An introduction to numeric palindrome

An introduction to numeric palindrome

Numeric palindrome or palindromic numbers are numbers that remains the same when its digits are reversed e.g $111,88,151$ and so on. 

The word palindrome literally means a word whose spelling is unchanged when its letters are reversed e.g racecar. Palindromic numbers can be categorized into 

palindromic primes and palindromic squares. 

In any base there are infinitely many palindromic numbers, since in any base the infinite sequence of numbers written  as $101,1001,10001,100001$ etc. consists solely of palindromic numbers.

The decimal numbers(base 10) are palindromic, all decimals with one digit are palindrome e.g $\{0,1,2,3,4,5,6,7,8,9\}$, there are 9 palindromic numbers with $2$-digits in the decimal system e.g $\{11,22,33,44,55,66,77,88,99\}$, there are $90$ palindromic numbers with $3$-digits e.g $\{101,111,121,131,141,...,989,999\}$ and there are $90$ palindromic numbers with four digits e.g $\{1001,1111,1221,1331,1441,1551,...,9779,9889,9999\}$, soin total there are $199$ palindromic numbers below $10^4$ i.e $10,000$.

In addition to numeric palindrome, there is a form of palindromic numbers known as "delayed palindrome", this is obtained by simply reversing a non-palindromic number and adding the result to the original number, if a palindrome is not obtained, the process is repeated until it gives a palindromic number e.g 

Let $190$ be a non-palindrome and $091$ be the reversal of $190$, add $190$ and $091$, if it doesn't give a palindrome, reverse the result and add again:
- $190+091=281$ non-palindromic number
- $281+182=463$ non-palin
dromic number
- $463+364=827$ non-palindromic number
- $827+728=1555$ non-palindromic number
- $1555+5551=7106$ non-palindromic number
- $7106+6017=13123$ non-palindromic number
- $13123+32131=45254$, this is now a palindrome because it can be reversed and $45254$ will still be obtained.

Although, limitations to this process exists since it is not tested for all non-palindromic numbers and there is no confirmed formula that proves its existence for all non-palindromes.

The number $196$ does not yield a palindrome even after $700,000,000$ iterations, numbers of this nature are known as the Lychrel number. Mathematicians have taken up the task of iterating non-palindromic numbers to obtain very large delayed palindromes. 

The largest known most delayed palindrome ever iterated is $1,999,291,987,030,606,810$.

To understand how deep palindromes can be operated, we must understand that palindromic numbers can be applied to other systems of numbers and bases, such as natural numbers, binary numbers and  numbers in bases; $7,18$ and $24$.

In the binary numbers systme the $0,1,11,101,111,1001,1111,10001,10101, 11011,$

$11111,100001,...$ are all palindromic. 

In base $18$, some powers of $7$ are palindromic while in base $24$ the first $8$ powers of five are palindromic.