Palindromic Numbers - The Numbers That Mirror Themselves

Palindrome derived from the Greek word “palindromus” which literally mean “to run back again “can be in word or numerical. A Palindrome is a word, sentence, phrase, and letters and so on, that reads the same from left or right. Examples are Abba, Bob, nun, noon, refer, dad, mum, level, rear, did, civic, dud, eke, deed, and so on.

Numeral palindrome or Palindromic numbers are numbers that remains unchanged when their respective digit are reversed, i.e A positive integer is said to be a palindromic number or, shortly, a palindrome if its leftmost digit is the same as its rightmost digit, its second digit from the left is equal to its second digit from the right, and so on.

For Example, 1001, 1111, 1221, 121, 1234321, 16461 etc.

Some palindromic primes are- 2, 3, 5, 7, 11, 101, 131, 151, and many more.

Some palindromic square numbers are- 0, 1, 4, 9, 121, 484, 676, 10201, 12321, and many more.

Palindromes are not just patterns of symmetry; they are like mirrors built into language and numbers which reflects a balance, where the beginning and the end become indistinguishable.

Interestingly, palindromes are base-dependent. A number that is a palindrome in base 10 system may lose its symmetry or palindrome nature in another base system, which keep on reminding us that mathematical beauty depends on different perspective.

There are also many interesting unsolved questions related to palindromes. For example, the famous “reverse-and-add” process—where you take a number, reverse its digits, and add it to the original number, then repeat this process until it produces a palindromic number. For instance, start with 47:

47 + 74 = 121

Which is already a palindromic number; In some cases, it takes more steps. For example, starting with 89:

89 + 98 = 187
187 + 781 = 968
968 + 869 = 1837
1837 + 7381 = 9218
9218 + 8129 = 17347
(continuing this process)
eventually gives 8813200023188 (a palindromic number)

However, for some numbers, such as 196, it has not yet been proven whether this process will ever result in a palindrome.