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Palindromic number

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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.

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 zeroes, followed by a 1) consists of palindromic numbers only.

Contents

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]
with, as usual, 0 ≤ ai < b for all i and ak ≠ 0. Then n is palindromic if and only if ai = ak−i 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:

Decimal palindromic numbers

All numbers in base 10 with one digit are palindromic ones. The number of palindromic numbers with two digits is 9:
.
There are 90 palindromic numbers with three digits:
and also 90 palindromic numbers with four digits:
,
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 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
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 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  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 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



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.



See also



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