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