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{{Infobox integer sequence
| con_number         = Infinite
| first_terms        = 2, 3, 5, 7, [[11 (number)|11]], [[101 (number)|101]], [[131 (number)|131]], [[151 (number)|151]]
| largest_known_term = 10<sup>1888529</sup> - 10<sup>944264</sup> - 1	
| OEIS               = A002385
| OEIS_name          = Palindromic primes: prime numbers whose decimal expansion is a palindrome
}}

In mathematics, a '''palindromic prime''' (sometimes called a '''palprime'''<ref>{{cite web |title=World of Palindromic Primes |last=De Geest |first=Patrick |url=https://www.worldofnumbers.com/palpri.htm |website=World!Of Numbers |access-date=1 April 2023}}</ref>) is a [[Prime number|prime number]] that is also a [[Palindromic number|palindromic number]]. Palindromicity depends on the [[Radix|base]] of the number system and its notational conventions, while primality is independent of such concerns. The first few [[Decimal|decimal]] palindromic primes are:
:2, 3, 5, 7, [[11 (number)|11]], [[101 (number)|101]], [[131 (number)|131]], [[151 (number)|151]], 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … {{OEIS|id=A002385}}

Except for 11, all palindromic primes have an [[Parity (mathematics)|odd]] number of digits, because the divisibility test for 11 tells us that every palindromic number with an [[Parity (mathematics)|even]] number of digits is a multiple of 11.  It is not known if there are infinitely many palindromic primes in base 10. The largest known {{As of|2021|10|lc=on}} is
:10<sup>1888529</sup> - 10<sup>944264</sup> - 1.
which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov.<ref>Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=53 ''The Top Twenty: Palindrome'']</ref> On the other hand, it is known that, for any base, [[Almost all|almost all]] palindromic numbers are [[Composite number|composite]],<ref>William D. Banks, Derrick N. Hart, Mayumi Sakata, February 1, 2008 [https://arxiv.org/abs/math/0405056 "Almost All Palindromes Are Composite"]</ref> i.e. the ratio between palindromic composites and all palindromes less than ''n'' tends to 1.

==Other bases==

In binary, the palindromic primes include the [[Mersenne prime]]s and the Fermat primes. All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are divisible by 3. The [[Integer sequence|sequence]] of binary palindromic primes begins (in binary):
:11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ... {{OEIS|id=A117697}}

The palindromic primes in [[Duodecimal|base 12]] are: (using A and B for ten and eleven, respectively)
:2, 3, 5, 7, B, 11, 111, 131, 141, 171, 181, 1B1, 535, 545, 565, 575, 585, 5B5, 727, 737, 747, 767, 797, B1B, B2B, B6B, ...

The palindromic prime numbers can also be generated based on Smarandache function  (Kempner function) using prime number algorithm.<ref>{{cite web|url=http://jurnalsaintek.uinsby.ac.id/index.php/mantik/issue/view/85  |date=June 2022|title=Palindromes in Some Smarandache-Type Functions |publisher=Jurnal Matematika MANTIK Vol. 8, No. 1, 2022, pp.1-9, by Hary Gunarto, S.M.S. Islam and A.A.K. Majumdar|access-date=8 October 2023}}</ref>

==Property==

Due to the [[Unsolved:Superstition|superstitious]] significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after Belphegor, one of the seven princes of [[Religion:Hell|Hell]]. Belphegor's Prime consists of the number [[666 (number)#Number of the beast|666]], on either side enclosed by [[13 (number)#Unlucky 13|thirteen]] zeroes and a one. Belphegor's Prime is an example of a '''beastly palindromic prime''' in which a prime ''p'' is palindromic with 666 in the center. Another beastly palindromic prime is 700666007.<ref>See Caldwell, ''Prime Curios!'' (CreateSpace, 2009) p. 251, quoted in {{cite news | last =Wilkinson | first =Alec | title =The Pursuit of Beauty | newspaper =The New Yorker | date =February 2, 2015 | url =http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty | access-date = January 29, 2015}}</ref>

Ribenboim defines a '''triply palindromic prime''' as a prime ''p'' for which: ''p'' is a palindromic prime with ''q'' digits, where ''q'' is a palindromic prime with ''r'' digits, where ''r'' is also a palindromic prime.<ref>[[Biography:Paulo Ribenboim|Paulo Ribenboim]], ''The New Book of Prime Number Records''</ref> For example, ''p'' = 10<sup>11310</sup> + 4661664{{e|5652}} +&thinsp;1, which has ''q'' = 11311 digits, and 11311 has ''r'' = 5 digits. The first (base-10) triply palindromic prime is the 11-digit number 10000500001. It is possible that a triply palindromic prime in base 10 may also be palindromic in another base, such as base 2, but it would be highly remarkable if it were also a triply palindromic prime in that base as well.

== Palindromic Prime in Decimal Expansion of Pi ==
On June 8, 2022 Google cloud announced<ref>{{Cite web |title=Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud |url=https://cloud.google.com/blog/ |access-date=2022-10-13 |website=Google Cloud Blog |language=en}}</ref> that they have calculated 100 Trillion digits of pi using [[Chudnovsky algorithm|y-cruncher]] on their cloud platform. The largest Palindromic prime appearing in the known decimal expansion of pi is 9609457639843489367549069.<ref>{{Cite web |title=Sigma Geek |url=https://sigmageek.com/challenge_results/1656603146901x235034290182684670 |access-date=2022-10-13 |website=SigmaGeek |language=en}}</ref>

== See also ==
* [[666 (number)|666 (number)]]

== References ==
{{Reflist}}

{{Prime number classes}}

[[Category:Base-dependent integer sequences]]
[[Category:Classes of prime numbers]]
[[Category:Unsolved problems in mathematics]]

{{Sourceattribution|Palindromic prime}}