Palindromes

A palindrome is a word, sentence or number that reads the same forwards or backwards.
Examples of palindrome words are "racecar", "radar", "kayak", and the longest length palindrome word in the English language is "detartrated".

Since this is a mathematics website, we will mainly be concerned with numbers that are palindromes.
We recently lived through 2 palindromic years: 1991 and 2002.
Unfortunately, if you are anxiously waiting for the next palindromic years, those won't occur until 2992 and 3003.

There are 9 palindromes from 1 through 9 and 9 palindromes from 11 through 99.
There are 90 palindromes from 11 through 99 and 90 from 101 through 999.
Basically, this pattern continues throughout all the higher numbers.

Palindromic numbers are of great interest to recreational mathematicians.
For example, a problem in recreational mathematics might require finding numbers that are both squares and palindromic.
Here are the first 28 palindromic squares.
 ** Number Number² 1 1² = 1 2 2² = 4 3 3² = 9 4 11² = 121 5 22² = 484 6 26² = 676 7 101² = 10201 8 111² = 12321 9 121² = 14641 10 202² = 40804 11 212² = 44944 12 264² = 69696 13 307² = 94249 14 836² = 698896 15 1001² = 1002001 16 1111² = 1234321 17 2002² = 4008004 18 2285² = 5221225 19 2636² = 6948496 20 10001² = 100020001 21 10101² = 102030201 22 10201² = 104060401 23 11011² = 121242121 24 11111² = 123454321 25 11211² = 125686521 26 20002² = 400080004 27 20102² = 404090404 28 22865² = 522808225

As for palindromic primes, here are the first 110.
Except for 11, notice that all these primes have an odd number of digits?
That is because every even-digit palindrome is a multiple of 11.
 1 to 5 2   3   5   7   11 6 to 10 101   131   151   181   191 11 to 15 313   353   373   383   727 16 to 20 757   787   797   919   929 21 to 25 10301   10501   10601   11311   11411 26 to 30 12421   12721   12821   13331   13831 31 to 35 13931   14341   14741   15451   15551 36 to 40 16061   16361   16561   16661   17471 41 to 45 17971   18181   18481   19391   19891 46 to 50 19991   30103   30203   30403   30703 51 to 55 30803   31013   31513   32323   32423 56 to 60 33533   34543   34843   35053   35153 61 to 65 35353   35753   36263   36563   37273 66 to 70 37573   38083   38183   38783   39293 71 to 75 70207   70507   70607   71317   71917 76 to 80 72227   72727   73037   73237   73637 81 to 85 74047   74747   75557   76367   76667 86 to 90 77377   77477   77977   78487   78787 91 to 95 78887   79397   79697   79997   90709 96 to 100 91019   93139   93239   93739   94049 101 to 105 94349   94649   94849   94949   95959 106 to 110 96269   96469   96769   97379   97579 111 to 114 97879   98389   98689   1003001
Notice that there are only 113 prime pallindromic numbers that are less than a milllion. In fact, palindromic numbers are almost always composite numbers, (non-prime numbers).

* * * * * * * * * * * * * * * * * * * *

Pehaps the most intriguing aspect of palindromic numbers is the reverse and add function.
This is done by taking a number, reversing its digits, then adding both numbers to make a third number.
For example, let's take 29, reverse its digits, then add both numbers.
 29 92 121
We can see that after just 1 "reverse and add function", we arrive at 121, which is a palindrome.

Does this work with all numbers?
Let's try 48.
 48 84 132 231 363
With 48, it requires two "reverse and add functions" (called "iterations"), to reach a palindrome.

We'll try 68.
 68 86 154 605 506 1111
Now, it requires three iterations to reach a palindrome.

Looking at the numbers we just calculated, we might assume that 2 digit numbers require just a few iterations to reach a palindrome but this is not the case.
Let's see what happens when we use 89 at the start of the "reverse and add function".

 89 98 Sum 1 187 ******* 781 Sum 2 968 ******* 869 Sum 3 1837 ******* 7381 Sum 4 9218 ******* 8129 Sum 5 17347 ******* 74371 Sum 6 91718 ******* 81719 Sum 7 173437 ******* 734371 Sum 8 907808 ******* 808709 Sum 9 1716517 ******* 7156171 Sum 10 8872688 ******* 8862788 Sum 11 17735476 ******* 67453771 Sum 12 85189247 ******* 74298158 Sum 13 159487405 ******* 504784951 Sum 14 664272356 ******* 653272466 Sum 15 1317544822 ******* 2284457131 Sum 16 3602001953 ******* 3591002063 Sum 17 7193004016 ******* 6104003917 Sum 18 13297007933 ******* 33970079231 Sum 19 47267087164 ******* 46178076274 Sum 20 93445163438 ******* 83436154439 Sum 21 176881317877 ******* 778713188671 Sum 22 955594506548 ******* 845605495559 Sum 23 1801200002107 ******* 7012000021081 Sum 24 8813200023188

Surprisingly, the number 89 reaches a palindrome after 24 iterations!
In fact, of all the numbers less than 10,000, 89 requires the most iterations to become a palindrome.
In other words, the number 89 is the most delayed palindromic number less than 10,000.

Listed below are the "most delayed palindromic numbers" based on the number of digits.
3 digit numbers, 4 digit numbers, 8 digit numbers, etc. are not listed because these numbers require fewer than 24 iterations.
(For example, the 3 digit number 187 requires only 23 iterations and the 4 digit number 1,297 only needs 21.)
 Digits Number Iterations 2 89 24 5 10,911 55 6 150,296 64 7 9,008,299 96 9 140,669,390 98 10 1,005,499,526 109 11 10,087,799,570 149 13 1,600,005,969,190 188 15 100,120,849,299,260 201 17 10,442,000,392,399,960 236 19 1,186,060,307,891,929,990 261 23 12,000,700,000,025,339,936,491 288

* * * * * * * * * * * * * * * * * * * *

By now, you might be wondering if all numbers subjected to the reverse and add process eventually reach a palindrome.
Presently, the answer is not known.
If such a number is ever discovered, it will be called a Lychrel number.
The number 196 stands the best chance of never reaching a palindrome because at the moment, 196 has undergone over a billion iterations without yielding a palindrome.