Problems & Puzzles: Puzzles

Puzzle 796. Looking for certain palprimes As you know, "A palindromic number or numeral palindrome is a number that remains the same when its digits are reversed". I propose you to construct a special kind of numeral palindrome, and test for the primality of it. Construct the palindrome the following way: a) concatenate the natural numbers from 1 to n, obtaining A b) reverse A obtaining B c) concatenate A and B omitting the first digit of B, obtaining the palindrome P(n) Q. What is the smallest such prime P(n)?

---

Contribution came from Jan van Delden

***

Jan wrote:

There are no prime solutions (n<=6794, digits(P(n))<=52137).
The answer to your question is therefore: I don’t know.

 

But if we change the last step c) to:

c) concatenate B and A omitting the first digit of A, obtaining the palindrome Q(n)

 

There are 3 prime solutions (n<=6801, digits(Q(n))<=52193).
Q(31) 105 digits, Q(59) 217 digits and Q(113) 461 digits are certified prime.
The smallest solution is:

13039282726252423222120291817161514131211101987654321234567891011
1213141516171819202122232425262728293031

***