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".
As far as a palindrome sentence, there is Adam's humorous introduction to Eve, "Madam I'm Adam".
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).
* * * * * * * * * * * * * * * * * * * *
The Reverse and Add Function
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.
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.
With 48, it requires two "reverse and add functions" (called "iterations"), to reach a palindrome.
We'll try 68.
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.
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