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A hierarchy of double, quadruple and octuple primes

Jon Rokne
Jon Rokne
   | 10 janv. 2024
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Article Category: Original Study
Publié en ligne: 10 janv. 2024
Pages: 251 - 262
Reçu: 13 oct. 2023
Accepté: 25 déc. 2023
DOI: https://doi.org/10.2478/ijmce-2024-0019
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© 2024 Jon Rokne, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

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-center q + 26 q + 28 q + 32 q + 34
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 octuple with o-center= o = 1006320.

type of center center label center value factored center
d-center d11 1006302 (2,1),(3,1),(11,1),(79,1),(193,1)
q-center q1 1006305 (3,1),(5,1),(73,1),(919,1)
d-center d12 1006308 (2,2),(3,2),(27953,1)
o-center o 1006320 (2,4),(3,1),(5,1),(7,1),(599,1)
d-center d21 1006332 (2,2),(3,1),(17,1),(4933,1)
q-center q2 1006335 (3,2),(5,1),(11,1),(19,1),(107,1)
d-center d22 1006338 (2,1),(3,1),(179,1),(937,1)

Primality o − 1 and o + 1 for select cases of octuple primes.

o =o-center prime o − 1 prime o + 1
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

Pure and impure octuples up to 1011.

Number of octuple primes Pure octuple primes Octuple primes with o − 1 prime Octuple primes with o + 1 prime Octuple primes with o − 1 and o + 1 prime
267 187 34 47 3

Pure and impure octuples up to 1010.

Number of octuple primes Pure octuple primes Octuple prime with o − 1 prime Ocuple primes with o + 1 prime Ocuple primes with o − 1 and o + 1 prime
65 42 9 14 0

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

index qn−1 − 4 qn−1 − 3 qn−1 − 2 qn−1 − 1 qn−1 qn−1 + 1 qn−1 + 2 qn−1 + 3 qn−1 + 4
divis unsp. 2,3 unsp. 2 3,5 2,7 unsp. 2,3 unsp.

Divisibility by 3.

index q − 9 q − 8 q − 7 q − 6 q − 5 q − 4 q − 3 q − 2 q − 1 q q + 1 q + 2 q + 3 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 around qprime q.

index q − 7 q − 6 q − 5 q − 4 q − 3 q − 2 q − 1 q q + 1 q + 2 q + 3 q + 4
divis 2 3 2,5 P 2,3 P 2 3,5 2 P 2,3 P

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

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

index qn+1 − 4 qn+1 − 3 qn+1 − 2 qn+1 − 1 qn+1 qn+1 + 1 qn+1 + 2 qn+1 + 3 qn+1 + 4
divis unsp. 2,3 unsp. 2,7 3,5 2 unsp. 2,3 unsp.

Quadruple prime example.

index 10 11 12 13 14 15 16 17 18 19 20
divis E P d1 P E q-center E P d2 P E

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

Double prime example.

index d − 3 d − 2 d − 1 d d + 1 d + 2 d + 3 d + 4 d + 5
divis E P dcenter P E E
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