# Palindromic number

A palindromic number is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters.

Palindromic numbers receive most attention in the realm of recreational mathematics. A typical problem asks for numbers that possess a certain property and are palindromic. For instance,

Buckminster Fuller referred to palindromic numbers as Scheherazade numbers in his book Synergetics, because Scheherazade was the name of the story-telling wife in the 1001 Arabian Nights.

## Formal definition

Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n > 0 in base b ≥ 2, where it is written in standard notation with k+1 digits ai as:

$\displaystyle n=\sum_{i=0}^ka_ib^i$

with, as usual, 0 ≤ ai < b for all i and ak ≠ 0. Then n is palindromic if and only if ai = aki for all i. Zero is written 0 in any base and is also palindromic by definition.

An alternative but equivalent definition is as follows. In an arbitrary but fixed base b, a number n is palindromic if and only if:

• n consists of a single digit, or
• n consists of two equal digits, or
• n consists of three or more digits, the first and last digits are equal, and the number obtained by stripping the first and last digits off n is itself palindromic.

## Decimal palindromic numbers

All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two digits is 9:

{11, 22, 33, 44, 55, 66, 77, 88, 99}.

There are 90 palindromic numbers with three digits:

{101, 111, 121, 131, 141, 151, 161, 171, 181, 191, ..., 909, 919, 929, 939, 949, 959, 969, 979, 989, 999}

and also 90 palindromic numbers with four digits:

{1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, ..., 9009, 9119, 9229, 9339, 9449, 9559, 9669, 9779, 9889, 9999},

so there are 199 palindromic numbers below 104. Below 105 there are 1099 palindromic numbers and for other exponents of 10n we have: 1999, 10999, 19999, 109999, 199999, 1099999, ... (sequence A070199 in OEIS). For some types of palindromic numbers these values are listed below in a table. Here 0 is included.

 101 102 103 104 105 106 107 108 109 1010 n natural 9 90 199 1099 1999 10999 19999 109999 199999 n even 5 9 49 89 489 + + + + + n odd 5 10 60 110 610 + + + + + n perfect square 3 6 13 14 19 + + n prime 4 5 20 113 781 5953 n square-free 6 12 67 120 675 + + + + + n non-square-free (μ(n)=0) 3 6 41 78 423 + + + + + n square with prime root 2 3 5 n with an even number of distinct prime factors (μ(n)=1) 2 6 35 56 324 + + + + + n with an odd number of distinct prime factors (μ(n)=-1) 5 7 33 65 352 + + + + + n even with an odd number of prime factors n even with an odd number of distinct prime factors 1 2 9 21 100 + + + + + n odd with an odd number of prime factors 0 1 12 37 204 + + + + + n odd with an odd number of distinct prime factors 0 0 4 24 139 + + + + + n even squarefree with an even number of distinct prime factors 1 2 11 15 98 + + + + + n odd squarefree with an even number of distinct prime factors 1 4 24 41 226 + + + + + n odd with exactly 2 prime factors 1 4 25 39 205 + + + + + n even with exactly 2 prime factors 2 3 11 64 + + + + + n even with exactly 3 prime factors 1 3 14 24 122 + + + + + n even with exactly 3 distinct prime factors n odd with exactly 3 prime factors 0 1 12 34 173 + + + + + n Carmichael number 0 0 0 0 0 1+ + + + + n for which σ(n) is palindromic 6 10 47 114 688 + + + + +

## Other bases

Palindromic numbers can be considered in other numeral systems than decimal. For example, the binary palindromic numbers are:

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, …

or in decimal: 0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, … (sequence A006995 in OEIS). The Mersenne primes form a subset of the binary palindromic primes.

Generally, a number that is palindromic in one base is not palindromic in another base; for instance, 1646110 = 404D16. (The subscripts indicate radices, so n16 means n written in hexadecimal.) However, some numbers are copalindromic in several bases. The number 10510, for example, is palindromic in five bases: 12214 = 1518 = 7714 = 5520 = 3334. The year 1991 is palindromic in both decimal and hexadecimal (7C7).

In base 18, some powers of seven are palindromic:

73 =     111
74 =     777
76 =   12321
79 = 1367631



Any number n is palindromic in all bases b with b ≥ n + 1 (trivially so, because n is then a single-digit number), and also in base n−1 (because n is then 11n−1). A number that is non-palindromic in all bases 2 ≤ b < n − 1 is called a strictly non-palindromic number.