- 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, … .

# Extensive Definition

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. The first palindromic numbers are:

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,

- the palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, … ()
- the palindromic square numbers are 0, 1, 4, 9, 121, 484, 676, 10201, 12321, … ()

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 Nights.

It is fairly straightforward to appreciate that
in any base
there are infinitely
many palindromic numbers, since in any base the infinite
sequence of numbers
written (in that base) as 101, 1001, 10001, etc. (in which the nth
number is a 1, followed by n zeros, followed by a 1) consists of
palindromic numbers only.

## 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:- n=\sum_^ka_ib^i

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 are palindromic. The number of palindromic numbers with two digits is 9:- .

- ,

101 102 103 104 105 106 107 108 109
1010 n natural 10
19 109 199 1099 1999 10999 19999 109999 199999 n even 5 9 49 89 489 889 4889 8889
48889 88889 n odd 5 10 60
110 610 1110 6110 11110 61110 111110 n square 4 7
14 15 20 31 n cube 3 4 5
7 8 n prime 4 5 20
113 781 5953 n squarefree
6 12 67 120 675 1200 6821 12160 + + n non-squarefree (μ(n)=0)
4 7 42 79 424 799 4178 7839 + + n square with prime root 2 3 5 n
with an even number of distinct prime
factors (μ(n)=1) 2 6 35 56 324 583 3383 6093 + + n with an odd
number of distinct prime factors (μ(n)=-1) 4 6 32 64 351 617 3438
6067 + + n even with an odd number of prime factors 1 2 9 21 100
180 1010 6067 + + n even with an odd number of distinct prime
factors 3 4 21 49 268 482 2486 4452 + + n odd with an odd number of
prime factors 3 4 23 43 251 437 2428 4315 + + n odd with an odd
number of distinct prime factors 4 5 28 56 317 566 3070 5607 + + n
even squarefree with an even number of (distinct) prime factors 1 2
11 15 98 171 991 1782 + + 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 39 205 303 1768 2403 + + n even with exactly
2 prime factors 2 3 11 64 413 + + n even with exactly 3 prime
factors 1 3 14 24 122 179 1056 + + + n even with exactly 3 distinct
prime factors 0 1 18 44 250 390 2001 + + + n odd with exactly 3
prime factors 0 1 12 34 173 348 1762 + + + n Carmichael
number 0 0 0 0 0 1 1 1 1 1 n for which σ(n) is
palindromic 6 10 47 114 688 1417 5683 + + +

## 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, …

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 number 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 And in
base 24
the first eight powers of five are palindromic as well:

51 = 5 52 = 11 53 = 55 54 = 121 55 = 5A5 56 =
1331 57 = 5FF5 58 = 14641 5A = 15AA51 5C = 16FLF61

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.

## Lychrel process

Non-palindromic numbers can be paired with
palendromic ones via a series of operations. First, the
non-palindromic number is reversed and the result is added to the
original number. If the result is not a palindromic number, this is
repeated until it gives a palindromic number.

It is not known whether all non-palindromic
numbers can be paired with palindromic numbers in this way. While
no number has been proven to be unpaired, many do not appear to be.
For example, 196 does not yield a palindrome even after 700,000,000
iterations. Any number that never becomes palindromic in this way
is known as a Lychrel
number.

## Colloquial terms in other languages

- In Catalonia, and through Catalan influence in Spain and by extension in the Spanish-speaking countries and Portugal (through Spanish), exists the specific word capicúa, origined by the two Catalan words "cap", that means head, and "cúa", that means tail. The conjunction "i" (and) joins the three words in a new one that explains in itself the symmetry. This word is widely used in the colloquial language, rather than palindromic number (mathematics) or palindrome, for words, phrases or others.

## See also

## External links

palindromicity in Catalan: Capicua

palindromicity in Czech: Palindromické
číslo

palindromicity in German: Zahlenpalindrom

palindromicity in Spanish: Capicúa

palindromicity in French: Nombre
palindrome

palindromicity in Korean: 대칭수

palindromicity in Italian: Numero
palindromo

palindromicity in Lombard: Nümar palindrum

palindromicity in Malayalam: പാലിന്ഡ്രോം
സംഖ്യ

palindromicity in Japanese: 回文数

palindromicity in Polish: Liczba
palindromiczna

palindromicity in Portuguese: Capicua

palindromicity in Slovenian: Palindromno
število

palindromicity in Swedish: Palindromtal

palindromicity in Chinese: 回文数