Problems & Puzzles: Puzzles

Puzzle 11.- Distinct, Increasing & Decreasing Gaps

Another way of viewing primes is by its gaps. A Gap between consecutive primes is defined this way:

Gap = g i = p i+1 - p i

With only one exception, all the gaps are even numbers. I have been calculating the earliest occurrence of certain conditions of the gaps, namely :

  1. The first N distinct consecutive gaps
  2. The first N increasing consecutive gaps
  3. The first N decreasing consecutive gaps

Here are the results

"Primes & Gaps for the first N distinct consecutive Gaps"

N Prime (gap) Prime…
2 2 (1) 3 (2) 5
3 17 ( 2 ) 19 ( 4 ) 23 ( 6 ) 29
4 83 ( 6 ) 89 ( 8 ) 97 ( 4 ) 101 ( 2 ) 103
5 113 ( 14 ) 127 ( 4 ) 131 ( 6 ) 137 ( 2 ) 139 ( 10 ) 149
6 491 ( 8 ) 499 ( 4 ) 503 ( 6 ) 509 ( 12 ) 521 ( 2 ) 523 ( 18 ) 541
7 1367 ( 6 ) 1373 ( 8 ) 1381 ( 18 ) 1399 ( 10 ) 1409 ( 14 ) 1423 ( 4 ) 1427 ( 2 ) 1429
8 1801 ( 10 ) 1811 ( 12 ) 1823 ( 8 ) 1831 ( 16 ) 1847 ( 14 ) 1861 ( 6 ) 1867 ( 4) 1871 ( 2 ) 1873
9 5869 ( 10 ) 5879 ( 2 ) 5881 ( 16 ) 5897 ( 6 ) 5903 ( 20 ) 5923 ( 4 ) 5927 ( 12) 5939 ( 14 ) 5953 ( 28 ) 5981
10 15919 ( 4 ) 15923 ( 14 ) 15937 ( 22 ) 15959 ( 12 ) 15971 ( 2 ) 15973 ( 18 ) 15991 ( 10 ) 16001 ( 6 ) 16007 ( 26 ) 16033 ( 24 ) 16057
11 34883 ( 14 ) 34897 ( 16 ) 34913 ( 6 ) 34919 ( 20 ) 34939 ( 10 ) 34949 ( 12 ) 34961 ( 2 ) 34963 ( 18 ) 34981 ( 42 ) 35023 ( 4 ) 35027 ( 24 ) 35051
12 70639 ( 18 ) 70657 ( 6 ) 70663 ( 4 ) 70667 ( 20 ) 70687 ( 22 ) 70709 ( 8 ) 70717 ( 12 ) 70729 ( 24 ) 70753 ( 16 ) 70769 ( 14 ) 70783 ( 10 ) 70793 ( 30 ) 70823
13 70657 ( 6 ) 70663 ( 4 ) 70667 ( 20 ) 70687 ( 22 ) 70709 ( 8 ) 70717 ( 12 ) 70729 ( 24 ) 70753 ( 16 ) 70769 ( 14 ) 70783 ( 10 ) 70793 ( 30 ) 70823 ( 18 ) 70841( 2 ) 70843
14 214867 ( 16 ) 214883 ( 8 ) 214891 ( 22 ) 214913 ( 26 ) 214939 ( 4 ) 214943 ( 24 ) 214967 ( 20 ) 214987 ( 6 ) 214993 ( 58 ) 215051 ( 12 ) 215063 ( 14 ) 215077 ( 10 ) 215087 ( 36 ) 215123 ( 18 ) 215141
15 214867 ( 16 ) 214883 ( 8 ) 214891 ( 22 ) 214913 ( 26 ) 214939 ( 4 ) 214943 ( 24 ) 214967 ( 20 ) 214987 ( 6 ) 214993 ( 58 ) 215051 ( 12 ) 215063 ( 14 ) 215077 ( 10 ) 215087 ( 36 ) 215123 ( 18 ) 215141 ( 2 ) 215143
16 2515871 ( 2 ) 2515873 ( 6 ) 2515879 ( 18 ) 2515897 ( 10 ) 2515907 ( 14 ) 2515921 ( 30 ) 2515951 ( 16 ) 2515967 ( 54 ) 2516021 ( 36 ) 2516057 ( 20 ) 2516077 ( 12 ) 2516089 ( 34 ) 2516123 ( 74 ) 2516197 ( 4 ) 2516201 ( 8 ) 2516209 ( 24 ) 2516233
17 3952733 ( 26 ) 3952759 ( 4 ) 3952763 ( 8 ) 3952771 ( 42 ) 3952813 ( 60 ) 3952873 ( 36 ) 3952909 ( 10 ) 3952919 ( 18 ) 3952937 ( 30 ) 3952967 ( 14 ) 3952981 ( 16 ) 3952997 ( 12 ) 3953009 ( 2 ) 3953011 ( 6 ) 3953017 ( 34 ) 3953051 ( 20 ) 3953071 ( 22 ) 3953093
18 13010143 ( 4 ) 13010147 ( 12 ) 13010159 ( 32 ) 13010191 ( 6 ) 13010197 ( 16 ) 13010213 ( 80 ) 13010293 ( 24 ) 13010317 ( 22 ) 13010339 ( 72 ) 13010411 ( 8 ) 13010419 ( 28 ) 13010447 ( 30 ) 13010477 ( 84 ) 13010561 ( 2 ) 13010563 ( 46 ) 13010609 ( 14 ) 13010623 ( 10 ) 13010633 ( 26 ) 13010659
19 30220163 ( 8 ) 30220171 ( 36 ) 30220207 ( 12 ) 30220219 ( 54 ) 30220273 ( 4 ) 30220277 ( 50 ) 30220327 ( 16 ) 30220343 ( 18 ) 30220361 ( 26 ) 30220387 ( 10 ) 30220397 ( 32 ) 30220429 ( 28 ) 30220457 ( 14 ) 30220471 ( 22 ) 30220493 ( 6 ) 30220499 ( 2 ) 30220501 ( 70 ) 30220571 ( 20 ) 30220591 ( 42 ) 30220633
20 60155567 ( 30 ) 60155597 ( 14 ) 60155611 ( 48 ) 60155659 ( 28 ) 60155687 ( 56 ) 60155743 ( 16 ) 60155759 ( 2 ) 60155761 ( 46 ) 60155807 ( 12 ) 60155819 ( 38 ) 60155857 ( 4 ) 60155861 ( 6 ) 60155867 ( 32 ) 60155899 ( 24 ) 60155923 ( 10 ) 60155933 ( 18 ) 60155951 ( 8 ) 60155959 ( 22 ) 60155981 ( 20 ) 60156001 ( 36 ) 60156037
36 N=36
1625800359439 (22) 1625800359461 (50) 1625800359511 (102) 1625800359613 (40) 1625800359653 (26) 1625800359679 (82) 1625800359761 (96) 1625800359857 (72) 1625800359929 (32) 1625800359961 (112) 1625800360073 (36) 1625800360109 (2) 1625800360111 (30) 1625800360141 (48) 1625800360189 (70) 1625800360259 (8) 1625800360267 (24) 1625800360291 (16) 1625800360307 (62) 1625800360369 (34) 1625800360403 (6) 1625800360409 (12) 1625800360421 (20) 1625800360441 (52) 1625800360493 (14) 1625800360507 (60) 1625800360567 (42) 1625800360609 (88) 1625800360697 (74) 1625800360771 (10) 1625800360781 (38) 1625800360819 (4) 1625800360823 (56) 1625800360879 (58) 1625800360937 (134) 1625800361071 (28) 1625800361099 ( Gennady Gusev, 12/5/01, of course that he produced all the series from N=21 to N=36, but I have published only the last )

"Primes & Gaps for the first N increasing consecutive Gaps"

N Prime (gap) Prime…
2 2 (1) 3 (2) 5
3 17 ( 2 ) 19 ( 4 ) 23 ( 6 ) 29
4 347 ( 2 ) 349 ( 4 ) 353 ( 6 ) 359 ( 8 ) 367
5 2903 ( 6 ) 2909 ( 8 ) 2917 ( 10 ) 2927 ( 12 ) 2939 ( 14 ) 2953
6 15373 ( 4 ) 15377 ( 6 ) 15383 ( 8 ) 15391 ( 10 ) 15401 ( 12 ) 15413 ( 14 ) 1542
7 128981 ( 2 ) 128983 ( 4 ) 128987 ( 6 ) 128993 ( 8 ) 129001 ( 10 ) 129011 ( 12 ) 129023 ( 14 ) 129037
8 1319407 ( 4 ) 1319411 ( 8 ) 1319419 ( 10 ) 1319429 ( 14 ) 1319443 ( 16 ) 1319459 ( 18 ) 1319477 ( 32 ) 1319509 ( 34 ) 1319543
9 17797517(2) 17797519(4) 17797523(8) 17797531(10) 17797541(12) 17797553(20) 17797573(28) 17797601(42) 17797643(50) 17797693
10 94097537(2) 94097539(4) 94097543(8) 94097551(10) 94097561(12) 94097573(14) 94097587(16) 94097603(18) 94097621(30) 94097651(32) 94097683
11 6927837557 (2) 6927837559 (4) 6927837563 (8) 6927837571 (12) 6927837583 (16) 6927837599 (18) 6927837617 (24) 6927837641 (32) 6927837673 (40) 6927837713 (44) 6927837757 (70) 6927837827, by Gennady Gusev
12 48486712783 (4) 48486712787 (6) 48486712793 (8) 48486712801 (10) 48486712811 (12) 48486712823 (14) 48486712837 (16) 48486712853 (18) 48486712871 (20) 48486712891 (22) 48486712913 (36) 48486712949 (38) 48486712987 by Gennady Gusev
13 N=13
968068681511 (8) 968068681519 (10) 968068681529 (14) 968068681543 (18) 968068681561 (22) 968068681583 (26) 968068681609 (28) 968068681637 (30) 968068681667 (36) 968068681703 (44) 968068681747 (46) 968068681793 (48) 968068681841 (50) 968068681891 ( Gennady Gusev, 12/5/01)
14 N=14
1472840004017 (2) 1472840004019 (4) 1472840004023 (6) 1472840004029 (8) 1472840004037 (10) 1472840004047 (12) 1472840004059 (14) 1472840004073 (28) 1472840004101 (30) 1472840004131 (38)1472840004169 (48) 1472840004217 (64) 1472840004281 (66) 1472840004347 (74) 1472840004421 ( Gennady Gusev, 12/5/01)

 

"Primes & Gaps for the first N decreasing consecutive Gaps"

N Prime (gap) Prime…
2 7 ( 4 ) 11 ( 2 ) 13
3 31 ( 6 ) 37 ( 4 ) 41 ( 2 ) 43
4 1637 ( 20 ) 1657 ( 6 ) 1663 ( 4 ) 1667 ( 2 ) 1669
5 1831 ( 16 ) 1847 ( 14 ) 1861 ( 6 ) 1867 ( 4 ) 1871 ( 2 ) 1873
6 74653 ( 34 ) 74687 ( 12 ) 74699 ( 8 ) 74707 ( 6 ) 74713 ( 4 ) 74717 ( 2 ) 74719
7 322171 ( 22 ) 322193 ( 20 ) 322213 ( 16 ) 322229 ( 8 ) 322237 ( 6 ) 322243 ( 4) 322247 ( 2 ) 322249
8 5051309 ( 32 ) 5051341 ( 28 ) 5051369 ( 14 ) 5051383 ( 10 ) 5051393 ( 8 ) 5051401 ( 6 ) 5051407 ( 4 ) 5051411 ( 2 ) 5051413
9 11938793 ( 60 ) 11938853 ( 38 ) 11938891 ( 28 ) 11938919 ( 14 ) 11938933 ( 10 ) 11938943 ( 8 ) 11938951 ( 6 ) 11938957 ( 4 ) 11938961 ( 2 ) 11938963
10 245333159 ( 54 ) 245333213 ( 20 ) 245333233 ( 18 ) 245333251 ( 16 ) 245333267 ( 14 ) 245333281 ( 12 ) 245333293 ( 10 ) 245333303 ( 8 ) 245333311 ( 6 ) 245333317 ( 4 ) 245333321
11 245333159 ( 54 ) 245333213 ( 20 ) 245333233 ( 18 ) 245333251 ( 16 ) 245333267 (14 ) 245333281 ( 12 ) 245333293 ( 10 ) 245333303 ( 8 ) 245333311 ( 6 ) 24533331 7 ( 4 ) 245333321 ( 2 ) 245333323
12 130272314561 (96) 130272314657 (44) 130272314701 (40) 130272314741 (38) 130272314779 (28) 130272314807 (20) 130272314827 (12) 130272314839 (10) 130272314849 (8) 130272314857 (6) 130272314863 (4) 130272314867 (2) 130272314869 by Gennady Gusev 
13 N=13
1273135176799 (72) 1273135176871 (60) 1273135176931 (46) 1273135176977 (44) 1273135177021 (42) 1273135177063 (36) 1273135177099 (34) 1273135177133 (24) 1273135177157 (12) 1273135177169 (8)1273135177177 (6) 1273135177183 (4) 1273135177187 (2) 1273135177189 ( Gennady Gusev, 12/5/01)

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