Odd and even symmetric prime constellations

Rokne, Jon

Odd and even symmetric prime constellations

International Journal of Mathematics and Computer in Engineering

Volume 3 (2025): Issue 2 (December 2025)

By: Jon Rokne

Open Access

|Sep 2024

Figures & Tables

Divisibility relative to center prime - odd case - rel. index −1 divisible by 3.

Divisibility relative to composite center - even base case.

Divisibility relative to composite center - even second case.

Proof of impossibility of suggestion for Table 20.

Table for a possible 16-tuple based on octuples.

Potential triple prime, C(13, 3)_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6
constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆	p₃
divisors/prime	P	2	C	2	C	2	P	2	C	2	C	2	P

The first few symmetric sixtuple constellations, third case, are given with C(29, 6)_

constellation	p₁	p₂	p₃	p₄	p₅	p₆

sixtuple prime 1	151	157	163	167	173	179
sixtuple prime 2	20101	20107	20113	20117	20123	20129
sixtuple prime 3	128461	128467	128473	128477	128483	128489
sixtuple prime 4	297601	297607	297613	297617	297623	297629
sixtuple prime 5	350431	350437	350443	350447	350453	350459
sixtuple prime 6	354301	354307	354313	354317	354323	354329
sixtuple prime 7	531331	531337	531343	531347	531353	531359

The three 10-tuples found with rel_ indices (−19, −17, −13, −11, −1, 1, 11, 13, 17, 19) when entries with indices (−7, 7) in the table in Figure 1 are composite_

10-tuple center	factors of center	entry index −7	entry index +7
39713433690	(2, 3, 5, 7, 23, 8222243)	composite	composite
66419473050	(2, (3, 2), (5, 2), (7, 2), 3012221)	composite	composite
71525244630	(2, 3, 5, (7, 3), 6950947)	composite	composite

The four 10-tuples found when rel_ indices (−19, −17, −13, −11, −7, 7, 11, 13, 17, 19) are specified with (−1, 1) in the table in Figure 1 being composite_

10-tuple center	factors of center	entry index −1	entry index +1
30	(2, 3, 5)	composite	composite
1864508550	(2, 3, 5, 5, 241, 151577)	composite	composite
4763132670	(2, 3, 5, 7193, 22073)	composite	composite
5302859550	2, 3, 5, 5, 167, 211691	composite	composite

First and last computed cases for constellation in Table 23_

Parameter	value	value	value	value	value	value	value	value

first case	p₁	p₂	p₃	p₄	p₅	p₆	p₇	p₈
first case	344231	344237	344243	344251	344253	344259	344263	344267

last case	p₁	p₂	p₃	p₄	p₅	p₆	p₇	p₈
last case	944554301	944554307	944554313	944554321	944554323	944554329	944554333	944554337

The first 10-tuple C(39, 10) prime constellation, center c = 39713433690_

rel. index	−19	−17	−13	−11	−1
prime	p₁	p₂	p₃	p₄	p₅
value	39713433671	39713433673	39713433677	39713433679	39713433689
rel. index	1	11	13	17	19
prime	p₆	p₇	p₈	p₉	p₁₀
value	39713433691	39713433701	39713433703	39713433707	39713433709

Triple prime (47, 53, 59) with m = 13_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val	val

constellation	47	48	49	50	51	52	53	54	55	56	57	58	59
factors	47	2⁴ * 3	7⁷	2 * 5²	3 * 17	2² * 13	53	2 * 3³	5 * 11	2³ * 7	3 * 19	2 * 29	59

Summarizing the even symmetric constellations, smallest extent so far_

configurations	value of m	number of cases	m is minimal ?	comment
2-tuples (double primes)	3	no limit	YES obvious	extensively studied
4-tuples (quadruple primes)	9	166 cases up to 10⁷	YES as shown in section 4.
6-tuples (sixtuple primes)	17	18 cases up to 10⁷	YES as shown in section 6	The cases with m < 17 were shown to be not possible using divisibility by 2, 3, 5, 7.
8-tuples (octuple primes)	27	28 cases up to 10¹⁰	Not verified	The cases with m = 27 have the smallest extent found so far
10-tuples	35	2 cases up to 10¹⁰	Not verified	The cases with m = 35 were the minimal extent found so far

Some 4-tuple symmetric primes, C(9, 4)_

constellations	p₁	p₂	p₃	p₄	q

quadruple prime, special case 1	5	7	11	13	9
quadruple prime 2	11	13	17	19	15
quadruple prime 3	101	103	107	109	105
quadruple prime 4	191	193	197	199	195
quadruple prime 5	821	823	827	829	825
quadruple prime 6	1481	1483	1487	1489	1485
quadruple prime 7	1871	1873	1877	1879	1875

9-tuple prime, C(120, 9)_

Parameter	value	value	value	value

constellation	p₁	p₂	p₃	p₄
location	c − 60	c − 42	c − 30	c − 18
divisors/prime	12383210011	12383210029	12383210041	12383210053

center	p₅
location	c
prime	12383210071

constellation	p₆	p₇	p₈	p₉
location	c + 18	c + 30	c + 42	c + 60
divisors/prime	12383210089	12383210101	12383210113	12383210131

Divisibility of symmetric sixtuple prime constellations with C(27, 6), even base case_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−13	−12	−11	−10	−9	−8	−7	−6	−5	−4	−3	−2
constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆
divisibility	prime	2, 3		2, 5	3	2	prime	2, 3	5	2	3	2

rel. index	−1	0	1
constellation	p₃	e₇	p₄
divisibility	prime	2, 3, 5	prime

rel. index	2	3	4	5	6	7	8	9	10	11	12	13
constellation	e₈	c₅	e₉	c₆	e₁₀	p₅	e₁₁	c₇	e₁₂	c₈	e₁₃	p₆
divisibility	2	3	2	5	2, 3	prime	2	3	2, 5		2, 3	prime

Centers of the 5 10-tuple constellations C(47, 10)_

1011208680

2233694520

4143953640

6486125010

Divisibility of symmetric octuple prime constellation - second case_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−19	−18	−17	−16	−15	−14	−13	−12	−11	−10	−9	−8
constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆
divis	prime	2, 3	?	2	3, 5	2	prime	2, 3	?	2, 5	3	2

rel. index	−7	−6	−5	−4	−3	−2
constellation	p₃	e₇	c₅	e₈	c₆	e₉
divis	prime	2	5	2	3	2

rel. index	−1	0	1
constellation	p₄	e₁₀	p₅
divis	prime	2, 3, 5	prime

rel. index	2	3	4	5	6	7
constellation	e₁₁	c₇	e₁₂	c₈	e₁₃	p₆
divis	2	3	2	5	2, 3	prime

rel. index	8	9	10	11	12	13	14	15	16	1	18	19
constellation	e₁₄	c₉	e₁₅	c₁₀	e₁₆	p₇	e₁₇	c₁₁	e₁₈	c₁₂	e₁₉	p₈
divis	2	3	2, 5	?	2, 3	prime	2	3, 5	2	?	2, 3	prime

The first few symmetric eight-tuple constellations C(27, 8) with rel_ indices at (−13, −11, −7, −1, 1, 7, 11, 13)_

constellation	p₁	p₂	p₃	p₄	p₅	p₆	p₇	p₈

eight-tuple 1	17	19	23	29	31	37	41	43
eight-tuple 2	1277	1279	1283	1289	1291	1297	1301	1303
eight-tuple 3	113147	113149	113153	113159	113161	113167	113171	113173
eight-tuple 4	2580647	2580649	2580653	2580659	2580661	2580667	2580671	2580673
eight-tuple 5	20737877	20737879	20737883	20737889	20737891	20737897	20737901	20737903

Summarizing the odd symmetric constellations, smallest extent so far_

constellations	value of m	number of cases	is minimal ?	comment
3-tuples (triple primes)	13	758163 up to 10⁹	YES by construction	m = 7, 11 not possible by symmetry, m = 5, 7 not possible by construction
5-tuples (qunintuple primes)	37	124 up to 10⁷	YES as shown in section 5	The cases with m < 37 were shown to be not possible using Theorem 3
7-tuples (qunintuple primes)	73	124 up to 10⁹	NO
9-tuples	121	124 up to 2 * 10¹⁰	NO	121 is the smallest extent found so far

Case_1: possible symmetric quintuple prime_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−18	−17	16	−15	−14	−13	−12	−11	−10	−9	−8	−7
constellation	p₁	e₁	c₁	e₂	c₂	e₃	c₃	e₄	c₄	e₅	c₅	e₆
divisors/prime	prime	2		2		2		2		2		2

rel. index	−6	−5	−4	−3	−2	−1
constellation	p₂	e₇	c₆	e₈	c₇	e₉
divisors/prime	prime	2		2		2

rel. index	0
center	p₃
prime	prime

rel. index	1	2	3	4	5	6
constellation	e₁₀	c₈	e₁₁	c₉	e₁₂	p₄
divisors/prime	2		2		2	prime

rel. index	7	8	9	10	11	12	13	14	15	16	17	18
constellation	e₁₃	c₁₀	e₁₄	c₁₁	e₁₅	c₁₂	e₁₆	c₁₃	e₁₇	c₁₄	e₁₈	p₅
divisors/prime	2		2		2		2		2		2	prime

Divisibility of symmetric 4-tuple prime C(3, 9) by 2, 3 and 5_

Parameter	value	value	value	value	value	value	value	value	value

rel. index	−4	−3	−2	−1	0	1	2	3	4
constellation	p₁	e₁	p₂	e₂	c	e₃	p₃	e₄	p₄
divisors/prime	P	2, 3	P	2	3, 5	2	P	2, 3	P

Divisibility of symmetric six-tuple prime constellations - C(17, 6)_ Center configuration is even second case p3, e4, c2, e5, p4_

Parameter	value	value	value	value	value	value

rel. index	−8	−7	−6	−5	−4	−3
constellation	p₁	e₁	c₁	e₂	p₂	e₃
divisibility/prime	prime	2, 3, 7		2, 5	prime	2

center	p₃	e₄	c₂	e₅	p₄
rel. index	−2	−1	0	1	2
constellation	prime	2, 3	3, 5, 7	2	prime

rel. index	3	4	5	6	7	8
constellation	e₆	p₅	e₇	c₃	e₈	p₆
divisibility/prime	2	prime	2, 3, 7		2	prime

Potential triple prime, C(9, 3)_

Parameter	value	value	value	value	value	value	value	value	value

rel. index	−4	−3	−2	−1	0	1	2	3	4
constellation	p₁	e₁	c₁	e₂	p₂	e₃	c₂	e₄	p₃
divisors/prime	P	2	C	2	P	2	C	2	P

The first few examples of computations according to Table 1_

The pair of double primes.	p₁	p₂	p₃	p₄

The next double primes.					c₆	c₇

sequence 1	11	13	17	19	29	31
sequence 2	18041	18043	18047	18049	18059	18061
sequence 3	97841	97843	97847	97849	97859	97861
sequence 4	165701	165703	165707	165709	165719	165721
sequence 5	392261	392263	392267	392269	392279	392281
sequence 6	663581	663583	663587	663589	663599	663601
sequence 7	1002341	1002343	1002347	1002349	1002359	1002361
sequence 8	1068701	1068703	1068707	1068709	1068719	1068721

The first few symmetric sixtuple constellations C(27, 6), even base case_

constellation	p₁	p₂	p₃	p₄	p₅	p₆

sixtuple prime 1	587	593	599	601	607	613
sixtuple prime 2	19457	19463	19469	19471	19477	19483
sixtuple prime 3	101267	101273	101279	101281	101287	101293
sixtuple prime 4	179807	179813	179819	179821	179827	179833
sixtuple prime 5	193367	193373	193379	193381	193387	193393

The first few symmetric sixtuple constellations with C(17, 6) are given here_

constellation	p₁	p₂	p₃	p₄	p₅	p₆

sixtuple prime 1	7	11	13	17	19	23
sixtuple prime 2	97	101	103	107	109	113
sixtuple prime 3	16057	16061	16063	16067	16069	16073
sixtuple prime 4	19417	19421	19423	19427	19429	19433
sixtuple prime 5	43777	43781	43783	43787	43789	43793
sixtuple prime 6	1091257	1091261	1091263	1091267	1091269	1091273

The first few symmetric seventuple constellations C(61, 7)_

constellation	p₁	p₂	p₃	p₄	p₅	p₆	p₇

seventuple 1	12003179	12003185	12003191	12003197	12003209	12003227	12003221
seventuple 2	14907619	14907625	14907631	14907637	14907649	14907667	14907661
seventuple 3	19755271	19755277	19755283	19755289	19755301	19755319	19755313

Centers of the last 3 10-tuple constellations C(47, 10)_

967352040

4407582630

Divisibility of symmetric sixtuple prime constellations - with C(29, 6)_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆
divisibility	prime	2	3	2	5	2, 3	prime	2	3	2		2, 3

constellation	p₃	e₇	c₅	e₈	p₄
divisibility	prime	2	3, 5	2

constellation	e₉	c₆	e₁₀	c₇	e₁₁	p₅	e₁₂	c₈	e₁₃	c₉	e₁₄	p₆
divisibility	2, 3	2	2, 5	3	2	prime	2, 3	5	2	3	2	prime

Case_2: possible symmetric quintuple primes_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−18	−17	16	−15	−14	−13	−12	−11	−10	−9	−8	−7
constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆
divisors/prime	prime	2		2		2	prime	2		2		2

rel. index	−6	−5	−4	−3	−2	−1
constellation	c₅	e₇	c₆	e₈	c₇	e₉
divisors/prime		2		2		2

rel. index	0
center	p₃
prime	prime

rel. index	1	2	3	4	5	6
constellation	e₁₀	c₈	e₁₁	c₉	e₁₂	c₁₀
divisors/prime	2		2		2

rel. index	7	8	9	10	11	12	13	14	15	16	17	18
constellation	e₁₃	c₁₁	e₁₄	c₁₂	e₁₅	p₄	e₁₆	c₁₃	e₁₇	c₁₄	e₁₈	p₅
divisors/prime	2		2		2	prime	2		2		2	prime

Divisibility of symmetric octuple prime constellations_

Parameter	val	val	val	val	val	val	val	val	val	val

rel. index	−19	−18	−17	−16	−15	−14	−13	−12	−11	−10
constellation	p₁	e₁	p₂	e₂	c₁	e₃	p₃	e₄	p₄	e₅
divis	prime	2, 3	prime	2	3, 5	2, 7	prime	2, 3	prime	2, 5

rel. index	−9	−8	−7	−6	−5	−4	−3	−2	−1	0
constellation	c₂	e₆	c₃	e₇	c₄	e₈	c₅	e₉	c₆	e₁₀
divis	3	2	7	2, 3	5	2	3	2	?	2, 3, 5, 7

rel. index	1	2	3	4	5	6	7	8	9	10
constellation	c₇	e₁₁	c₈	e₁₂	c₉	e₁₃	c₁₀	e₁₄	c₁₁	e₁₅
divis	?	2	3	2	5	2, 3	7	2	3	2, 5

rel. index	11	12	13	14	15	16	17	18	19	20
constellation	p₅	e₁₆	p₆	e₁₇	c₁₂	e₁₈	p₇	e₁₉	p₈	e₂₀
divis	prime	2, 3	prime	2, 7	3, 5	2	prime	2, 3	prime	2, 5

Some Case_2 C(37, 5) 5-tuple symmetric primes_

constellation	p₁	p₂	p₃	p₄	p₅

quintuple 1	18713	18719	18731	18743	18749
quintuple 2	25603	25609	25621	25633	25639
quintuple 3	28051	28057	28069	28081	28087
quintuple 4	31033	31039	31051	31063	31069
quintuple 5	97423	97429	97441	97453	97459
quintuple 6	103651	103657	103669	103681	103687

The two 10-tuples found up to 30 * 1010 with rel_ indices (−17, −13, −11, −1, 1, 11, 13, 17) when entries with indices (−19, 19) in the table in Figure 1 are composite_

10-tuple center	factors of center	entry index −1	entry index +1
30	(2, 3, 5)	prime	composite
113160	(2(3), 3, 5, 23, 41)	composite	composite

Possible symmetric quintuple prime C(25, 5)_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−12	−11	−10	−9	−8	−7	−6	−5	−4	−3	−2	−1
constellation	p₁	e₁	c₁	e₂	c₂	e₃	p₂	e₄	c₃	e₅	c₄	e₆
divisors/prime	prime	2		2		2	prime	2		2		2

rel. index	0
center	p₃
prime	prime

rel. index	1	2	3	4	5	6	7	8	9	10	11	12
constellation	e₇	c₅	e₈	c₆	e₉	p₄	e₁₀	c₇	e₁₁	c₈	e₁₂	p₅
divisors/prime	2		2		2	prime	2		2		2	prime

With rhe rel_ indices (−30, −18, −12, 0, 12, 18, 30) for possible symmetric seven-tuple prime constellations_

Parameter	val	val	val	val	val	val	val	val	val	val	val	val

rel. index	−30	−29	−28	−27	−26	−25	−24	−23	−22	−21	−20	19
constellation	p₁	e₁	c₁	e₂	c₂	e₃	c₃	e₄	c₄	e₅	c₅	e₆

rel. index	−18	−17	−16	−15	−14	−13	−12	−11	−10	−9	−8	7
constellation	p₂	e₇	c₆	e₈	c₇	e₉	p₃	e₁₀	c₈	e₁₁	c₉	e₁₂

rel. index	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5
constellation	c₁₀	e₁₃	c₁₁	e₁₄	c₁₂	e₁₅	p₄	e₁₆	c₁₃	e₁₇	c₁₄	e₁₈

rel. index	6	7	8	9	10	11	12	13	14	15	16	17
constellation	c₁₅	e₁₉	c₁₆	e₂₀	c₁₇	e₂₁	p₅	e₂₂	c₁₈	e₂₃	c₁₉	e₂₄

rel. index	18	19	20	21	22	23	24	25	26	27	28	29
constellation	p₆	e₂₅	c₂₀	e₂₆	c₂₁	e₂₇	c₂₂	e₂₈	c₂₃	e₂₉	c₂₄	e₃₀

rel. index	30
constellation	p₇

Closest double prime to a constellation formed by a double prime pair_

sequence	p₁	d₁	p₂	e₁	c₁	e₂	p₃	d₂	p₄	e₃	c₂
divisors/prime	prime	2, 3	prime	2	3, 5	2	prime	2, 3	prime	2, 5	3
sequence	e₄	c₃	e₅	c₄	e₆	c₅	e₇	c₆	e₈	c₇	e₉
divisors/prime	2		2, 3	5	2	3	2		2, 3, 5		2

Divisibility of possible symmetric six-tuple prime constellation - C(15, 6)_ Even base case_

Parameter	value	value	value	value	value	value

constellation	p₁	e₁	p₂	e₂	c₁	e₃
divisibility/prime	prime	2, 3	prime	2		2

center	p₃	e₄	p₄
constellation	prime	2, 3	prime

constellation	e₅	c₂	e₆	p₅	e₇	p₆
divisibility/prime	2		2	prime	2	prime

Divisibility of symmetric seventuple prime constellations C(61, 7)_

rel. index

−30

−29

−28

−27

−26

−25

−24

−23

−22

−21

−20

constellation

p₁

e₁

c₁

e₂

c₂

e₃

c₃

e₄

c₄

e₅

c₅

e₆

divisibility

prime

2, 3

rel. index

−18

−17

−16

−15

−14

−13

−12

−11

−10

−9

−8

constellation

p₂

e₇

c₆

e₈

c₇

e₉

p₃

e₁₀

c₈

e₁₁

c₉

e₁₂

divisibility

prime

2, 3

prime

2, 3

rel. index

−6

−5

−4

−3

−2

−1

constellation

c₁₀

e₁₃

c₁₁

e₁₄

c₁₂

e₁₅

p₄

e₁₆

c₁₃

e₁₇

c₁₄

e₁₈

divisibility

2, 3

prime

2, 3

rel. index

constellation

c₁₅

e₁₉

c₁₆

e₂₀

c₁₇

e₂₁

p₅

e₂₂

c₁₈

e₂₃

c₁₉

e₂₄

divisibility

2, 3

prime

2, 3

rel. index

constellation

p₆

e₂₅

c₂₀

e₂₆

c₂₁

e₂₇

c₂₂

e₂₈

c₂₃

e₂₉

c₂₄

e₃₀

divisibility

prime

2, 3

rel. index

constellation

p₇

divisibility

prime

DOI: https://doi.org/10.2478/ijmce-2025-0020 | Journal eISSN: 2956-7068

Journal RSS Feed

Language: English

Page range: 267 - 292

Submitted on: Apr 29, 2024

Accepted on: Sep 5, 2024

Published on: Sep 20, 2024

Published by: Harran University

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

Prime,

symmetric prime constellations

Related subjects:

Computer sciences,

Computer sciences, other,

Engineering,

Introductions and overviews,

Physics,

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

Previous article Volume 3 (2025): Issue 2 (December 2025)

Odd and even symmetric prime constellations

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Potential triple prime, C(13, 3)_

The first few symmetric sixtuple constellations, third case, are given with C(29, 6)_

The three 10-tuples found with rel_ indices (−19, −17, −13, −11, −1, 1, 11, 13, 17, 19) when entries with indices (−7, 7) in the table in Figure 1 are composite_

The four 10-tuples found when rel_ indices (−19, −17, −13, −11, −7, 7, 11, 13, 17, 19) are specified with (−1, 1) in the table in Figure 1 being composite_

First and last computed cases for constellation in Table 23_

The first 10-tuple C(39, 10) prime constellation, center c = 39713433690_

Triple prime (47, 53, 59) with m = 13_

Summarizing the even symmetric constellations, smallest extent so far_

Some 4-tuple symmetric primes, C(9, 4)_

9-tuple prime, C(120, 9)_

Divisibility of symmetric sixtuple prime constellations with C(27, 6), even base case_

Centers of the 5 10-tuple constellations C(47, 10)_

Divisibility of symmetric octuple prime constellation - second case_

The first few symmetric eight-tuple constellations C(27, 8) with rel_ indices at (−13, −11, −7, −1, 1, 7, 11, 13)_

Summarizing the odd symmetric constellations, smallest extent so far_

Case_1: possible symmetric quintuple prime_

Divisibility of symmetric 4-tuple prime C(3, 9) by 2, 3 and 5_

Divisibility of symmetric six-tuple prime constellations - C(17, 6)_ Center configuration is even second case p3, e4, c2, e5, p4_

Potential triple prime, C(9, 3)_

The first few examples of computations according to Table 1_

The first few symmetric sixtuple constellations C(27, 6), even base case_

The first few symmetric sixtuple constellations with C(17, 6) are given here_

The first few symmetric seventuple constellations C(61, 7)_

Centers of the last 3 10-tuple constellations C(47, 10)_

Divisibility of symmetric sixtuple prime constellations - with C(29, 6)_

Case_2: possible symmetric quintuple primes_

Divisibility of symmetric octuple prime constellations_

Some Case_2 C(37, 5) 5-tuple symmetric primes_

The two 10-tuples found up to 30 * 1010 with rel_ indices (−17, −13, −11, −1, 1, 11, 13, 17) when entries with indices (−19, 19) in the table in Figure 1 are composite_

Possible symmetric quintuple prime C(25, 5)_

With rhe rel_ indices (−30, −18, −12, 0, 12, 18, 30) for possible symmetric seven-tuple prime constellations_

Closest double prime to a constellation formed by a double prime pair_

Divisibility of possible symmetric six-tuple prime constellation - C(15, 6)_ Even base case_

Divisibility of symmetric seventuple prime constellations C(61, 7)_