Problems & Puzzles: Puzzles

Puzzle 1057 A cycle of ten primes

Paolo Lava sent the following nice puzzle:


Let us consider primes with 2 or more digits. Now add 1 to all digits (0->1, 1->2, 2->3, , 9->0) except to the rightmost one.

Take note if it is another prime and repeat the process. Of course, we can do that at maximum 9 times: at the tenth step we have again the starting number.

I found 1261444763 that is the least prime that produces other 9 different primes: 2372555873, 3483666983, 4594777093, 5605888103, 6716999213, 7827000323, 8938111433, 9049222543, 0150333653.

Q1: Send the next 3 starting primes of this type.

Q2. Send the largest starting primes you can get.


During the week 2-8 Oct. 2021, contributions came from Giorgos Kalogeropoulos, Oscar Volpatti, Emmanuel Vantieghem.

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Giorgos wrote:

Q1: 
18406732501
1019815085831
1031002008137
 
Q2: The largest starting primes I got are 1476673596143, 149.8234818341 and 12534707867987.

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Oscar wrote:

The next 3 starting primes are:
18406732501
1019815085831
1031002008137
Exaustive search up to 10^14:
38 starting primes with 13 digits (best: 1991056946149)
67 starting primes with 14 digits (best: 19679541017219)
See attached file "P1057ov.txt", also containing some larger starting primes.
25 digits:
1111111111110304474579319,
1111111111117991191742641,
1111111111119573773813953.
26 digits:
11111111111112938673467077.

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Emmanuel wrote:

The biggest starting prime I could find is  18406732501
It generates the nine primes , 29517843611, 30628954721, 41739065831, 52840176941, 63951287051, 74062398161, 85173409271, 96284510381, 7395621491.

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