Problems & Puzzles: Puzzles

Puzzle 954. Growing by the center, sequence of primes Carlos Rivera poses the following puzzle: Here we deal with sequence of primes that grow for the center, two digits xy at a time, without repeating these xy values. Example: Starting with the two digits prime: "11" 11 1021 100621 10012621 ... where the inserted xy digits are: 02, 06, 12, and so on... My largest sequence is one starting with the two digits prime 31, and finishing with a 118 digits prime: 39999889999348944532221473565546713568117208260779315204232452113850013 24767179248444034591098680567392557391849262731 Q. Send your largest solution.

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Contributions came from Emmanuel Vantieghem, Giovanni Resta, Dmitry Kamenetsky, Vicente Felipe Izquierdo.

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Emmanuel wrote on May 22, 2019

My best 120 digits prime is :

5222020011211100131633093251536469453685483982499837677093553289176614
30726445796520453083612539663153283905119745522409.
Inserted numbers :
20, 24, 22, 02, 25, 5, 04, 17, 19, 21, 11, 15, 10, 9, 03, 18, 32, 13, 65, 31, 33, 6, 96, 39, 23, 55, 12, 51, 36, 63, 48, 60, 93, 45, 54, 30, 62, 85, 56, 49, 87, 35, 94, 84, 26, 42, 97, 90, 83, 34, 71, 66, 76, 77, 01, 99, 38, 52, 53

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Giovanni wrote on May 23, 2019

The largest prime I found for Puzzle 954 has 130 digits:

300010000809569664782543135217163618996158473794355134838108764835700
7889334904744162664560321024985086526833863896328591762174301.

The search space is so large that I just scratched it, so I'm rather 
confident there are much longer solutions.

I've also investigated the restricted case in which
all the "centers" we add have to be distinct 2-digits primes (11, 13, 
..., 97),

In this case it is easy to explore all the possibilities, and the 
largest prime is 85356231189279373911973799.

If we allow repetitions of the "centers", then we can reach 
411838643875146272183193393119993777379311.

If we allow also 02, 03, 05, 07, as the centers then the maximal values 
are 3309271405676812739397119337937271 and 
5277943160464386195600450673933513771971113111771333, respectively.

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Dmitry wrote on May 24, 2019

Here is a solution with no repeats and length 126

460145797841244017149096916235785777610874390428966130
305273299431044509338087774486855233731869319111605902871130629062454673

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Vicente wrote on May 24, 2019

My best try is:

50000061113229948736201744515899876456514095240648357193330486957334290729
44372083531563874161316419858745606051282340469209

beginning with the 59 and length = 124

The xy numbers inserted in order are:

59,00,02,09,06,04,60,14,13,12,38,22,21,95,90,46,80,76,35,64,27,08,15,78,49,41,54,16,
51,83,91,96,81,74,67,48,53,66,55,11,43,05,93,58,20,42,07,63,44,84,39,52,77,10,99,32,
34,33,03,47,85,69

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On May 31, 2019, Seth Troisi wrote:

126 digit

6072011000387449513928348637227844528824841904311696943936320654217332
75621005715839472868430802566655741019216134203958003967

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