What are Palindromes?
Palindromes are very special kind of numbers. Typically a palindrome can be described as a number, word, sentence, etc. which reads same forward and backward. Specifically with regards to numbers, Palindromes are numbers which are symmetrical, i.e. they remain the same even when their digits are reversed.
For example 14641 is a Palindrome. In fact all the single digit numbers and numbers with same digit repeated are palindromes. So all numbers like 1,2,3…8,9,11,22,99,111,etc. are palindromes.
Properties of Palindrome Numbers
Reverse a non-palindromic number and add it to the original number. We will get a palindromic number by repeating this process. We may even get a palindromic number in first go. For example, let the original number be 37 (non-palindromic). Add reverse of it 73 to 37, we get 110 (not a palindromic number). Therefore repeat the process. 110 + 011 = 121 (palindromic number). Another example, 16+61 = 77 (palindromic number).
Any number that never becomes palindromic in this way is known as Lychrel Number. The most famous Lychrel number is 196. Check out the calculations for yourself!
A palindromic number in one base may or may not be palindromic in any other base. For example, 1991 is palindromic in both decimal and hexadecimal (7C7)
Certain powers of palindromes made up of digit 1,2 and at times 3 are mostly palindromes.
- 11^2 = 121
- 22^2 = 484
There are, however, an infinite number of cases as demonstrated here:
- 11^2 = 121, 101^2 = 10201, 1001^2 = 1002001, 10001^2 = 100020001, etc.
- 22^2 = 484, 202^2 = 40804, 2002^2 = 4008004, 20002^2 = 400080004, etc.
All even digit palindromes are divisible by 11. There are many prime palindrome numbers also like 101, 131, 151, 181, and 191
Similar to palindromic numbers, 1089 and 6174 (Kaprekar Constant) have beautiful properties
Palindrome Challenge for You
Based on the above knowledge take this very interesting palindrome challenge –
- Give 2 examples of known “Lychrel Number”, other than 196.
- Give me the most recent palindromic date.
- Most of us have lived through two palindrome years, 1991 being the last one. Only 11 years separate 1991 and 2002. Most palindrome years are separated by 110 years.Has there ever been a time when two palindrome years have been separated by less than 11 years? (Here I am not talking about single digit palindromes)
There are many other interesting features related to palindromes. Share some special feature which you could figure out.