Siamese Primes: n2 -2 & n2 +2
This is an example of how prime number patterns can be deciphered using the Prime Spiral Sieve
as an analytical tool; in this case Siamese Primes.
Given the fact that all potential prime numbers (p) > 5 are p ≡ 1, 7, 11, 13, 17, 19, 23 or 29 (modulo 30) [dubbed "prime roots," for convenience],
one need simply test all odd numbers in the intervals between the prime roots (3, 5, 9, 15, 21, 25 and 27) against the formula for Siamese Primes
(n2 -2 & n2 +2), and in turn test these results modulo 30 to see which ones possess prime roots, i.e., are potentially prime. Computing
accordingly, the following numbers test positive for prime roots:
n ≡ 3 (modulo 30) (generates mod30, 7 & 11)
n ≡ 9 (modulo 30) (generates mod30, 19 & 23)
n ≡ 15 (modulo 30) (generates mod30, 13 & 17)
n ≡ 21 (modulo 30) (generates mod30, 19 & 23)
n ≡ 27 (modulo 30) (generates mod30, 7 & 11)
Reconfiguring the above into a number sequence for potential values of n we get: 3 {+6+6+6+6+6} {repeat ... ∞} or, simplifying,
6n+3.
The matrix below lists the first 40 integers in the sequence:
Values for n = Potential Generators of Siamese Primes
6n+3
3 |
+6 |
9 |
+6 |
15 |
+6 |
21 |
+6 |
27 |
+6 |
33 |
+6 |
39 |
+6 |
45 |
+6 |
51 |
+6 |
57 |
+6 |
63 |
+6 |
69 |
+6 |
75 |
+6 |
81 |
+6 |
87 |
+6 |
93 |
+6 |
99 |
+6 |
105 |
+6 |
111 |
+6 |
117 |
+6 |
123 |
+6 |
129 |
+6 |
135 |
+6 |
141 |
+6 |
147 |
+6 |
153 |
+6 |
159 |
+6 |
165 |
+6 |
171 |
+6 |
177 |
+6 |
183 |
+6 |
189 |
+6 |
195 |
+6 |
201 |
+6 |
207 |
+6 |
213 |
+6 |
219 |
+6 |
225 |
+6 |
231 |
+6 |
237 |
+6 |
In the knowledge that all potential prime numbers stair-step in intervals of 30, one need simply take
each root value for n (3, 9, 15, 21 & 27) and repeatedly add 30 (i.e., n + 30; n + 60; n + 90 ...) then test the results for
primality as shown in the matrix below, or, alternatively, starting with n = 0 test every 6n+3, as sequenced in the matrix above.
Doing so yields results consistent with all n2 -2 & n2 +2 = Siamese Primes listed by the
OEIS Foundation, with values of n ranging from 3 to 4305.
[Note: Of these the author contributed n = 3621; n = 3807; and n = 4305, which he found readily employing this method.]
Also shown below are the modulo 30 results for all the Siamese Prime pairs themselves.
Root 3 |
|
n |
n2 |
|
n2 -2 |
Prime? |
mod30 |
|
n2 +2 |
Prime? |
mod30 |
|
Siamese? |
n = 3 |
|
3 |
9 |
|
7 |
yes |
7 |
|
11 |
yes |
11 |
|
Yes |
n +30 |
|
33 |
1089 |
|
1087 |
yes |
7 |
|
1091 |
yes |
11 |
|
Yes |
+30 |
|
63 |
3969 |
|
3967 |
yes |
7 |
|
3971 |
no |
11 |
|
No |
+30 |
|
93 |
8649 |
|
8647 |
yes |
7 |
|
8651 |
no |
11 |
|
No |
+30 |
|
123 |
15129 |
|
15127 |
no |
7 |
|
15131 |
yes |
11 |
|
No |
test +30...n |
|
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
|
273 |
74529 |
|
74527 |
yes |
7 |
|
74531 |
yes |
11 |
|
Yes |
↓ |
|
303 |
91809 |
|
91807 |
yes |
7 |
|
91811 |
yes |
11 |
|
Yes |
↓ |
|
513 |
263169 |
|
263167 |
yes |
7 |
|
263171 |
yes |
11 |
|
Yes |
↓ |
|
573 |
328329 |
|
328327 |
yes |
7 |
|
328331 |
yes |
11 |
|
Yes |
↓ |
|
1113 |
1238769 |
|
1238767 |
yes |
7 |
|
1238771 |
yes |
11 |
|
Yes |
↓ |
|
1143 |
1306449 |
|
1306447 |
yes |
7 |
|
1306451 |
yes |
11 |
|
Yes |
↓ |
|
1233 |
1520289 |
|
1520287 |
yes |
7 |
|
1520291 |
yes |
11 |
|
Yes |
↓ |
|
1563 |
2442969 |
|
2442967 |
yes |
7 |
|
2442971 |
yes |
11 |
|
Yes |
↓ |
|
1953 |
3814209 |
|
3814207 |
yes |
7 |
|
3814211 |
yes |
11 |
|
Yes |
↓ |
|
2133 |
4549689 |
|
4549687 |
yes |
7 |
|
4549691 |
yes |
11 |
|
Yes |
↓ |
|
2283 |
5212089 |
|
5212087 |
yes |
7 |
|
5212091 |
yes |
11 |
|
Yes |
↓ |
|
3093 |
9566649 |
|
9566647 |
yes |
7 |
|
9566651 |
yes |
11 |
|
Yes |
↓ |
|
3453 |
11923209 |
|
11923207 |
yes |
7 |
|
11923211 |
yes |
11 |
|
Yes |
Root 9 |
|
n |
n2 |
|
n2 -2 |
Prime? |
mod30 |
|
n2 +2 |
Prime? |
mod30 |
|
Siamese? |
n = 9 |
|
9 |
81 |
|
79 |
yes |
19 |
|
83 |
yes |
23 |
|
Yes |
n +30 |
|
39 |
1521 |
|
1519 |
no |
19 |
|
1523 |
yes |
23 |
|
No |
+30 |
|
69 |
4761 |
|
4759 |
yes |
19 |
|
4763 |
no |
23 |
|
No |
+30 |
|
99 |
9801 |
|
9799 |
no |
19 |
|
9803 |
yes |
23 |
|
No |
+30 |
|
129 |
16641 |
|
16639 |
no |
19 |
|
16643 |
no |
23 |
|
No |
test +30...n |
|
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
|
309 |
95481 |
|
95479 |
yes |
19 |
|
95483 |
yes |
23 |
|
Yes |
↓ |
|
429 |
184041 |
|
184039 |
yes |
19 |
|
184043 |
yes |
23 |
|
Yes |
↓ |
|
609 |
370881 |
|
370879 |
yes |
19 |
|
370883 |
yes |
23 |
|
Yes |
↓ |
|
1239 |
1535121 |
|
1535119 |
yes |
19 |
|
1535123 |
yes |
23 |
|
Yes |
↓ |
|
1749 |
3059001 |
|
3058999 |
yes |
19 |
|
3059003 |
yes |
23 |
|
Yes |
↓ |
|
1839 |
3381921 |
|
3381919 |
yes |
19 |
|
3381923 |
yes |
23 |
|
Yes |
↓ |
|
2589 |
6702921 |
|
6702919 |
yes |
19 |
|
6702923 |
yes |
23 |
|
Yes |
↓ |
|
3549 |
12595401 |
|
12595399 |
yes |
19 |
|
12595403 |
yes |
23 |
|
Yes |
Root 15 |
|
n |
n2 |
|
n2 -2 |
Prime? |
mod30 |
|
n2 +2 |
Prime? |
mod30 |
|
Siamese? |
n = 15 |
|
15 |
225 |
|
223 |
yes |
13 |
|
227 |
yes |
17 |
|
Yes |
n +30 |
|
45 |
2025 |
|
2023 |
no |
13 |
|
2027 |
yes |
17 |
|
No |
+30 |
|
75 |
5625 |
|
5623 |
yes |
13 |
|
5627 |
no |
17 |
|
No |
+30 |
|
105 |
11025 |
|
11023 |
no |
13 |
|
11027 |
yes |
17 |
|
No |
+30 |
|
135 |
18225 |
|
18223 |
yes |
13 |
|
18227 |
no |
17 |
|
No |
test +30...n |
|
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
|
1035 |
1071225 |
|
1071223 |
yes |
13 |
|
1071227 |
yes |
17 |
|
Yes |
↓ |
|
2715 |
7371225 |
|
7371223 |
yes |
13 |
|
7371227 |
yes |
17 |
|
Yes |
↓ |
|
2955 |
8732025 |
|
8732023 |
yes |
13 |
|
8732027 |
yes |
17 |
|
Yes |
↓ |
|
3555 |
12638025 |
|
12638023 |
yes |
13 |
|
12638027 |
yes |
17 |
|
Yes |
↓ |
|
4305 |
18533025 |
|
18533023 |
yes |
13 |
|
18533027 |
yes |
17 |
|
Yes |
Root 21 |
|
n |
n2 |
|
n2 -2 |
Prime? |
mod30 |
|
n2 +2 |
Prime? |
mod30 |
|
Siamese? |
n = 21 |
|
21 |
441 |
|
439 |
yes |
19 |
|
443 |
yes |
23 |
|
Yes |
n +30 |
|
51 |
2601 |
|
2599 |
no |
19 |
|
2603 |
no |
23 |
|
No |
+30 |
|
81 |
6561 |
|
6559 |
no |
19 |
|
6563 |
yes |
23 |
|
No |
+30 |
|
111 |
12321 |
|
12319 |
no |
19 |
|
12323 |
yes |
23 |
|
No |
+30 |
|
141 |
19881 |
|
19879 |
no |
19 |
|
19883 |
no |
23 |
|
No |
test +30...n |
|
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
|
441 |
194481 |
|
194479 |
yes |
19 |
|
194483 |
yes |
23 |
|
Yes |
↓ |
|
561 |
314721 |
|
314719 |
yes |
19 |
|
314723 |
yes |
23 |
|
Yes |
↓ |
|
1071 |
1147041 |
|
1147039 |
yes |
19 |
|
1147043 |
yes |
23 |
|
Yes |
↓ |
|
1311 |
1718721 |
|
1718719 |
yes |
19 |
|
1718723 |
yes |
23 |
|
Yes |
↓ |
|
1611 |
2595321 |
|
2595319 |
yes |
19 |
|
2595323 |
yes |
23 |
|
Yes |
↓ |
|
2211 |
4888521 |
|
4888519 |
yes |
19 |
|
4888523 |
yes |
23 |
|
Yes |
↓ |
|
2721 |
7403841 |
|
7403839 |
yes |
19 |
|
7403843 |
yes |
23 |
|
Yes |
↓ |
|
3081 |
9492561 |
|
9492559 |
yes |
19 |
|
9492563 |
yes |
23 |
|
Yes |
↓ |
|
3621 |
13111641 |
|
13111639 |
yes |
19 |
|
13111643 |
yes |
23 |
|
Yes |
Root 27 |
|
n |
n2 |
|
n2 -2 |
Prime? |
mod30 |
|
n2 +2 |
Prime? |
mod30 |
|
Siamese? |
n = 27 |
|
27 |
729 |
|
727 |
yes |
7 |
|
731 |
no |
11 |
|
No |
n +30 |
|
57 |
3249 |
|
3247 |
yes |
7 |
|
3251 |
no |
11 |
|
No |
+30 |
|
87 |
7569 |
|
7567 |
no |
7 |
|
7571 |
no |
11 |
|
No |
+30 |
|
117 |
13689 |
|
13687 |
no |
7 |
|
13691 |
yes |
11 |
|
No |
+30 |
|
147 |
21609 |
|
21607 |
no |
7 |
|
21611 |
yes |
11 |
|
No |
test +30...n |
|
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
↓ |
|
↓ |
↓ |
|
237 |
56169 |
|
56167 |
yes |
7 |
|
56171 |
yes |
11 |
|
Yes |
↓ |
|
387 |
149769 |
|
149767 |
yes |
7 |
|
149771 |
yes |
11 |
|
Yes |
↓ |
|
447 |
199809 |
|
199807 |
yes |
7 |
|
199811 |
yes |
11 |
|
Yes |
↓ |
|
807 |
651249 |
|
651247 |
yes |
7 |
|
651251 |
yes |
11 |
|
Yes |
↓ |
|
897 |
804609 |
|
804607 |
yes |
7 |
|
804611 |
yes |
11 |
|
Yes |
↓ |
|
1617 |
2614689 |
|
2614687 |
yes |
7 |
|
2614691 |
yes |
11 |
|
Yes |
↓ |
|
1737 |
3017169 |
|
3017167 |
yes |
7 |
|
3017171 |
yes |
11 |
|
Yes |
↓ |
|
1827 |
3337929 |
|
3337927 |
yes |
7 |
|
3337931 |
yes |
11 |
|
Yes |
↓ |
|
3807 |
14493249 |
|
14493247 |
yes |
7 |
|
14493251 |
yes |
11 |
|
Yes |