Problems & Puzzles: Puzzles

 Puzzle 348. 414347 prime string Peter Sysel posed the following nice puzzle: Three subsequent primes (41, 43 & 47) in their concatenated form (414347) - and its evenly decomposed sums 414+347, 41+43+47, 4+1+4+3+4+7 are all primes. Question:  What is the next prime string with the same property?

Contributions came from Jeff Helee and Carlos Rivera

Jeff wrote:

The next solution for puzzle 348 appears to be

701, 709 and 719.

There doesn't seem to be any solutions for 4-digit primes but there are a few for 5-digit and 6-digit primes. The ones I found are

11171 11173 11177
11257 11261 11273
12071 12073 12097
12433 12437 12451
15647 15649 15661
18671 18679 18691
20249 20261 20269
20947 20959 20963
32309 32321 32323
36011 36013 36017
41203 41213 41221
42859 42863 42899
56299 56311 56333
64693 64709 64717
67049 67057 67061
70537 70549 70571
77093 77101 77137
77141 77153 77167

166693 166703 166723
238397 238417 238423
358327 358331 358349
482593 482597 482621

and that's it for primes <10^6.

***

Carlos Rivera wrote:

One solution with consecutive primes of lenght L prime. Example: for L=43:

1000000000000000000000000000000000009067413
1000000000000000000000000000000000009067423
1000000000000000000000000000000000009067443

form the following four prime, according to the rules of the puzzle:

100000000000000000000000000000000000906741310000000000000000000000000000000000090674231000000000000000000000000000000000009067443
3000000000000000000000000000000000027202279
3571
97

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