4 consecutive primes as near to each other as they can possibly be is called a prime 4-tuplet or prime quadruplet or prime decade because, for those starting greater than 10, the last digits (the one's place, unit's place, least significant digit) have the pattern "1 3 7 9"; that is, they all fall in one decade. it is never "3 7 9 1" nor "7 9 1 3" nor "9 1 3 7".
prime gaps of length 10 are aesthetically the dual.
inspired by memorizing primes in order.
below are decades and gaps up to 3011.
the first three prime quadruplets are the weird ones, not having last digits "1 3 7 9".
97 101 103 107 109 113 is a prime 6-tuplet containing two prime 5-tuplets and (only) one prime 4-tuplet. the next prime 5-tuplet is 1481 1483 1487 1489 1493. the next prime 6-tuplet is 16057 16061 16063 16067 16069 16073. prime decades generally must be separated by a multiple of 30 (future post bqazxnke).
we give the gap size only if greater than 10.
decade 2 3 5 7
decade 3 5 7 11
decade 5 7 11 13
decade 11 13 17 19
decade 101 103 107 109
gap 113 127 size 14
gap 139 149
gap 181 191
decade 191 193 197 199
gap 199 211 size 12
gap 211 223 size 12
gap 241 251
gap 283 293
gap 293 307 size 14
gap 317 331 size 14
gap 337 347
gap 409 419
gap 421 431
gap 467 479 size 12
gap 509 521 size 12
gap 523 541 size 18
gap 547 557
gap 577 587
gap 619 631 size 12
gap 631 641
gap 661 673 size 12
gap 691 701
gap 709 719
gap 773 787 size 14
gap 787 797
gap 797 809 size 12
gap 811 821
decade 821 823 827 829
gap 829 839
gap 839 853 size 14
gap 863 877 size 14
gap 887 907 size 20
gap 919 929
gap 953 967 size 14
gap 997 1009 size 12
gap 1021 1031
gap 1039 1049
gap 1051 1061
gap 1069 1087 size 18
gap 1129 1151 size 22
gap 1153 1163
gap 1171 1181
gap 1201 1213 size 12
gap 1237 1249 size 12
gap 1249 1259
gap 1259 1277 size 18
gap 1307 1319 size 12
gap 1327 1361 size 34
gap 1381 1399 size 18
gap 1399 1409
gap 1409 1423 size 14
gap 1459 1471 size 12
gap 1471 1481
decade 1481 1483 1487 1489
gap 1499 1511 size 12
gap 1511 1523 size 12
gap 1531 1543 size 12
gap 1583 1597 size 14
gap 1627 1637
gap 1637 1657 size 20
gap 1669 1693 size 24
gap 1699 1709
gap 1709 1721 size 12
gap 1723 1733
gap 1759 1777 size 18
gap 1789 1801 size 12
gap 1801 1811
gap 1811 1823 size 12
gap 1831 1847 size 16
gap 1847 1861 size 14
decade 1871 1873 1877 1879
gap 1879 1889
gap 1889 1901 size 12
gap 1913 1931 size 18
gap 1933 1949 size 16
gap 1951 1973 size 22
gap 2017 2027
gap 2029 2039
gap 2039 2053 size 14
gap 2053 2063
gap 2069 2081 size 12
decade 2081 2083 2087 2089
gap 2089 2099
gap 2099 2111 size 12
gap 2113 2129 size 16
gap 2143 2153
gap 2161 2179 size 18
gap 2179 2203 size 24
gap 2221 2237 size 16
gap 2251 2267 size 16
gap 2297 2309 size 12
gap 2311 2333 size 22
gap 2357 2371 size 14
gap 2399 2411 size 12
gap 2423 2437 size 14
gap 2447 2459 size 12
gap 2477 2503 size 26
gap 2503 2521 size 18
gap 2521 2531
gap 2557 2579 size 22
gap 2579 2591 size 12
gap 2593 2609 size 16
gap 2621 2633 size 12
gap 2633 2647 size 14
gap 2647 2657
gap 2719 2729
gap 2731 2741
gap 2753 2767 size 14
gap 2767 2777
gap 2777 2789 size 12
gap 2803 2819 size 16
gap 2819 2833 size 14
gap 2861 2879 size 18
gap 2887 2897
gap 2917 2927
gap 2927 2939 size 12
gap 2939 2953 size 14
gap 2971 2999 size 28
gap 3001 3011
gaps become increasingly common as primes thin out, so we stop listing them. here are just decades from 3000 to a million:
decade 3251 3253 3257 3259
decade 3461 3463 3467 3469
decade 5651 5653 5657 5659
decade 9431 9433 9437 9439
decade 13001 13003 13007 13009
decade 15641 15643 15647 15649
decade 15731 15733 15737 15739
decade 16061 16063 16067 16069
decade 18041 18043 18047 18049
decade 18911 18913 18917 18919
decade 19421 19423 19427 19429
decade 21011 21013 21017 21019
decade 22271 22273 22277 22279
decade 25301 25303 25307 25309
decade 31721 31723 31727 31729
decade 34841 34843 34847 34849
decade 43781 43783 43787 43789
decade 51341 51343 51347 51349
decade 55331 55333 55337 55339
decade 62981 62983 62987 62989
decade 67211 67213 67217 67219
decade 69491 69493 69497 69499
decade 72221 72223 72227 72229
decade 77261 77263 77267 77269
decade 79691 79693 79697 79699
decade 81041 81043 81047 81049
decade 82721 82723 82727 82729
decade 88811 88813 88817 88819
decade 97841 97843 97847 97849
decade 99131 99133 99137 99139
decade 101111 101113 101117 101119
decade 109841 109843 109847 109849
decade 116531 116533 116537 116539
decade 119291 119293 119297 119299
decade 122201 122203 122207 122209
decade 135461 135463 135467 135469
decade 144161 144163 144167 144169
decade 157271 157273 157277 157279
decade 165701 165703 165707 165709
decade 166841 166843 166847 166849
decade 171161 171163 171167 171169
decade 187631 187633 187637 187639
decade 194861 194863 194867 194869
decade 195731 195733 195737 195739
decade 201491 201493 201497 201499
decade 201821 201823 201827 201829
decade 217361 217363 217367 217369
decade 225341 225343 225347 225349
decade 240041 240043 240047 240049
decade 243701 243703 243707 243709
decade 247601 247603 247607 247609
decade 247991 247993 247997 247999
decade 257861 257863 257867 257869
decade 260411 260413 260417 260419
decade 266681 266683 266687 266689
decade 268811 268813 268817 268819
decade 276041 276043 276047 276049
decade 284741 284743 284747 284749
decade 285281 285283 285287 285289
decade 294311 294313 294317 294319
decade 295871 295873 295877 295879
decade 299471 299473 299477 299479
decade 300491 300493 300497 300499
decade 301991 301993 301997 301999
decade 326141 326143 326147 326149
decade 334421 334423 334427 334429
decade 340931 340933 340937 340939
decade 346391 346393 346397 346399
decade 347981 347983 347987 347989
decade 354251 354253 354257 354259
decade 358901 358903 358907 358909
decade 361211 361213 361217 361219
decade 375251 375253 375257 375259
decade 388691 388693 388697 388699
decade 389561 389563 389567 389569
decade 392261 392263 392267 392269
decade 394811 394813 394817 394819
decade 397541 397543 397547 397549
decade 397751 397753 397757 397759
decade 402131 402133 402137 402139
decade 402761 402763 402767 402769
decade 412031 412033 412037 412039
decade 419051 419053 419057 419059
decade 420851 420853 420857 420859
decade 427241 427243 427247 427249
decade 442571 442573 442577 442579
decade 444341 444343 444347 444349
decade 452531 452533 452537 452539
decade 463451 463453 463457 463459
decade 465161 465163 465167 465169
decade 467471 467473 467477 467479
decade 470081 470083 470087 470089
decade 477011 477013 477017 477019
decade 490571 490573 490577 490579
decade 495611 495613 495617 495619
decade 500231 500233 500237 500239
decade 510611 510613 510617 510619
decade 518801 518803 518807 518809
decade 536441 536443 536447 536449
decade 536771 536773 536777 536779
decade 539501 539503 539507 539509
decade 549161 549163 549167 549169
decade 559211 559213 559217 559219
decade 563411 563413 563417 563419
decade 570041 570043 570047 570049
decade 572651 572653 572657 572659
decade 585911 585913 585917 585919
decade 594821 594823 594827 594829
decade 597671 597673 597677 597679
decade 607301 607303 607307 607309
decade 622241 622243 622247 622249
decade 626621 626623 626627 626629
decade 632081 632083 632087 632089
decade 632321 632323 632327 632329
decade 633461 633463 633467 633469
decade 633791 633793 633797 633799
decade 654161 654163 654167 654169
decade 657491 657493 657497 657499
decade 661091 661093 661097 661099
decade 663581 663583 663587 663589
decade 664661 664663 664667 664669
decade 666431 666433 666437 666439
decade 680291 680293 680297 680299
decade 681251 681253 681257 681259
decade 691721 691723 691727 691729
decade 705161 705163 705167 705169
decade 715151 715153 715157 715159
decade 734471 734473 734477 734479
decade 736361 736363 736367 736369
decade 739391 739393 739397 739399
decade 768191 768193 768197 768199
decade 773021 773023 773027 773029
decade 795791 795793 795797 795799
decade 803441 803443 803447 803449
decade 814061 814063 814067 814069
decade 822161 822163 822167 822169
decade 823721 823723 823727 823729
decade 829721 829723 829727 829729
decade 833711 833713 833717 833719
decade 837071 837073 837077 837079
decade 845981 845983 845987 845989
decade 854921 854923 854927 854929
decade 855731 855733 855737 855739
decade 857951 857953 857957 857959
decade 875261 875263 875267 875269
decade 876011 876013 876017 876019
decade 881471 881473 881477 881479
decade 889871 889873 889877 889879
decade 907391 907393 907397 907399
decade 930071 930073 930077 930079
decade 938051 938053 938057 938059
decade 946661 946663 946667 946669
decade 954971 954973 954977 954979
decade 958541 958543 958547 958549
decade 959471 959473 959477 959479
decade 976301 976303 976307 976309
decade 978071 978073 978077 978079
decade 983441 983443 983447 983449
prime decades can only occur in decades 10*(3*n + 1) + {1,3,7,9}. here are "composite decades" when all numbers in a candidate prime decade are composite:
1331 = 11^3
1333 = 31 * 43
1337 = 7 * 191
1339 = 13 * 103
1961 = 37 * 53
1963 = 13 * 151
1967 = 7 * 281
1969 = 11 * 179
2321 = 11 * 211
2323 = 23 * 101
2327 = 13 * 179
2329 = 17 * 137
2561 = 13 * 197
2563 = 11 * 233
2567 = 17 * 151
2569 = 7 * 367
2981 = 11 * 271
2983 = 19 * 157
2987 = 29 * 103
2989 = 7^2 * 61
3281 = 17 * 193
3283 = 7^2 * 67
3287 = 19 * 173
3289 = 11 * 13 * 23
3971 = 11 * 19^2
3973 = 29 * 137
3977 = 41 * 97
3979 = 23 * 173
4031 = 29 * 139
4033 = 37 * 109
4037 = 11 * 367
4039 = 7 * 577
4061 = 31 * 131
4063 = 17 * 239
4067 = 7^2 * 83
4069 = 13 * 313
4181 = 37 * 113
4183 = 47 * 89
4187 = 53 * 79
4189 = 59 * 71
4301 = 11 * 17 * 23
4303 = 13 * 331
4307 = 59 * 73
4309 = 31 * 139
4571 = 7 * 653
4573 = 17 * 269
4577 = 23 * 199
4579 = 19 * 241
4841 = 47 * 103
4843 = 29 * 167
4847 = 37 * 131
4849 = 13 * 373
6401 = 37 * 173
6403 = 19 * 337
6407 = 43 * 149
6409 = 13 * 17 * 29
6431 = 59 * 109
6433 = 7 * 919
6437 = 41 * 157
6439 = 47 * 137
6641 = 29 * 229
6643 = 7 * 13 * 73
6647 = 17^2 * 23
6649 = 61 * 109
7091 = 7 * 1013
7093 = 41 * 173
7097 = 47 * 151
7099 = 31 * 229
7271 = 11 * 661
7273 = 7 * 1039
7277 = 19 * 383
7279 = 29 * 251
7421 = 41 * 181
7423 = 13 * 571
7427 = 7 * 1061
7429 = 17 * 19 * 23
8021 = 13 * 617
8023 = 71 * 113
8027 = 23 * 349
8029 = 7 * 31 * 37
8471 = 43 * 197
8473 = 37 * 229
8477 = 7^2 * 173
8479 = 61 * 139
8651 = 41 * 211
8653 = 17 * 509
8657 = 11 * 787
8659 = 7 * 1237
8981 = 7 * 1283
8983 = 13 * 691
8987 = 11 * 19 * 43
8989 = 89 * 101
9071 = 47 * 193
9073 = 43 * 211
9077 = 29 * 313
9079 = 7 * 1297
9701 = 89 * 109
9703 = 31 * 313
9707 = 17 * 571
9709 = 7 * 19 * 73
9911 = 11 * 17 * 53
9913 = 23 * 431
9917 = 47 * 211
9919 = 7 * 13 * 109
two decades in a row can have the same non-empty prime pattern (same last digits) only when one decade is a candidate (but failed) prime decade 10*(3*n + 1). here are those repeated patterns, with the candidate (but failed) prime decade marked "decade".
373 379 decade
383 389
787
797 decade
1249 decade
1259
1399 decade
1409
1543 1549 decade
1553 1559
1801
1811 decade
2341 2347
2351 2357 decade
2767
2777 decade
2887
2897 decade
2917
2927 decade
3109 decade
3119
3313 3319 decade
3323 3329
4363 decade
4373
4441 4447
4451 4457 decade
4993 4999 decade
5003 5009
5323 decade
5333
5581
5591 decade
6091
6101 decade
6163 decade
6173
6247
6257 decade
6481
6491 decade
6961 6967
6971 6977 decade
7069 decade
7079
7177
7187 decade
7507
7517 decade
8209 decade
8219
8563 decade
8573
8731 8737
8741 8747 decade
8941
8951 decade
9511
9521 decade
9613 9619 decade
9623 9629
9733 9739 decade
9743 9749
Pari/GP code for prime decades:
sz=4; width=8; a=List(vector(sz)); forprime(p=0, 10^5, listpop(a, 1); listput(a, p); if(a[1] && (a[sz]-a[1]<=width), print1("decade");for(i=1, sz, print1(" ",a[i])); print))
to explore prime constellations other than prime quadruplets (sz=4, width=8), set (sz,width) to one of the following: (2,2) (3,6) (4,8) (5,12) (6,16) (7,20) and more terms at OEIS A008407. (future work jqvjozns)
prime gaps:
old=2; forprime(p=0, 3012, d=p-old; if(d>=10, print1("gap "old" "p); if(d>10, print(" size "d), print())); old=p)
composite decades:
decade=[1,3,7,9]; for(i=0, 100, d=30*i+10; z=1; for(j=1, 4, if(isprime(d+decade[j]), z=0; break)); if(z, print(); for(j=1, 4 ,e=d+decade[j]; print(e" ", factor(e)))))
consecutive decades having the same prime pattern (but not both empty):
decade=[1,3,7,9];
pattern(n)=my(s=0); for(j=0, 3, i=4-j; s*=2; s+=isprime(n+decade[i])); s;
listdecade(n,trail)=for(i=n, n+9, if(isprime(i), print1(" "i))); print(trail);
for(i=0, 400, n=10*(3*i+1); p=pattern(n); if(!p,next); if(pattern(n-10)==p, listdecade(n-10," "); listdecade(n," decade"); print()); if(pattern(n+10)==p, listdecade(n," decade "); listdecade(n+10,""); print()))
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