A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191,
313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385;
Beiler 1964, p. 228). The number of palindromic primes less than a given number
are illustrated in the plot above. The number of palindromic numbers having 
, 2, 3, ... digits are 4, 1, 15, 0,
93, 0, 668, 0, 5172, 0, ... (OEIS A016115;
De Geest) and the total number of palindromic primes less than 10, 
, 
, ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251).
Gupta (2009) has computed the numbers of palindromic primes up to 
.
The following table lists palindromic primes in various small bases.
 | OEIS | base- palindromic primes |
| 2 | A117697 | 11, 101, 111, 10001, 11111, 1001001, 1101011, ... |
| 3 | A117698 | 2, 111, 212, 12121, 20102, 22122, ... |
| 4 | A117699 | 2, 3, 11, 101, 131, 323, 10001, 11311, 12121, ... |
| 5 | A117700 | 2, 3, 111, 131, 232, 313, 414, 10301, 12121, 13331, ... |
| 6 | A117701 | 2, 3, 5, 11, 101, 111, 141, 151, 515, ... |
| 7 | A117702 | 2, 3, 5, 131, 212, 313, 515, 535, 616, ... |
| 8 | A006341 | 2, 3, 5, 7, 111, 131, 141, 161, 323, ... |
| 9 | A117703 | 2, 3, 5, 7, 131, 151, 212, 232, 272, 414, ... |
| 10 | A002385 | 2, 3, 5, 7, 11, 101, 131, 151, 181, ... |
Banks et al. (2004) proved that almost all palindromes (in any base) are composite, with the precise statement being
 |
(1)
|
where 
is the number of palindromic primes 
and 
is the number of palindromic numbers 
.
The sum of the reciprocals of the palindromic primes converges to 
(OEIS A118064)
a number sometimes known as Honaker's constant (Rivera), where the value computed
using all palindromic primes 
is 1.32398... (M. Keith).
The first few palindromic primes formed by taking 
digits in the decimal expansion
of pi and reflecting about the last digit are 3, 313, 31415926535897932384626433833462648323979853562951413,
... (OEIS A039954; Caldwell). These numbers
are prime for 
,
2, 27, 151, 461, 2056, ... (OEIS A119351),
with no others for 
(E. W. Weisstein, Mar. 21, 2009).
The first few 
such that both 
and 
are palindromic (where 
is the 
th prime) are given by 1, 2, 3, 4, 5, 8114118, ... (OEIS A046942; Rivera), corresponding to 
of 2, 3, 5, 7, 11, 143787341 (OEIS A046941;
Rivera).
Palindromic primes of the form
 |
(2)
|
for 
include 5, 181, 313, 3187813, ... (OEIS A050239;
De Geest, Rivera), which occur for 
, 9, 12, 1262, ... (OEIS A050236;
De Geest, Rivera), with no others for 
and 
(De Geest).
As of Nov. 2014, the largest proven palindromic prime is
 |
(3)
|
which has 
decimal digits (http://primes.utm.edu/top20/page.php?id=53#records).
." 13 Mar 2009.