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An original piece of work by Jake Harry. Read it with pleasure as I did.

A New Method of Generating Palindromes
Written by Jake Harry
December 2011

Firstly, the amount of interest you have shown in the subject of palindromic numbers is awesome. They are indeed interesting numbers. However, the way in which we go about finding these numbers does not seem to be the best method. It seems to be incomplete. I am assuming that in the beginning it was noticed that if one uses the reversal/add technique on most integers that it would yield a palindromic number. Some numbers may take a few steps (or a rather large amount of steps when dealing with larger numbers), but for most of the integers below 10.000 it only took between 2 and 5 steps. Thus, the reversal/add method was adopted as the definitive/best way of turning a number into a palidrome.

Yet, I believe I have come to find a new method, a more reliable method, of turning a number into a palidrome. This method works for every number that the reversal/add technique works for (well, the ones that I've manually tested, but I am confident) as well as the infamous lychrel numbers such as 196 including also numbers that take a great amount of steps. From the numbers I have tested it seems to take about 3 steps, one more then the reversal/add technique, but is more reliable.

Here is the method:

I'll start by using the number 186. -First you have to add the lowest interval of 10 that raises the sum to the next 100. For example, 186 + 10 = 196. Therefore the lowest interval of 10 would be 20. so 186 +20 =206.

Step 1- Add lowest interval of 10 that raises the number to the next 100.

-Now we have 206. You then add the digits of the orignal number and of this new number. 1 + 8 +6 =15 and 2 + 0 + 6 = 8. Subtract the new number digit sum (8) from the original digit sum (15) which is 7. Then add 7 to the new number of 206 which is 213.

Step 2- Add the difference in digit sums between original number and new number.

-Now we just do what the traditional process which is reversing the number and adding. so 213 reversed is 312 + 213 =525.

Step 3- Reversal/addition

-Step 4- Repeat reversal/addition if neccessary

I will do 196 for you.

-The lowest interval of 10 to raise 196 to the next hundred is 10. 196 + 10 = 206.

-The difference in digit sums between 196 and 206 is 8. 206 + 8 = 214.

-Use the reversal/addition method 214 + 412 = 626.

3 steps made 196 palindromic. This also works well with the other lychrel numbers as well as numbers that take many steps.

-I will also show one more example with a fairly large number. I will do 10,309,988. -The lowest interval of 10 to raise this number to the next hundred is 20. 10,309,988 + 20 = 10,310,008

-The difference in digit sums between the original number and the new number is 25. 10,310,008 + 25 = 10,310,033.

-Use the reversal/addition method. 10,310,033 + 330,013,01 = 43,311,334.

10,309,988 became palindromic at 43,311,334 after 3 steps.

*A note about step number 2 in this process .I found that for many (not all, as numbers that end in the high 900s are a bit pesky, but a vast majority) of the numbers, if it ends in the '90's (such as 698,) when added with the lowest interval of 10 that raises it to the next 100 (in this case 10), the digit sum of the new number subtracted from the original number will always be the same depending on which interval of 10 you use. This case we used 10. 6 plus 9 plus 7 = 22. ...(697 +10 = 707)... This is already a palindromic number, but to show the point..7 plus 0 plus 7 =14. 22 - 14 = 8.

*Most numbers ending in the 90s when added with the interval 10, the digit sum difference between the new and orginial numbers will be 8. Most numbers ending in the 80s when added with the interval 20, the digit sum difference between the new and original numbers will be 7. Most numbers ending in the 70s when added with the interval 30, the digit sum difference is 6. As an example, 10,087,799,570 when added with the lowest interval of 10 that raises it to the next hundred (30) will yield 6 as the difference in digit sums between the original and new numbers.

This goes all the way down to numbers ending in the 'teen numbers. Those added with the interval of 90 come out to have digit sum differences of 0.

However, as stated the difference in digit sums may not always be a fixed number, but it does not matter. In my process when you add the difference in digit sums to the new number, no matter what number that digit sum difference is, and then do the reversal/addition method, you will get a palindrome in relatively few steps.

I thank you for your time in reading this and I hope you come to pursue making this system be known. Thank you, i appreciate it.

-Jake Harry