Odd and even symmetric prime constellations

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

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

90

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)_

60

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

19

constellation

p1

e1

c1

e2

c2

e3

c3

e4

c4

e5

c5

e6

divisibility

prime

2, 3

2

3

2

2, 3

2

3

2

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

divisibility

prime

2, 3

2

3

2

prime

2, 3

2

3

2

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

divisibility

2, 3

2

3

2

prime

2, 3

2

3

2

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

divisibility

2, 3

2

3

2

prime

2, 3

2

3

2

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

divisibility

prime

2, 3

2

3

2

2, 3

2

3

2

rel. index

30

constellation

p7

divisibility

prime