundreds has a cousin and a couple of other primes at the start and a cousin at the end, but besides that there are two systems 937,941,947,953 and 967,971,977,983 that are direct repetitions of one another modulo thirty. This second instance of repetition of a pattern in the prime gaps, similar to what occurred in the eight hundreds, is another trick you can use to aid in the memorization of all primes less then a thousand.
907, 911 919 929 937, 941, 947, 953 967, 971, 977, 983 991, 997References:
The Distribution of Prime Numbers by Dimitris Koukoulopoulos
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