## New chains of prime numbers

If we find 4 prime numbers (p1, p2, p3, p4) such as p2=p1+2, p3=2p1+1, p4 = 2p2-1, we form an ensemble of prime numbers with 2 twins (Twin), 1 Sophie-Germain (S.G.), 1 Cunningham chain of second kind of length 1 (Cunn.), according to the diagram below:

The first example is quite simple [5,7   11,13].
These groups of prime numbers that I call " BiTwins " are rather easy to find (and are somewhat aesthetically pleasing!). Indeed, because of their connections, the constituent numbers have simple linear forms. A small calculation shows that p1 is form 30a-1 (for p1>5), p2 = 30a+1, p3 = 60a-1, p4 = 60a+1. In base 10 representation their final digits must be 9 and 1.
The following examples can be found immediately:
[29,31  59,61]    [659,661  1319,1321]    [809,811  1619,1621]    [2129,2131  4259,4261]  ...  ...
... [2003999,2004001  4007999,4008001] ... ...

Generalization :
It is possible to continue the process to obtain a chain of several " BiTwins " (perhaps as long as one wants), according to the diagram below:

It will be noted that one obtains Cunningham chains of both first and second kinds of length = (number of BiTwin 'links' +1). The calculations yield simple linear forms for n:  210*a (for 2 links) 2310*a+0,420,1890 (for 3 links) etc...

For 2 links smallest is: [211049,211051   422099,422101   844199,844201]

For 3 links smallest is: [253679,253681   507359,507361   1014719,1014721   2029439,2029441]

...

Questions arise:

• infinity of the numbers with 1, 2, 3... links?
• infinity of the links ?
• density of the numbers (see table below)?
• possibility of finding very large " BiTwins "? To test p1, p2, p3, p4, it is enough that p2 is first to be able to use the tests in p+1 or p-1, for p1, p3, p4 and p2 is form 30*a+1.

- is (30*a) easily decomposable in prime factors for n large?
- can p2 be put in the form k*2m+1 to be able to use Proth's theorem of Proth? That is possible, but unfortunately seldom (10 times for n<6*1010) :

 Rank "BiTwins"   k*2m+/-1    k*2m+1+/-1 1 855*210-1, 855*210+1     855*211-1, 855*211+1 2 5565*213-1, 5565*213+1     5565*214-1, 5565*214+1 3 4935*216-1, 4935*216+1     4935*217-1, 4935*217+1 4 6105*216-1, 6105*216+1     6105*217-1, 6105*217+1 5 735*220-1, 735*220+1     735*221-1, 735*221+1 6 31215*215-1, 31215*215+1     31215*216-1, 31215*216+1 7 47745*218-1, 47745*218+1     47745*219-1, 47745*219+1 8 80985*218-1, 80985*218+1     80985*219-1, 80985*219+1 9 50505*220-1, 50505*220+1     50505*221-1, 50505*221+1 10 54645*220-1, 54645*220+1     54645*221-1, 54645*221+1 See Records

- Can p2 be put in the p*2b3c5d+1 form with p prime, in order to use a primality test (n-1) more general ? It is also possible and the "BiTwins" are a lot more easy to find  ( 197 pour n<107):

 Rank "BiTwins"  p*2b3c5d+-1 p 1 659,661   1319,1321 11 2 2129,2131   4259,4261 71 3 2549,2551   5099,5101 17 4 3329,3331   6659,6661 37 5 3389,3391   6779,6781 113

We can also find "BiTwins" of this form with p=1 (3 pour n<109):

 Rank "BiTwins"  2b3c5d+-1 "BiTwins"  2b3c5d+-1 1 29,31   59,61 2.3.5-1, 2.3.5+1     22.3.5-1, 22.3.5+1 2 809,811   1619,1621 2.34.5-1, 2.34.5+1     22.34.5-1, 22.34.5+1 3 431999999,432000001   863999999,864000001 210.33.56-1, 210.33.56+1    211.33.56-1, 211.33.56+1

Some other numerical results: :

 Rank "BiTwins"  1 link 1 5,7   11,13 2 29,31   59,61 3 659,661   1319,1321 4 809,811   1619,1621 5 2129,2131   4259,4261 6 2549,2551   5099,5101 7 3329,3331   6659,6661 8 3389,3391   6779,6781 9 5849,5851   11699,11701 10 6269,6271   12539,12541 See Records

 Rank "BiTwins"  2 links 1 211049,211051   422099,422101   844199,844201 2 248639,248641   497279,497281   994559,994561 3 253679,253681   507359,507361   1014719,1014721 4 410339,410341   820679,820681   1641359,1641361 5 507359,507361   1014719,1014721   2029439,2029441 6 605639,605641   1211279,1211281   2422559,2422561 7 1121189,1121191   2242379,2242381   4484759,4484761 8 1138829,1138831   2277659,2277661   4555319,4555321 9 1262099,1262101   2524199,2524201   5048399,5048401 10 2162579,2162581   4325159,4325161   8650319,8650321 See Records

 Rank "BiTwins"  3 links 1 253679,253681    507359,507361    1014719,1014721     2029439,2029441 2 1138829,1138831   2277659,2277661   4555319,4555321    9110639,9110641 3 58680929,58680931   117361859,117361861   234723719,234723721   469447439,469447441 4 90895769,90895771   181791539,181791541   363583079,363583081   727166159,727166161 See Records

 n<= "BiTwins"  1 link "BiTwins"  2 links "BiTwins"  3 links 10 1 0 0 102 2 0 0 103 4 0 0 104 10 0 0 105 29 0 0 106 144 6 1 107 752 15 2 108 4390 52 4

Generalized BiTwins :
If we replace the Cunningham chain of first and second kind respectively by Generalized Cunningham chains we can have Generalized BiTwins, according to the diagram below :

Created by  Henri Lifchitz : September , 23  1998, last modification: August, 04  2001.