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)

Maybe you would like to extend this tables.

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