A palindromic number is a number (in some base ) that is the same when written forwards or backwards, i.e., of the form . The first few palindromic numbers are therefore are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, ... (Sloane's A002113). The number of palindromic numbers less than a given number are illustrated in the plot above.

The numbers of palindromic numbers less than 10, , , ... are 9, 18, 108, 198, 1098, 1998, 10998, ... (Sloane's A050250). This sequence is given by the closed-form formula

(1)

Banks et al. (2004) proved that almost all palindromes (in any base) are composite, with the precise statement being

(2)

where is the number of palindromic primes and is the number of palindromic numbers .

The sum of the reciprocals of the palindromic numbers converges to a constant (Sloane's A118031; Rivera), where the value has been computed using all palindromic numbers is 3.370001832....

The first few for which the pronic number is palindromic are 1, 2, 16, 77, 538, 1621, ... (Sloane's A028336), and the first few palindromic numbers which are pronic are 2, 6, 272, 6006, 289982, ... (Sloane's A028337). The first few numbers whose squares are palindromic are 1, 2, 3, 11, 22, 26, ... (Sloane's A002778), and the first few palindromic squares are 1, 4, 9, 121, 484, 676, ... (Sloane's A002779).

There are no palindromic square -digit numbers for , 4, 8, 10, 14, 18, 20, 24, 30, ... (Sloane's A034822).

Numbers that are not the sum of two palindromes (where 0 is itself considered a palindrome) are 21, 32, 43, 54, 65, 76, 87, 98, 201, 1031, ... (Sloane's A035137). Numbers that are not the difference of two palindromes are 1020, 1029, 1031, 1038, 1041, 1047, 1051, 1061, ... (Sloane's A104444).

Banks, W. D.; Hart, D. N.; and Sakata, M. "Almost All Palindromes Are Composite." Preprint ESI 1456 (2004). Vienna, Austria: The Erwin Schrödinger International Institute for Mathematical Physics. Feb. 5, 2004. ftp://ftp.esi.ac.at:/pub/Preprints/esi1456.pdf.

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Dr. Pete. "The Math Forum. Ask Dr. Math: Questions & Answers from Our Archives. Palindromic Numbers." http://mathforum.org/dr.math/problems/akyildiz1.4.98.html.

Dr. Rob. "The Math Forum. Ask Dr. Math: Questions & Answers from Our Archives. Palindromic Numbers." http://mathforum.org/dr.math/problems/stang4.8.14.97.html.

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Pappas, T. "Numerical Palindromes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, p. 146, 1989.

Rivera, C. "Problems & Puzzles: Puzzle 056-The Honaker's Constant." http://www.primepuzzles.net/puzzles/puzz_056.htm.

Sloane, N. J. A. Sequences A002113/M0484, A002385/M0670, A002778/M0907, A002779/M3371, A028336, A028337, A034822, A035137, A050250, and A118031 in "The On-Line Encyclopedia of Integer Sequences."

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Weisstein, Eric W. "Palindromic Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PalindromicNumber.html