Palindromic Prime (original) (raw)

Last Updated : 23 Jul, 2025

**Palindromic Primes are numbers that are both a Palindrome and a **Prime Number. A palindrome number is a number that reads the same forwards and backwards. In simple words, a **palindrome number is a number that remains the same when its digits are reversed.

**For Example- 121 is a Palindrome number that remains the same when its digits are reversed. As 121 is Prime, thus 121 is Palindromic Prime.

Table of Content

First Few Palindromic Primes
Belphegor's Prime
Largest Palindromic Prime
Facts about Palindromic Primes

**First Few Palindromic Primes are:

2, 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, . . .

**Some Examples of Palindromic Primes with various digits are:

**Digits	**Palindromic Prime
1	2, 3, 5, 7
2	11
3	101, 131, 151, 181, 191
4	3137, 7557
5	10301, 10501, 11311
6	9040409, 9374737
7	1234321, 7654567
8	1003003001, 1008888001
9	123454321, 1007557001

Belphegor's Prime

A Belphegor prime (also known as a Beelphegor prime) is a prime Belphegor number, i.e., a palindromic prime of the form " 1(000...)666(000...)1.

The General formula for Belphegor Primes are as follows:

B_n = 10^{2n+4} + 666\times10^{n+1} +1

The first few Belphegor primes are defined for n = 0, 13, 42, 506, 608, 2472:

**n (Number of Zeros)	**Belphegor Prime
0	6661
13	10000000000006660000000000001
42	1000000000000000000000000000000000000006660000000000000000000000000000000000001
506	100000000...666...00000001 (506 zeros)
608	100000000...666...00000001 (608 zeros)
2472	100000000...666...00000001 (2472 zeros)

Largest Palindromic Prime

The largest known **palindromic prime is a number with 474,501 digits. It was discovered in September 2020.

**Largest Palindromic Prime : 10^{237250} + 3 \times 10^{118624} + 1

Facts about Palindromic Primes

Some fun facts about Palindromic primes are:

Sum of the reciprocals of the palindromic primes converges to ≈ 1.3240 (Honaker's constant).
First few n such that both n and pn are palindromic (where pn is the nth prime) are given by 1, 2, 3, 4, 5, 8114118, 535252535, 4025062605204, . . . corresponding to pn 2, 3, 5, 7, 11, 143787341, 11853735811, 126537757735621, . .

Conclusion

**Palindromic primes are unique numbers that combine the properties of being prime and palindromic. They are rare and get harder to find as the numbers get bigger, yet they are interesting to mathematicians for their symmetry and complexity.

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