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, ... (Sloane's 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, ... (Sloane's A016115; De Geest) and the total number of palindromic primes
less than 10,  ,  , ... are 4,
5, 20, 20, 113, 113, 781, ... (Sloane's A050251). Gupta (2009) has computed the numbers of palindromic
primes up to  .
The following table lists palindromic primes in various small bases.
 | Sloane | 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  (Sloane's
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,
... (Sloane's A039954;
Caldwell). These numbers are prime for  , 2, 27, 151,
461, 2056, ... (Sloane's 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, ... (Sloane's A046942; Rivera), corresponding to  of 2, 3, 5,
7, 11, 143787341 (Sloane's A046941; Rivera).
Palindromic primes of the form
 |
(2)
|
for  include 5, 181, 313, 3187813, ...
(Sloane's A050239;
De Geest, Rivera), which occur for  , 9, 12, 1262,
... (Sloane's A050236;
De Geest, Rivera), with no others for  and
 (De Geest).
As of Feb. 2009, the largest proven palindromic prime is
 |
(3)
|
which has  decimal digits (http://primes.utm.edu/top20/page.php?id=53).
Banks, W. D.; Hart, D. N.; and Sakata, M. "Almost All Palindromes
Are Composite." Math. Res. Lett. 11, 853-868, 2004.
Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematical
Entertains. New York: Dover, 1964.
Caldwell, C. "The Top Twenty: Palindrome." http://primes.utm.edu/top20/page.php?id=53.
Caldwell, C. "Prime Curios!: 31415...51413 (53-digits)." http://primes.utm.edu/curios/page.php?curio_id=725.
De Geest, P. "Palindromic Numbers and Other Recreational Topics." http://www.worldofnumbers.com/index.shtml.
De Geest, P. "Palindromic Prime Statistics--The Table." http://www.worldofnumbers.com/palprim1.htm.
De Geest, P. "Palindromic Prime Page 3." http://www.worldofnumbers.com/palprim3.htm.
De Geest, P. "Palindromic Sums of Squares of Consecutive Integers." http://www.worldofnumbers.com/sumsquare.htm.
Gupta, S. S. "Palindromic Primes Up to  ." 13
Mar 2009. http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0903&L=nmbrthry&T=0&F=&S=&P=2104.
Jobling, P. "Re: Record Palindrome." 27 Dec 2005. http://groups.yahoo.com/group/primeform/message/6764.
Rivera, C. "Problems & Puzzles: Puzzle 014-Pal-Primes and Sum of Powers."
http://www.primepuzzles.net/puzzles/puzz_014.htm.
Rivera, C. "Problems & Puzzles: Puzzle 051-Pi Such that Pi is Palprime & i = Palindrome." http://www.primepuzzles.net/puzzles/puzz_051.htm.
Rivera, C. "Problems & Puzzles: Puzzle 056-The Honaker's Constant."
http://www.primepuzzles.net/puzzles/puzz_056.htm.
Sloane, N. J. A. Sequences A002385/M0670, A006341, A016115, A039954, A046941, A046942, A050251, A050236, A050239, A117697, A117698, A117699, A117700, A117701, A117702, A117703, A118064, and A119351 in "The On-Line Encyclopedia of Integer Sequences."
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