Problems & Puzzles: Puzzles

Puzzle 721 p(i)= DR(p(i-1))&(p(i-1)\10) From one similar puzzle in the Claudio Meller's site: Starting with a prime number p1 generate a sequence of k prime numbers p1, p2. ... pk such that each p(i) = DR(p(i-1))&(p(i-1)\10), where DR(x) means "the digital root of x" a&b means "concatenation of a and b", or "ab" Here is an example: k=7:  609999773,  560999977,  756099997,  775609999,  777560999,  577756099,  157775609. Q. Find sequences for k>7.

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Contributions came from Abhiram R. Devesh, Giovanni Resta and Emmanuel Vantieghem.

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Abhiram wrote:

I have scanned through all primes less than 3.4E10, I could not find any prime that can generate sequence length > 7. 
 

number of primes having the sequence length 7 (in the domain of my search) is 12 (other than the result you had provided)

  [1697377111 7169737711 4716973771 7471697377 4747169737 1474716973, 4147471697 ] 
  [ 3289133993 5328913399 7532891339 5753289133 1575328913 8157532891 4815753289 ] 
  [ 5097377993 5509737799 7550973779 5755097377 1575509737 4157550973 1415755097 ] 
  [ 7541779339 1754177933 2175417793 1217541779 8121754177 7812175417 7781217541 ] 
  [ 8189339311 1818933931 1181893393 1118189339 8111818933 7811181893 2781118189 ] 
  [ 8443971937 1844397193 4184439719 5418443971 1541844397 1154184439 4115418443 ] 
  [ 17263993777 71726399377 77172639937 77717263993 77771726399 27777172639 42777717263 ] 
  [ 17303393333 21730339333 12173033933 81217303393 48121730339 54812173033 15481217303 ] 
  [ 19589937739 71958993773 57195899377 75719589937 77571958993 77757195899 27775719589 ] 
  [ 22529733799 42252973379 84225297337 78422529733 77842252973 27784225297 12778422529 ] 
  [ 30501777193 73050177719 27305017771 42730501777 74273050177 77427305017 77742730501 ] 
  [ 32109919993 13210991999 81321099199 78132109919 57813210991 15781321099 11578132109 ] 

the chance of finding a prime with sequence length 8 seems low and there is need to scan a lot larger set of numbers.

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Giovanni wrote:

For Puzzle 721, I found the smallest primes
which start a chain of length 8, 9 and 10.

The chains are:

8 : 76613333393, 27661333339, 12766133333, 21276613333, 12127661333,
81212766133, 48121276613, 54812127661

9 : 217337133779, 821733713377, 782173371337, 778217337133, 777821733713, 277782173371, 127778217337, 112777821733, 411277782173

10: 1069307399777933, 2106930739977793, 1210693073997779,
8121069307399777, 7812106930739977, 7781210693073997,
7778121069307399, 7777812106930739, 5777781210693073, 1577778121069307

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Emmanuel wrote:

Here is the smallest  p1  that makes a chain of  k  primes :

   k = 8 : p1 = 76613333393

   k = 9 : p1 = 217337133779

For  k = 10, p1 will be bigger than 1.8*10^14.  Unless I find a better program, it can last a few months before I find  p1.

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