Palindromic Primes

2016-03-23, post № 110

mathematics, programming, Python, #generating, #numbers, #palindrome, #palindromes, #palindromic numbers, #prime numbers, #prime palindromes, #symmetric numbers, #symmetry

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 prime [1]. The number sadly has five distinct prime factors ($171717=3\cdot 7\cdot 13\cdot 17\cdot 37$).

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 $1003001-171717=831284$ 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

Footnotes

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