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A hierarchy of double, quadruple and octuple primes

International Journal of Mathematics and Computer in Engineering
Volume 2 (2024): Issue 2 (December 2024)
By:
Open Access
|Jan 2024

Figures & Tables

Fig. 1

The hierarchy formed from primes, double primes, qudruple primes and an octuple prime.
The hierarchy formed from primes, double primes, qudruple primes and an octuple prime.

Fig. 2

Divisors around specific qcenter q = 195.
Divisors around specific qcenter q = 195.

Fig. 3

Divisors around general qcenter q.
Divisors around general qcenter q.

Fig. 4

Expanded table in the neighborhood of the quadruple prime with qcenter 195, now including divisibility by 7.
Expanded table in the neighborhood of the quadruple prime with qcenter 195, now including divisibility by 7.

Fig. 5

Table of divisors for a general octuple where first quadruple has qcenter= q.
Table of divisors for a general octuple where first quadruple has qcenter= q.

Fig. 6

Octuple with ocenter o between o − 26 and o + 24.
Octuple with ocenter o between o − 26 and o + 24.

Primality of q + 26, q + 28, q + 32 and q + 34 for general q-center=q_

q-centerq + 26q + 28q + 32q + 34
195NOYESYESYES
825NOYESYESYES
1485YESNONONO
1875NOYESYESNO
2085YESYESNONO
21015NONONONO

The octuple with o-center= o = 1006320_

type of centercenter labelcenter valuefactored center
d-centerd111006302(2,1),(3,1),(11,1),(79,1),(193,1)
q-centerq11006305(3,1),(5,1),(73,1),(919,1)
d-centerd121006308(2,2),(3,2),(27953,1)
o-centero1006320(2,4),(3,1),(5,1),(7,1),(599,1)
d-centerd211006332(2,2),(3,1),(17,1),(4933,1)
q-centerq21006335(3,2),(5,1),(11,1),(19,1),(107,1)
d-centerd221006338(2,1),(3,1),(179,1),(937,1)

Primality o − 1 and o + 1 for select cases of octuple primes_

o =o-centerprime o − 1prime o + 1
1006320FalseFalse
2594970FalseTrue
3919230TrueFalse
9600570FalseTrue
10531080FalseFalse
157131660FalseTrue
179028780TrueFalse
211950270TrueFalse
255352230FalseFalse
267587880FalseFalse
724491390TrueFalse
871411380FalseFalse

Pure and impure octuples up to 1011_

Number of octuple primesPure octuple primesOctuple primes with o − 1 primeOctuple primes with o + 1 primeOctuple primes with o − 1 and o + 1 prime
26718734473

Pure and impure octuples up to 1010_

Number of octuple primesPure octuple primesOctuple prime with o − 1 primeOcuple primes with o + 1 primeOcuple primes with o − 1 and o + 1 prime
65429140

Divisors in the neighborhood of the first o-center o of a possible sixtentuple prime_

indexqn−1 − 4qn−1 − 3qn−1 − 2qn−1 − 1qn−1qn−1 + 1qn−1 + 2qn−1 + 3qn−1 + 4
divisunsp.2,3unsp.23,52,7unsp.2,3unsp.

Divisibility by 3_

indexq − 9q − 8q − 7q − 6q − 5q − 4q − 3q − 2q − 1qq + 1q + 2q + 3q+4
divis2,3 232P2,3P232P2,3P

Divisibility by 2, 3 and 5 around qprime q_

indexq − 7q − 6q − 5q − 4q − 3q − 2q − 1qq + 1q + 2q + 3q + 4
divis232,5P2,3P23,52P2,3P

Divisors in the neighborhood of the quadruple prime with qcenter 195, now including 7_

index185186187188189190191192193194
divis52,3 23,72,5P2,3P2
index195196197198199200201202203204
divis3,52,7P2,3P2,53272,3
index205206207208209210211212213214
divis5232 2,3,5,7P232
index215216217218219220221222223224
divis52,37232,5 2,3P2,7
index225226227228229230231232233234
divis3,52P2,3P2,53,72PE,3
index235236237238239240241242243244
divis5232,7P2,3,5P232

Divisors in the neighborhood of the second o-center on of a possible sixtentuple prime_

indexqn+1 − 4qn+1 − 3qn+1 − 2qn+1 − 1qn+1qn+1 + 1qn+1 + 2qn+1 + 3qn+1 + 4
divisunsp.2,3unsp.2,73,52unsp.2,3unsp.

Quadruple prime example_

index1011121314151617181920
divisEPd1PEq-centerEPd2PE

Divisors of centers_

centers divisible byspan divisible bycomment
dcenter11*2verified
qcenter31*2*3verified
ocenter151*2*3*5verified
scenter1051*2*3*5*7hypothetical

Double prime example_

indexd − 3d − 2d − 1dd + 1d + 2d + 3d + 4d + 5
divis EPdcenterPE E
DOI: https://doi.org/10.2478/ijmce-2024-0019 | Journal eISSN: 2956-7068
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Language: English
Page range: 251 - 262
Submitted on: Oct 13, 2023
Accepted on: Dec 25, 2023
Published on: Jan 10, 2024
Published by: Harran University
In partnership with: Paradigm Publishing Services
Publication frequency: 2 issues per year
Keywords:
Prime,
double prime,
quadruple prime,
octuple prime,
calculations
Related subjects:
Computer sciences,
Computer sciences, other,
Engineering,
Introductions and overviews,
Engineering, other,
Mathematics,
General mathematics,
Physics,
Physics, other

© 2024 Jon Rokne, published by Harran University
This work is licensed under the Creative Commons Attribution 4.0 License.

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