Jonathan. Frech’s WebBlog

Palindromic Primes (#110)

Jonathan Frech

TheOnlinePhotographer has published a post to celebrate 171717 comments and was amused by the num­ber’s symmetry.
A great comment by Lynn pointed out that this num­ber is indeed an interesting num­ber but not symmetrical.
Symmetrical numbers or words — also called palindromes — are defined as being the same read forwards or backwards. Examples for palindromic words are “radar”, “noon” or “level”. Palindromic numbers are 3, 404 or 172271.

Lynn then went further and checked if 171717 is at least a prime⁠¹. The num­ber sadly has five distinct prime factors ($171717=3\cdot 7\cdot 13\cdot 17\cdot 37$$171717=3\cdot 7\cdot 13\cdot 17\cdot 37$).

So Lynn wondered what the next palindromic prime would be.
To answer this ques­tion, I wrote this little Python program to check for palindromic primes. The first 120 palindromic primes are shown below.
Based on this list, the smallest palindromic prime larger than 171717 is 1003001.

      3,       5,       7,      11,     101,     131,     151,     181,
    191,     313,     353,     373,     383,     727,     757,     787,
    797,     919,     929,   10301,   10501,   10601,   11311,   11411,
  12421,   12721,   12821,   13331,   13831,   13931,   14341,   14741,
  15451,   15551,   16061,   16361,   16561,   16661,   17471,   17971,
  18181,   18481,   19391,   19891,   19991,   30103,   30203,   30403,
  30703,   30803,   31013,   31513,   32323,   32423,   33533,   34543,
  34843,   35053,   35153,   35353,   35753,   36263,   36563,   37273,
  37573,   38083,   38183,   38783,   39293,   70207,   70507,   70607,
  71317,   71917,   72227,   72727,   73037,   73237,   73637,   74047,
  74747,   75557,   76367,   76667,   77377,   77477,   77977,   78487,
  78787,   78887,   79397,   79697,   79997,   90709,   91019,   93139,
  93239,   93739,   94049,   94349,   94649,   94849,   94949,   95959,
  96269,   96469,   96769,   97379,   97579,   97879,   98389,   98689,
1003001, 1008001, 1022201, 1028201, 1035301, 1043401, 1055501, 1062601, ...

Thus it takes $1003001-171717=831284$$1003001-171717=831284$ more comments to reach the closest palindromic prime.

The sequence of palindromic primes is num­ber A002385 in the On-line Encyclopedia of Integer Sequences (OEIS).

Source code: palindromic-primes.py

[1][2020-07-18] Meaning “prime num­ber”.