Problems & Puzzles:
A subset of A289351
sent the following nice puzzle:
Please have a look to
thinking to a subset: consider reversible zeroless
primes p such that both p and it reverse satisfy the
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
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?
What is the least prime of this kind with all the digits
from 1 to 9?
During the week 13-19 November, 2021, contributions came from Giorgos Kalogeropoulos,
Oscar Volpatti and Emmanuel Vantieghem.
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:
1111111111111111111 (All repunit primes
belong in this sequence. The greatest known has 8177207
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
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?
next solutions have 13 digits:
The smallest solution using all
non-zero digits is 7316857162579461377.
I found two pairs :
(7571772679, 9762771757) and
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
I did not find any solution with
less number of digits <= 13.