Problems & Puzzles: Puzzles

Puzzle 1063 A subset of A289351 Paolo Lava sent the following nice puzzle: Please have a look to A289351. I’m thinking to a subset: consider reversible zeroless primes p such that both p and it reverse satisfy the process in A289351 "Starting from one digit move right by x steps, being x the value of the digit. If the steps go beyond the LSD they continue from the left side. Then repeat the process from the reached digit. The sequence lists the numbers such that all the digits are touched just one time and the last run end in the initial digit" Here are listed only the lesser between p and its reverse. I tested up to 10^10. 11, 13, 17, 37, 79, 1151, 1559, 3373, 11161, 16661, 72227, 72727, 94949, 1181881, 1881881, 9222229, 9929999, 11111911, 11191991, 11919991, 19111991, 91111199, 1152692257, 1526922571 Q1. Other primes of this kind? Q2. What is the least prime of this kind with all the digits from 1 to 9?

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During the week 13-19 November, 2021, contributions came from Giorgos Kalogeropoulos, Oscar Volpatti and Emmanuel Vantieghem.

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Giorgos wrote:

Q1. The only approach that returned results in finding other primes of this kind was by searching palindromic primes.

The following numbers belong in the sequence although they are not consecutive terms. There may exist smaller terms: 

3394152514933

37733683538633773

1111111111111111111 (All repunit primes belong in this sequence. The greatest known has 8177207 digits)

1473394152514933741

3649463661663649463

7362559535359552637

9441396831386931449

 

The only non-palindromic prime which was very close to be a term was 1368149383397.

This number is prime and the process works in both directions but unfortunately the reverse is not prime

 

 

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Oscar wrote:

Paolo Lava's list contains many palprimes, including 11.
Every repunit satisfies the given process, so any prime repunit is a solution too.
Hence the list also contains R(19), R(23), R(317), R(1031).
Further PRP candidates currently known: R(49081), R(86453), R(109297), R(270343), R(5794777), R(8177207).

What about non-repunit solutions?
 

After 1526922571, next solutions have 13 digits:

1772716924793, 1827182972297, 3394152514933.

The smallest solution using all non-zero digits is 7316857162579461377.

 

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Emmanuel wrote;

I found two pairs :

(7571772679, 9762771757)  and  (7718849969, 9699488177).
 

I spent the rest of the week searching for a pair in which the nine digits are used.

Despite the fact that the sum of the digits of the sought-for numbers must be a multiple of their length, 

I did not find any solution with less number of digits <= 13.

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