Problems & Puzzles: Puzzles

 Puzzle 617. Concatenating & superposing This is a variation of the Puzzle 830 from the always interesting Claudio Meller's site. This is our puzzle: You are able to use all the primes composed by two digits (from 11 to 97). Starting with a selected initial prime, you are able to append another prime making superposition of the last/right digit of the concatenation previously produced and the first/left digit of the new & unused prime. Each concatenation must form a prime integer. Example: if the first prime is 41, then you can append 19 producing 419, which is a prime too. And this is the end because you can not append any more primes because 4191 & 4197 are composites. Q1. Can you produce the largest sequence of primes & the largest ending prime using this procedure? Q2. Redo Q1 using all the primes of 3 digits.

Contributions came from Torbjörn Alm, Giovanni Resta.

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Torbjörn Alm & Giovanni Resta wrote:

Q1: There are 2 solutions for 2 digits, both with chain length = 4:
31379  (31 13 37 78)
61979  (61 19 97 79)

Q2: 3 solutions for 3 digits, length=17:

21157278797570947610317197127515109
(211 157 727 787 797 757 709 947 761 103 317 719 971 127 751 151 109 )

21157278797570947610317197127519139
(211 157 727 787 797 757 709 947 761 103 317 719 971 127 751 191 139 )

85313175149537911991941517347574359
(853 313 317 751 149 953 379 911 199 919 941 151 173 347 757 743 359 )

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Torbjörn added:

I did an attempt to find a solution with 4-digit primes.
The maximum one is not feasible to find, would need eons and terabyte disks
to find. So I did a run, saving the first 2500 intermediary results in every step.
Here is a solution, using the first prime as a starting point.

100900715902904916112915195123768712947967983990199337123147130703949
735112334370174142913346767316748145376768187966145985730909157937142
3433257621597297459127927639851093049
(1009 9007 7159 9029 9049 9161 1129 9151 1951 1237 7687 7129 9479 9679 9839 9901 1993 3371 1231 1471 1307 7039 9497 7351 1123 3343 3701 1741 1429 9133 3467 7673 3167 7481 1453 3767 7681 1879 9661 1459 9857 7309 9091 1579 9371 1423 3433 3257 7621 1597 7297 7459 9127 7927 7639 9851 1093 3049)

It is 58 primes in the chain.

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