Number theory is one of the oldest mathematical disciplines. It deals with the properties of integers [1, 2]. One of the central topics in number theory is the factorization of the numbers and if a number cannot be factored into a product of two or more integers then it is a prime number. A prime is, therefore, an integer that is only divisible by integers 1 and the number itself. Methods for establishing whether a number is a prime or a composite number in a consecutive sequence of integers include several methods such as Eratosthenes, Selberg, Brun and others [3].
Primes have many interesting properties that have been studied since antiquity and a number of papers and books have been written about them [4,5,6,7,8,9,10]. A computational perspective on primes is provided in the book
Prime patterns have been of interest to a number of authors [18, 19] and prime constellations have been discussed by, for example [20]. Both prime patterns and prime constellations are concerned with particular configurations of primes. In this paper a new pattern forming a hierarchy of groups of primes is depicted as a binary tree as shown in Figure 1 where each level of the tree represents particular constellations.
The hierarchy consists of constellations of double primes some of which form the basis for quadruple primes which are then, in some cases, the basis for octuple primes where the exact nature of these constellations will be discussed in the sequel. Known and new properties for these constellations of primes are proven and examples of quadruple and octuple primes are exhibited. A conjecture for the existence of 16-tuple primes is stated as well.
The paper is in parts experimental mathematics [21, 22] where interesting numerical facts are established using a computer and in parts theoretical mathematics when some properties of these prime constellations are proven.
Applications of the results in this paper might include cryptography.
By “center” of a prime
The letter E denotes “even”.
The letter P will denote “prime”.
The term “index” denotes a number being considered.
The abbreviation “divis” is short for divisor.
“dcenter” denotes the center of a double prime.
“qcenter” denotes the center of a quadruple prime.
“ocenter” denotes the center of octuple primes.
“scenter” denotes the center of a sextuple prime if it exists.
“tuple” denotes any of the double, quadruple or octuple primes.
“span” denotes the distance between centers of defining tuples in the hierarchy of tuples.
A double prime, also called a prime pair or twin primes, is a set of two primes separated by an even non-prime which is called the dcenter of the double prime. The first double primes are
The following simple result is well known, but established here as well.
Double prime example.
index
divis
E
P
dcenter
P
E
E
In Table 1
From this theorem it follows that if the dcenter of a double prime is
Double primes have been studied by a number of authors and for some background on these primes see for example [23] and [24, 25].
A quadruple prime is a set of two double primes with dcenters
Computationally exhibiting the following case of a two double primes with dcenters at
Quadruple prime example.
index | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
divis | E | P | P | E | q-center | E | P | P | E |
Note that it is easily shown that two double primes separated by an even number is not possible hence the separation would have to be at least 3 numbers.
The first case of a quadruple prime has dcenters
Examples of quadruple primes are found, for example, in [26].
Before continuing it should be noted that given a consecutive list of
There are some results for quadruple primes that are now discussed in the following theorem.
This follows from the remark above for double primes. As a result the sequence numbers from Divisibility by 3. Consider the numbers Viewing Table 3 it follows that:
- - - - Hence It follows immediately from 2) that
index
q+4
divis
2,3
2
3
2
P
2,3
P
2
3
2
P
2,3
P
Divisibility by 2, 3 and 5 is now shown in Table 4.
Divisibility by 2, 3 and 5 around qprime
index | ||||||||||||
divis | 2 | 3 | 2,5 | P | 2,3 | P | 2 | 3,5 | 2 | P | 2,3 | P |
Consider now the quadruple prime with qcenter
For a general quadruple prime with qcenter
In the figures there are two framed sequences. The red-framed sequence centered on the qcenter 195 runs from 190 to 199 see Figure 2, or in terms of distances to a general qcenter
Given the sequence of divisors shown in Figure 2 it is clear that the first possible quadruple prime following the quadruple prime with qcenter 195 would have to have a qcenter at 225. In this particular case 221 is undefined, 223 a prime, 227 a prime and 229 a prime. This means that if 221 had been prime then 225 would have been the qcenter of a new quadruple prime as indicated by the green frame.
The green frame in table in Figure 3 shows the general case indicating that if
Primality of
q-center | ||||
---|---|---|---|---|
195 | NO | YES | YES | YES |
825 | NO | YES | YES | YES |
1485 | YES | NO | NO | NO |
1875 | NO | YES | YES | NO |
2085 | YES | YES | NO | NO |
21015 | NO | NO | NO | NO |
The following result places some restrictions on the possible quadruple primes with qcenter
Consider now the sequence
This leaves
Note that the theorem does not imply that if
The table in Figure 2 can now be expanded with added information to the table in Figure 4.
Divisors in the neighborhood of the quadruple prime with qcenter 195, now including 7.
index
185
186
187
188
189
190
191
192
193
194
divis
5
2,3
2
3,7
2,5
P
2,3
P
2
index
195
196
197
198
199
200
201
202
203
204
divis
3,5
2,7
P
2,3
P
2,5
3
2
7
2,3
index
205
206
207
208
209
210
211
212
213
214
divis
5
2
3
2
2,3,5,7
P
2
3
2
index
215
216
217
218
219
220
221
222
223
224
divis
5
2,3
7
2
3
2,5
2,3
P
2,7
index
225
226
227
228
229
230
231
232
233
234
divis
3,5
2
P
2,3
P
2,5
3,7
2
P
E,3
index
235
236
237
238
239
240
241
242
243
244
divis
5
2
3
2,7
P
2,3,5
P
2
3
2
In the table in Figure 2 when starting from the quaduple prime center
The question is therefore, are there quadruple primes
The question is now: do octuple primes defined in this manner exist and if so, what specific properties do they have? It should be noted that the term octuple prime has been used before in a slightly different context, see for example [24]. It is used in this context to denote the progression of particular tuples from primes to double primes then quadruple primes and finally octuple primes.
As shown computationally in [26] octuple primes exist and the first octuple prime has o-center=
The octuple with o-center=
type of center | center label | center value | factored center |
---|---|---|---|
d-center | 1006302 | (2,1),(3,1),(11,1),(79,1),(193,1) | |
q-center | 1006305 | (3,1),(5,1),(73,1),(919,1) | |
d-center | 1006308 | (2,2),(3,2),(27953,1) | |
o-center | 1006320 | (2,4),(3,1),(5,1),(7,1),(599,1) | |
d-center | 1006332 | (2,2),(3,1),(17,1),(4933,1) | |
q-center | 1006335 | (3,2),(5,1),(11,1),(19,1),(107,1) | |
d-center | 1006338 | (2,1),(3,1),(179,1),(937,1) |
In the case of double primes there was one non-prime separating the two primes and in the case of quadruple primes there were three non-primes separating the double primes. In a sense the “gaps” between the previous sets of primes were small, i.e one and three steps. In the case of octuple primes the gap is much larger. For a given octuple center
Primality
prime |
prime |
|
---|---|---|
1006320 | False | False |
2594970 | False | True |
3919230 | True | False |
9600570 | False | True |
10531080 | False | False |
157131660 | False | True |
179028780 | True | False |
211950270 | True | False |
255352230 | False | False |
267587880 | False | False |
724491390 | True | False |
871411380 | False | False |
The octuples with composite
Pure and impure octuples up to 1010.
Number of octuple primes | Pure octuple primes | Octuple prime with |
Ocuple primes with |
Ocuple primes with |
---|---|---|---|---|
65 | 42 | 9 | 14 | 0 |
There are not many prime octuples. The results shown in Table 9 show the number of octuple primes of the different kinds for numbers up to 1010.
The table shows that there are no octuple primes where both
However, when the computations are performed up to 1011 the results are as in Table 10 disproving the conjecture.
Pure and impure octuples up to 1011.
Number of octuple primes | Pure octuple primes | Octuple primes with |
Octuple primes with |
Octuple primes with |
---|---|---|---|---|
267 | 187 | 34 | 47 | 3 |
The three examples where
The previous sections have shown that the centres of the double, quadruple and octuple primes form a sequence as shown in Table 11.
Divisors of centers.
centers divisible by | span divisible by | comment | |
---|---|---|---|
dcenter | 1 | 1*2 | verified |
qcenter | 3 | 1*2*3 | verified |
ocenter | 15 | 1*2*3*5 | verified |
scenter | 105 | 1*2*3*5*7 | hypothetical |
Since the ocentres of octuple are all divisible by 15 = 3 * 5 one might conjecture that there are sixtentuple primes composed of two octuple primes whose ocentres are separated by 210 = 2 * 3 * 5 * 7 and with a scenter divisible by 3 * 5 * 7 = 105. This table shows an unusual regularity in the first three entries rarely found in the literature on primes.
Considering that a specific octuple prime has the properties in the neighborhood as in Table 12 a possible second ocenter at a distance of 210 from the given ocenter would be as in Table 13.
Divisors in the neighborhood of the first o-center
index | |||||||||
divis | unsp. | 2,3 | unsp. | 2 | 3,5 | 2,7 | unsp. | 2,3 | unsp. |
Divisors in the neighborhood of the second o-center
index | |||||||||
divis | unsp. | 2,3 | unsp. | 2,7 | 3,5 | 2 | unsp. | 2,3 | unsp. |
In both o-center cases there are entries that are unspecified as far as the divisors 3, 5, 7 are concerned. This means that the possibility of these entries all being prime is not excluded.
For the specific case of o-center
There is no conflict of interest.
J.R.-Conceptualization, Methodology, Formal analysis, Writing-Review and Editing, Validation, Writing-Original Draft, Resources. All authors read and approved the final submitted version of this manuscript.
No funding was received to assist with the preparation of this manuscript.
Not applicable.
All data that support the findings of this study are included within the article.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.