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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 ''n''th number is a 1, followed by ''n'' zeroes, 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 Digit s ''a''''i'' as:
:n=\sum_{i=0}^ka_ib^i
with, as usual, 0 ≤ ''a''''i'' < ''b'' for all ''i'' and ''a''''k'' ≠ 0. Then ''n'' is palindromic if and only if ''a''''i'' = ''a''''k''−''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:
  • ''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, ... . 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 Factor s (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 System s 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, … . The Mersenne Prime s 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 ''n''16 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 11''n''−1). A number that is non-palindromic in all bases 2 ≤ ''b'' < ''n'' − 1 is called a Strictly Non-palindromic Number .


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