Palindrome

Palindrome

Definition: a word, number, phrase or other sequence of characters which reads the same from  the both forward (beginning) and backward (ending) positions ; is called a Palindrome.

 e.g. madam, 121, noon ,...etc.

Types of Palindrome:

There are several types of Palindrome.

 (1) Characters, word and line palindromes:
  Characters, word(s) and line(s) which reads same from the both forward and backward positions, are these types of palindromes.

 example: noon, madam, refer, level,...

Note: A sequence of characters (string) palindrome is called a string palindrome. e.g. madam.

(2) Sentence or Phrase palindromes:
 A sentence or a phrase which reads the same from the both forward and backward positions are these types of palindromes.

 example:

   "Rats live on no evil stars"

    " Step on no pets"

[Please remember that: spaces are included and capitization and spaces are to be ignored in sentence palindrome.]

 number:

 The number palindrome are  called  as "Palindromic number" or "numeral palindrome".

   e.g. 121, 11, 22, 8, ...etc.

  Now, we will discuss about the Palindromic number or numeral palindrome.
   

Palindromic number

Actually, a palindromic number is a number which remains unchanged when its digits are reversed.

   As we can see that if the digits of "121" are reversed we get "121", which remains unchanged. So, 121 is a palindromic number. But if we reverse the digits of "123" we get "321", which is a different number. So, 123 is not a palindromic number.
   How to check a number is palindromic or not:
 (1) Take the given number.
 (2) write the number from the ending position.
   (3) If the new number is equal to the original number, then the original number is palindromic; otherwise it is not a palindromic number.

      The first few decimal palindromic numbers are: 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,...202,212,...

      The palindromic prime numbers or palprimes( a prime number which is also a Palindromic number) are: 2,3,5,7,11,101,111,131,151,...

   The palindromic square numbers are: 1,4,9,121,484,676,...
   The palindromic cube numbers are:
   0,1,8,343,1331,...

    The binary palindromic numbers are: 0,1,11,101,111,1001,1111,...

   So, it is clear that, there are many palindromic numbers in different bases.
   Palindromic and anti palindromic polynomial:
   Let us consider a polynomial of degree n of the form:
    P= a(0)+a(1)x+a(2)x²+...+a(n)xⁿ.
  Now, P is called palindromic polynomial if,
 a(i)=a(n-i), for i=0,1,2,..,n ; and called anti - palindromic polynomial if, a(i)=-a(n-i), for, i=0,1,..,n.
Example:
  The polynomials, P(x)= (x+1)ⁿ is palindromic polynomial for all n. But the polynomials, R(x)=(x-1)ⁿ  is palindromic polynomial for even n and anti - palindromic polynomial for odd n.
  

  

 If you find out any incorrect information or know anything more about this , please write it in the comment section!