Palindromic number or numeral palindrome

Palindromic number or numeral palindrome

( AlomIivlomI sMiKAwvW)

Rajwinder Singh

 M.Sc. (Maths), MMC, M.Ed., M.A (Eco.)

Punjab Education Department

Palindrome numeral is the number that remains same even after reversing the digits .

( AlomIivlomI sMiKAwvW auh sMiKAvW hn ijnHW dy AMkW dw kRm aultx qy sMiKAw nhI bdldI)

e.g. 11, 22 33, 44, 55, 66, 77, 88, 99, 101, 111 121, 1331, and many more.

In recreational mathematics Palindrome numeral received most attention

Palindromic primes are

 2, 3, 5, 7, 11, 101, 131, 151, …

Palindromic square numbers are

0, 1, 4, 9, 121, 484, 676, 10201, 12321, …

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 (Using the Rule of product: 9 choices for the first digit - which determines the third digit as well - multiplied by 10 choices for the second digit):

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

There are  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 Squares	4		7		14	15	20		31
N Cube	3		4	5			7			8
N Prime	4	5	20		113		781		5953

Palindromic numbers Table

·         Palindromic squares:

 0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, ...

·         Palindromic cubes:

 0, 1, 8, 343, 1331, 1030301, 1367631, 1003003001, ...

·         Palindromic fourth powers:

0, 1, 14641, 104060401, 1004006004001, ...

The first nine terms of the sequence, ... form the palindromes

 ...

1² = 1

11² = 121

111² = 12321

1111² = 1234321

11111² = 123454321

The only known non-palindromic number whose cube is a palindrome is 2201,

 and it is a conjecture the fourth root of all the palindrome fourth powers are a palindrome with 100000...000001 (10ⁿ + 1).

G. J. Simmons conjectured there are no palindromes of form n^k for k > 4 (and n > 1)

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, …