Initial members of "L - consecutive prime k-tuplets with the smallest possible distance (D)"

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last updated: 12 February 2022

Consecutive Prime Twins
 L  D  Pattern Z  First initial members
 2  6  Z = 0, 6  [ 5 + d , d = 0, 2 ] + Z
 3  18  Z = 0, 6, 18
 Z = 0, 12, 18		  [ 11 + d , d = 0, 2 ] + Z
 [ 179 + d , d = 0, 2 ] + Z
 4  30  Z = 0, 6, 18, 30
 Z = 0, 12, 24, 30		  [ 11 + d , d = 0, 2 ] + Z
 [ 626597 + d , d = 0, 2 ] + Z
 5  36  Z = 0, 6, 18, 30, 36  [ 39713433671 + d , d = 0, 2 ] + Z
 6  48  Z = 0, 6, 18, 30, 36, 48
 Z = 0, 12, 18, 30, 42, 48 		 [ 1256522812841 + d , d = 0, 2 ] + Z
 [ 45183473856329 + d , d = 0, 2 ] + Z
 7  60  Z = 0, 6, 18, 30, 36, 48, 60
 Z = 0, 12, 24, 30, 42, 54, 60 		 [ 1135141716537971 + d , d = 0, 2 ] + Z
 [ 9725353586573267 + d , d = 0, 2 ] + Z
 8  84  Z = 0, 12, 24, 30, 42, 54, 60, 84
 Z = 0, 24, 30, 42, 54, 60, 72, 84 		 [ 969971533709583960197 + d , d = 0, 2 ] + Z
 [ 865740639783851560847 + d , d = 0, 2 ] + Z
 9  102  
 Z = 0, 12, 30, 42, 54, 60, 84, 90, 102
 Z = 0, 12, 24, 30, 42, 54, 60, 84, 102
 Z = 0, 18, 42, 48, 60, 72, 78, 90, 102
 Z = 0, 12, 18, 42, 48, 60, 72, 90, 102 		  found by Jörg Waldvogel & Peter Leikauf
 [ 35902987875008630158997 + d , d = 0, 2 ] + Z
 [ 670962238726376003928317 + d , d = 0, 2 ] + Z
 [ 1063660630652819772482009 + d , d = 0, 2 ] + Z
 [ 39582971901830749382519 + d , d = 0, 2 ] + Z

Consecutive Prime Triplets
 L  D  Pattern Z  First initial members
 2  6  Z = 0, 6
 Z = 0, 6
  [ 5 + d , d = 0, 2, 6 ] + Z
 [ 7 + d , d = 0, 4, 6 ] + Z
 3  30  Z = 0, 6, 30
 Z = 0, 24, 30
  [ 11 + d , d = 0, 2, 6 ] + Z
 [ 1230343 + d , d = 0, 4, 6 ] + Z
 4  60  Z = 0, 24, 54, 60
 Z = 0, 30, 36, 60
 Z = 0, 6, 36, 60
 Z = 0, 24, 30, 60
  [ 1367736257 + d , d = 0, 2, 6 ] + Z
 [ 3676712951 + d , d = 0, 2, 6 ] + Z
 [ 228847463227 + d , d = 0, 4, 6 ] + Z
 [ 65455701733 + d , d = 0, 4, 6 ] + Z
 5  84  Z = 0, 24, 54, 60, 84
 Z = 0, 24, 30, 60, 84
  [ 98538255329487707 + d , d = 0, 2, 6 ] + Z
 [ 1515672991269843583 + d , d = 0, 4, 6 ] + Z

Consecutive Prime Quadruplets
 L  D  Pattern Z  First initial members
 2  30  Z = 0, 30  [ 1006301 + d , d = 0, 2, 6, 8 ] + Z
 3  120  Z = 0, 30, 120
 Z = 0, 90, 120
  [ 282005261771 + d , d = 0, 2, 6, 8 ] + Z
 [ 31007639083781 + d , d = 0, 2, 6, 8 ] + Z
 4  210
 Z = 0, 30, 120, 210
 Z = 0, 90, 180, 210 		  found by Jörg Waldvogel & Peter Leikauf
 [ 3051450534439926131 + d , d = 0, 2, 6, 8 ] + Z
 [ 300000224101777931 + d , d = 0, 2, 6, 8 ] + Z

Consecutive Prime Quintuplets
 L  D  Pattern Z  First initial members
 2  90
 90		  Z = 0, 90
 Z = 0, 90 		 [ 11 + d , d = 0, 2, 6, 8, 12 ] + Z
 [ 7 + d , d = 0, 4, 6, 10, 12 ] + Z
 3  210  Z = 0, 90, 210
 Z = 0, 120, 210
 Z = 0, 90, 210
 Z = 0, 120, 210 		 [ 2478508456517985161 + d , d = 0, 2, 6, 8, 12 ] + Z
 [ 1830711463147481141 + d , d = 0, 2, 6, 8, 12 ] + Z
 [ 1044428370657137017 + d , d = 0, 4, 6, 10, 12 ] + Z
 [ 89997702037281547 + d , d = 0, 4, 6, 10, 12 ] + Z