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Old 2019-06-07, 16:24   #1
robert44444uk
 
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Jun 2003
Oxford, UK

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Default Prime gap following a twin pair

I'm a bit surprised to see that Sloane does not have the sequence 2,4,4,4,6,4,6,6,4,4..

These are the gaps between the larger of a pair of twins and the next prime. Results for all twins up to 10000 are shown below:

Col A & B = twin pair
Col C = the next prime
Col D - the gap to the next prime

There are no gaps 2mod6 other than the first !! All twins with primes >3 are 5mod6 and 1 mod6, where 1mod6 is the larger. Hence a gap of 2mod6 would provide the next prime as 3mod6, which is divisible by 3. Contradiction.

My hunch is that these gaps are larger than regular prime gaps...I wonder if they are?
Code:
3 5 7 2
5 7 11 4
11 13 17 4
17 19 23 4
29 31 37 6
41 43 47 4
59 61 67 6
71 73 79 6
101 103 107 4
107 109 113 4
137 139 149 10
149 151 157 6
179 181 191 10
191 193 197 4
197 199 211 12
227 229 233 4
239 241 251 10
269 271 277 6
281 283 293 10
311 313 317 4
347 349 353 4
419 421 431 10
431 433 439 6
461 463 467 4
521 523 541 18
569 571 577 6
599 601 607 6
617 619 631 12
641 643 647 4
659 661 673 12
809 811 821 10
821 823 827 4
827 829 839 10
857 859 863 4
881 883 887 4
1019 1021 1031 10
1031 1033 1039 6
1049 1051 1061 10
1061 1063 1069 6
1091 1093 1097 4
1151 1153 1163 10
1229 1231 1237 6
1277 1279 1283 4
1289 1291 1297 6
1301 1303 1307 4
1319 1321 1327 6
1427 1429 1433 4
1451 1453 1459 6
1481 1483 1487 4
1487 1489 1493 4
1607 1609 1613 4
1619 1621 1627 6
1667 1669 1693 24
1697 1699 1709 10
1721 1723 1733 10
1787 1789 1801 12
1871 1873 1877 4
1877 1879 1889 10
1931 1933 1949 16
1949 1951 1973 22
1997 1999 2003 4
2027 2029 2039 10
2081 2083 2087 4
2087 2089 2099 10
2111 2113 2129 16
2129 2131 2137 6
2141 2143 2153 10
2237 2239 2243 4
2267 2269 2273 4
2309 2311 2333 22
2339 2341 2347 6
2381 2383 2389 6
2549 2551 2557 6
2591 2593 2609 16
2657 2659 2663 4
2687 2689 2693 4
2711 2713 2719 6
2729 2731 2741 10
2789 2791 2797 6
2801 2803 2819 16
2969 2971 2999 28
2999 3001 3011 10
3119 3121 3137 16
3167 3169 3181 12
3251 3253 3257 4
3257 3259 3271 12
3299 3301 3307 6
3329 3331 3343 12
3359 3361 3371 10
3371 3373 3389 16
3389 3391 3407 16
3461 3463 3467 4
3467 3469 3491 22
3527 3529 3533 4
3539 3541 3547 6
3557 3559 3571 12
3581 3583 3593 10
3671 3673 3677 4
3767 3769 3779 10
3821 3823 3833 10
3851 3853 3863 10
3917 3919 3923 4
3929 3931 3943 12
4001 4003 4007 4
4019 4021 4027 6
4049 4051 4057 6
4091 4093 4099 6
4127 4129 4133 4
4157 4159 4177 18
4217 4219 4229 10
4229 4231 4241 10
4241 4243 4253 10
4259 4261 4271 10
4271 4273 4283 10
4337 4339 4349 10
4421 4423 4441 18
4481 4483 4493 10
4517 4519 4523 4
4547 4549 4561 12
4637 4639 4643 4
4649 4651 4657 6
4721 4723 4729 6
4787 4789 4793 4
4799 4801 4813 12
4931 4933 4937 4
4967 4969 4973 4
5009 5011 5021 10
5021 5023 5039 16
5099 5101 5107 6
5231 5233 5237 4
5279 5281 5297 16
5417 5419 5431 12
5441 5443 5449 6
5477 5479 5483 4
5501 5503 5507 4
5519 5521 5527 6
5639 5641 5647 6
5651 5653 5657 4
5657 5659 5669 10
5741 5743 5749 6
5849 5851 5857 6
5867 5869 5879 10
5879 5881 5897 16
6089 6091 6101 10
6131 6133 6143 10
6197 6199 6203 4
6269 6271 6277 6
6299 6301 6311 10
6359 6361 6367 6
6449 6451 6469 18
6551 6553 6563 10
6569 6571 6577 6
6659 6661 6673 12
6689 6691 6701 10
6701 6703 6709 6
6761 6763 6779 16
6779 6781 6791 10
6791 6793 6803 10
6827 6829 6833 4
6869 6871 6883 12
6947 6949 6959 10
6959 6961 6967 6
7127 7129 7151 22
7211 7213 7219 6
7307 7309 7321 12
7331 7333 7349 16
7349 7351 7369 18
7457 7459 7477 18
7487 7489 7499 10
7547 7549 7559 10
7559 7561 7573 12
7589 7591 7603 12
7757 7759 7789 30
7877 7879 7883 4
7949 7951 7963 12
8009 8011 8017 6
8087 8089 8093 4
8219 8221 8231 10
8231 8233 8237 4
8291 8293 8297 4
8387 8389 8419 30
8429 8431 8443 12
8537 8539 8543 4
8597 8599 8609 10
8627 8629 8641 12
8819 8821 8831 10
8837 8839 8849 10
8861 8863 8867 4
8969 8971 8999 28
8999 9001 9007 6
9011 9013 9029 16
9041 9043 9049 6
9239 9241 9257 16
9281 9283 9293 10
9341 9343 9349 6
9419 9421 9431 10
9431 9433 9437 4
9437 9439 9461 22
9461 9463 9467 4
9629 9631 9643 12
9677 9679 9689 10
9719 9721 9733 12
9767 9769 9781 12
9857 9859 9871 12
9929 9931 9941 10

Last fiddled with by robert44444uk on 2019-06-07 at 16:28
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