Problems & Puzzles: Puzzles

Puzzle 1174  Twin Prime sums between Twin Primes Paul Cleary sent the following nice puzzle: Twin Prime sums between Twin Primes. 5 + 7 = 12 is between 11 and 13. There are many examples like this. 211049 + 211051 = 422100 is between 422099 and 422101. 422099 + 422101 = 844200 is between 844199 and 844201. 253679 + 253681 = 507360 is between 507359 and 507361. 507359 + 507361 = 1014720 is between 1014719 and 1014721. 1014719 + 1014721 = 2029440 is between 2029439 and 2029441. 41887255409 + 41887255411 = 83774510820 is between 83774510819 and 83774510821. 83774510819 + 83774510821 = 167549021640 is between 167549021639 and 167549021641. 167549021639 + 167549021641 = 335098043280 is between 335098043279 and 335098043281. 335098043279 + 335098043281 = 670196086560 is between 670196086559 and 670196086561. Q1. Find the next chain of 4. Q2. Find and example with a chain of 5 or 6

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From May 4 to 10, 2024, contributions came from Michael Branicky, J-M Rebert, Giorgos Kalogeropoulos, Gennady Gusev, Adam Stinchcombe, Simon Cavegn, Emmanuel Vantieghem,

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Michael Branicky wrote:

This is studied in OEIS A068635, so the smallest chain of length 5 starts at 73768891456259...And subsequent chains of length 4 are in OEIS A237495, the next few (denoted by the smaller of the first twin prime) being 364223689829, 376655795669, 790031896499, 1558600513469, 2180283962009, 3266149150109, 4424063189699, 4655123392919, 6924093600269.

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J-M Rebert wrote:

A068635  Smaller of the first twin pair to yield a twin pair chain of length n. 
3, 5, 211049, 253679, 41887255409, 73768891456259 
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Giorgos Kalogeropoulos wrote:

Q1. The first 48 chains of 4 can be found in  A237495

Q2. The first chain of 5 can be found in A068635 and the starting twin prime is 73768891456259 - 73768891456261

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Gennady Gusev wrote:

These are Bi-Twin primes (OEIS: A068635).
Smaller of the first twin pair to yield a twin pair chain of length n:
3, 5, 211049, 253679, 41887255409, 73768891456259, 337190719854678689, 631627448582856455729, 112511682470782472978099.
3 Last are from http://www.primenumbers.net/Henri/fr-us/BiTwinRec.htm 

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Adam Stinchcombe wrote:

I found three sequences of length 5, starting at the primes (small value of the twin): 364223689829, 376655795669, and 41887255409

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Simon Cavegn wrote:

5 chain length: 1
211049 chain length: 2
253679 chain length: 3
41887255409 chain length: 4
364223689829 chain length: 4
376655795669 chain length: 4
790031896499 chain length: 4
1558600513469 chain length: 4
2180283962009 chain length: 4
3266149150109 chain length: 4
4424063189699 chain length: 4
4655123392919 chain length: 4
6924093600269 chain length: 4
7706450161409 chain length: 4
9792446379869 chain length: 4
14825914106849 chain length: 4
15049625144399 chain length: 4
15612571518389 chain length: 4
18228407987789 chain length: 4
20440411077239 chain length: 4
21313184572679 chain length: 4
23064487085639 chain length: 4
23895025594439 chain length: 4
24563120886149 chain length: 4
25156601854229 chain length: 4
26123339522279 chain length: 4
29444506291199 chain length: 4
30532179511679 chain length: 4
30747006069779 chain length: 4
32624618742869 chain length: 4
33575985536939 chain length: 4
37673114879399 chain length: 4
39174972582899 chain length: 4
40675882100699 chain length: 4
40867864897859 chain length: 4
40923336554099 chain length: 4
41267888598089 chain length: 4
42098142556469 chain length: 4
43157118974969 chain length: 4
43780826999909 chain length: 4
44282022534749 chain length: 4
45907497559769 chain length: 4
52873313326679 chain length: 4
52985062960829 chain length: 4
53660888230949 chain length: 4
53919110104859 chain length: 4
56617553909069 chain length: 4
59681676294239 chain length: 4
62175962852849 chain length: 4
63026707633889 chain length: 4
63179305229039 chain length: 4
70414918547819 chain length: 4
71258797299689 chain length: 4
73768891456259 chain length: 5
75779434206629 chain length: 4
77800553504519 chain length: 4

 

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Emmanuel Vantieghem wrote:

The next twin that generates four twins is  {364223689829,364223689831}
(it generates   {728447379659,728447379661},{1456894759319,1456894759321},{2913789518639,2913789518641},{5827579037279,5827579037281})
The next one is  {376655795669,376655795671}
(and generates  {753311591339,753311591341},{1506623182679,1506623182681},{3013246365359,3013246365361},{6026492730719,6026492730721}}

The first twin that generates five twins is  {73768891456259,73768891456261}
(it generates  {147537782912519,147537782912521},{295075565825039,295075565825041},{590151131650079,590151131650081},{1180302263300159,1180302263300161},{2360604526600319,2360604526600321}}

The first twin that generates at least six twins has smallest member > 33*10^14.
 

 

 

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