## Primes in network

I call prime numbers in network a sequence of primes in arithmetic and/or geometric progression with more than one dimension.

This sequence can be made of consecutive primes.

### "2D Primes":

For arithmetic progressions, the problem is to find primes with a gap g1 "horizontally" and g2 "vertically" :

The number at the top-left of tables indicates the number of primes, c indicates consecutive primes.

2D Twins :

 9 42 --> Twins 90 | V 17 59 101 9 42 --> 107 149 191 90 | V 19 61 103 197 239 281 109 151 193 Twins 199 241 283

2D Network :

 18 60 --> 798 | V - - 224033 - 224153 - 224711 224771 224831 224891 224951 - 225509 225569 225629 225689 225749 225809 226307 226367 226427 226487 226547 -

 19 90 --> 378 | V 1973 2063 2153 2243 2333 2423 2351 2441 2531 2621 2711 2801 2729 2819 2909 2999 3089 - - - - - 3467 3557

2D Staircase :

 7c 2-->(Twins) 4 |  V 5 7 11 13 17 19 23

If g1=2 we also have Twins in arithmetic progression of difference g2 :

 8c 10 --> 8 | V 67944073 67944083 67944091 67944101 67944109 67944119 67944127 67944137

 8c 2 --> (Twins) 28 | V 263872067 263872069 263872097 263872099 263872127 263872129 263872157 263872159

2D 4x2 :
 8 6 --> 30 | 11 17 23 29 41 47 53 59

 8c 6 --> 20 | 344231 344237 344243 344249 344251 344257 344263 344269

2D Triangular :

 6 6 --> 24 | V 13 37 43 61 67 73

 6c 6 --> 24 | V 4116419 4116443 4116449 4116467 4116473 4116479

### "3D Primes":

For arithmetic progressions, the problem is to find primes with a gap g1 "horizontally", g2 "vertically" and g3 "in depth".

Below, here is the smallest with 8 consecutive primes (g1=4,g2=6,g3=24) :

It is also possible to write them easily with symetric sequences of their differences, with the starting prime :

• "3D" : {13 ;4,2,4,14,4,2,4}
• "3D consecutive" : {853 ;4,2,4,14,4,2,4}

### Primes in network and k-tuplets :

We have the possibility to transform them in particular k-tuplets, for example :

• "2D Triangular consecutive" : {4116419 + 0,24,30,48,54,60}
• "3D" : {13 + 0,4,6,10,24,28,30,34}
• "3D consecutive" : {853 + 0,4,6,10,24,28,30,34}

### Twins in arithmetic progression (consecutive) :

If we consider the particular case g1=2, g2 increasing and the smallest p1,p2,p3 corresponding, we have 3 odd primes with the smallest gap (g1=2) and the biggest (g2). I call this triplet "quartered prime numbers".

 3c p1 p2 p3 g2 1 3 5 7 2 2 5 7 11 4* 3 29 31 37 6 4 137 139 149 10* 5 197 199 211 12* 6 521 523 541 18* 7 1667 1669 1693 24* 8 2969 2971 2999 28* 9 7757 7759 7789 30 10 12161 12163 12197 34 11 16139 16141 16183 42* 12 25469 25471 25523 52 13 40637 40639 40693 54 14 79697 79699 79757 58 15 149627 149629 149689 60 16 173357 173359 173429 70* 17 265619 265621 265703 82* 18 404849 404851 404941 90* 19 838247 838249 838349 100 20 1349531 1349533 1349651 118* 21 1895357 1895359 1895479 120* 22 5825999 5826001 5826127 126 23 10343759 10343761 10343903 142* 24 19918751 19918753 19918901 148 25 37369529 37369531 37369681 150 26 42082301 42082303 42082471 168 27 79167731 79167733 79167917 184* 28 151931909 151931911 151932103 192 29 191186249 191186251 191186447 196 30 192983849 192983851 192984059 208* 31

A strange coincidence appeares often. For the ticked g2 (*), we rediscover the first occurence of prime gaps ! Would it have any sort of immediate compensation between a very small gap and a very large one ?

With 2 Twins :

 4c p1 p2 p3 p4 g2 1 5 7 11 13 4 2 137 139 149 151 10 3 1931 1933 1949 1951 16 4 2969 2971 2999 3001 28 5 20441 20443 20477 20479 34 6 48677 48679 48731 48733 52 7 173357 173359 173429 173431 70 8 838247 838249 838349 838351 100 9 4297091 4297093 4297199 4297201 106 10 14982551 14982553 14982677 14982679 124 11 30781187 30781189 30781319 30781321 130 12 34570661 34570663 34570799 34570801 136 13 43891037 43891039 43891187 43891189 148 14 79167731 79167733 79167917 79167919 184 15 16

With 3 Twins :

 6c p1 p2 p3 p4 p5 p6 g2 1 5 7 11 13 17 19 4 2 4217 4219 4229 4231 4241 4251 10 3 208931 208933 208961 208963 208991 208993 28 4 27507827 27507829 27507869 27507871 27507911 27507913 40 5 120151859 120151861 120151919 120151921 120151979 120151981 58 6 7

With 4 Twins :

 8c p1 p2 p3 p4 p5 p6 p7 p8 g2 1 263872067 263872069 263872097 263872099 263872127 263872129 263872157 263872159 28 2 3

......

### Challenges :

Find prime numbers in network (consecutive or not)

• The longest chain of twins (*)
• A 2D network with a maximum of primes
• A 3D network with a maximum of primes
• ......

(*) On Mai 13, 2000 Paul Jobling has found 12 sets of 10 twin primes in (not consecutive) arithmetic progression :

i=0 to 9
(7146+i*7087)*17#+239670 +- 1
(27193+i*15352)*17#+39930 +- 1
(103299+i*8702)*17#+409602 +- 1
(240056+i*1185)*17#+399000 +- 1
(28070+i*24909)*17#+369168 +- 1
(43711+i*33725)*17#+160878 +- 1
(392688+i*2040)*17#+429018 +- 1
(263377+i*18202)*17#+359172 +- 1
(1952+i*70022)*17#+392772 +- 1
(104521+i*62645)*17#+353532 +- 1
(189110+i*89957)*17#+244200 +- 1
(419358+i*62769)*17#+424980 +- 1

You can also consult the next links :

Created by  Henri Lifchitz : October, 10 1999, last modification: June, 3  2000.