A palindromic prime is a prime number that is also a palindromic number in a specified base.
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
is illustrated in the plot above. The number of palindromic primes 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 palindromic numbers in any fixed integer base 
are composite. As
, their big-O
notation estimate is
 |
where 
is the number of palindromic primes 
and 
is the number of palindromic
numbers 
in base 
. The implied constant may depend
on 
.
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;
Honaker). These numbers are prime for 
, 2, 27, 151, 461, 2056, ... (OEIS A119351),
with no others reported for 
as of Mar. 21, 2009 (E. W. Weisstein).
The first few 
such that both 
and 
are palindromic (where 
is the 
th prime number) 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
 |
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 below 
reported by De Geest.
The 20 largest known palindromic primes listed by PrimePages are summarized in the following table.
." 13 Mar 2009.