Problems & Puzzles: Puzzles

Puzzle 1044. Sequence of primes concatenating primes... Jeff Heleen sent the following nice puzzle. Chain primes Start with any prime, P>=11. Find the next prime Q larger than P whose first digit is the same as the last digit of P. Join them together such that the last digit of P is the first digit of Q. Ie, don't repeat that digit. Check to see if the new number is prime. If it is, the new number becomes the new P. Continue until a composite is reached. For example, P=61. The next prime larger than 61 whose first digit is 1 is Q=101. So 61 & 101 = 6101. 6101 is prime so repeat. The next prime larger than 6101 whose first digit is 1 is Q=10007. So 6101 & 10007 = 61010007. This number is composite so the chain ends at k = 2 links. Obviously any individual prime is a k=1 link chain. Continuing for the first instance. Examples:  k=2, 11 prime         11 & 13 = 113 prime         113 & 307 = 11307 composite   k=3, 661 prime         661 & 1009 = 661009 prime         661009 & 900001 = 66100900001 prime         66100900001 & 100000000003 = 6610090000100000000003 composite   k=4, 11689 prime         11689 & 90001 = 116890001 prime         116890001 & 116890021 = 11689000116890021 prime         11689000116890021 & 11689000116890053 = 116890001168900211689000116890053 prime         116890001168900211689000116890053 & 300000000000000000000000000000023 =                11689000116890021168900011689005300000000000000000000000000000023 composite Q. Get solutions for K>4

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During the week 4-9 July 2021, contributions came from Simone Cavegn, Giogios Kalogeropoulos and Emmanuel Vantieghem

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Simone wrote:

k=5, 1351169 prime
1351169 & 9000011 = 1351169000011 prime
1351169000011 & 1351169000017 = 1351169000011351169000017 prime
1351169000011351169000017 & 7000000000000000000000061 = 1351169000011351169000017000000000000000000000061 prime
1351169000011351169000017000000000000000000000061 & 1351169000011351169000017000000000000000000000117 = 1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117 prime
1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117 & 7000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000309 =
1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117/
000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000309 composite

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Giorgios wrote.

k=5, 1351169 is prime

1351169 & 9000011 = 1351169000011 prime
1351169000011 & 1351169000017 = 1351169000011351169000017 prime
1351169000011351169000017 & 7000000000000000000000061 = 1351169000011351169000017000000000000000000000061 prime
1351169000011351169000017000000000000000000000061 & 1351169000011351169000017000000000000000000000117 = 1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117 prime  
1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117 & 7000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000309 = 
135116900001135116900001700000000000000000000006135116900001135116900001700000000000000000000011700000
0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000309 composite

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Emmanuel wrote:

The next prime giving  K = 5  is 1351169, giving further primes :
1351169000011
1351169000011351169000017
1351169000011351169000017000000000000000000000061
1351169000011351169000017000000000000000000000061351169000011351169000017000000000000000000000117.
the next concatenation is not prime,

I didn't find a solution for  K = 6  among the primes less than  16*10^9.

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