Palindrome Numbers

It is now the last month of 2002, the last palindrome year for 110 years. So I could not finish out the year without a tribute to palindrome numbers.

A recent column by Whitney Matheson at USA Today got me thinking about palindromes, words or phrases that spell the same thing forwards or backwards, and wondered if there might not be a math column in there as well.

A quick Google search led to World! of Numbers a website dedicated mostly to palindrome numbers.

What are Palindrome Numbers?

They are numbers that read the same forward or backward. Single digit numbers qualify (1,2,3,4,5,6,7,8, and 9), as do the 11 times tables through 9 (11,22,33,44,55,66,77,88, and 99). The smallest three digit palindrome is 101.

The first interesting property of palindrome numbers is that all palindrome numbers with an even number of digits are divisible by 11. Two digit numbers are obvious, but what about four digit numbers?

The first four digit palindrome is 1001 which is divisible by 11. The next four digit palindrome is 1111, which is also divisible by 11 and is 110 bigger than 1001. 110 is important, because not only is it divisible by 11, but every four digit palindrome is x(1001) + y(110), where x is any integer from 1 to 9, and y is any integer from 0 to 9.

Since 11 is a factor of both 1001 and 110, then all four digit primes are divisible by 11, and since 11 is the only common prime factor of 1001 (prime factors 7, 11, 13) and 110 (prime factors 2, 5, 11), it is the only number with this property.

It is not that difficult to expand this argument to 6 digits, 8 digits and so on. You can also prove that it is true for all cases algebraically, but I'll leave that up to you the reader to figure out.

Palindrome Primes?

So all even digit palindrome numbers are divisible by 11, and except for 11, all are composite numbers. But, what about palindrome numbers with odd digits? Of the first ten 3 digit palindromes, five of them are prime (101, 131, 151, 181, and 191). Is this a sign that there are a lot of palindrome primes?

Well of the 90 triple digit palindrome numbers, 15 of them are prime. Among the 900 5 digit palindromes, 93 are prime. Among the 9000 7 digit palindromes, 668 are prime. The ratio drops dramatically after that, so palindrome primes are in fact fairly rare compared to the number of primes to composites in general.

The largest known palindrome prime was discovered in 2001, has 39027 digits, and is equal to 10³⁹⁰²⁶ + 4538354 * 10¹⁹⁵¹⁰ + 1.

When do Powers of Palindromes result in Palindromes?

Another observation often made is that sometimes raising a palindrome number to a power results in a palindrome number. The most obvious example is 11*11 = 121. Then it turns out that 121*121 = 14641. How often does this happen?

Turns out this is kind of rare as well. First, it only happens when the digits involved are 0, 1, 2, or in extreme rare cases (once?) 3. Among three digit palindromes or less:

1*1=1
2*2=4
3*3=9
11*11=121
22*22=484
101*101=10201
111*111=12321 (for more fun try squaring 1111, 11111, etc.)
121*121=14641
202*202=40804
212*212=44944

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.

2002 will be the last palindrome year with this property until the year 10,001

More info on this can be found here.

The Palindrome Conjecture

A famous unsolved number problem called the "palindrome conjecture" says that you can start with any number greater than 10, reverse it, and add the two numbers. After a certain number of times repeating this process, you will come up with a palindrome:

1) 97+79=176
2) 176+671=847
3) 847+748=1595
4) 1595+5951=7546
5) 7546+6457=14003
6) 14003+30041=44044 which is a palindrome.*

The "palindrome conjecture" makes sense on at least a statistical level. If two numbers that are digit reverses of one another are added together, and you never once have to carry, the result will always be a palindrome number. By repeating this process, which adds the effect of averaging (lowering) the digits over time to the process, it makes sense that eventually the result will be an addition that never needs to be carried.

Strangely enough, certain numbers do not work, the smallest being 196, and no one can figure out why. Applying 196 to the palindrome conjecture has been tried, and tried, and tried, and tried with over 100 million iterations and results extending past the 60 million digit mark.

Once you get that far, the probability of 60 million digits all falling into line so that there are no carries is practically zero.

Other "base" numbers that (as far as anyone knows) do not go to palindromes are 879, 1997, 7059, 10553, 10563, 32419, 36973, and 43844. 

There are others. 196+691=887 which being part of the same string makes it also non-palindromic. But then 493, 592, 691, and 790, will all meet at 887 in the second iteration, and will result in no palindromic solution. I am curious if maybe all of these other "base" numbers eventually converge as well into one defiant path. More on this subject can be found here.

The current record for the longest time to converge belongs to the number 100,120,849,299,260 which after 201 iterations, converges to the palindrome number: 16616723795884852455598564101455698426624444977944442662489655410146589555425848859732761661 which incidentally is the largest palindrome number calculated by this method. Details can be found here.

One Final Note

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? (Yes, but I'll let you figure it out.) When was the last time it happened?

* Thanks to Reader Rich Helvey for pointing out a mistake I made on this page. (Yes, I too make math mistakes, just ask any of my former calculus teachers.)

He also points out:

And finally, the smallest number of years separating two palindrome years was between the single digit years.  Then there was a 2-year difference between 9 and 11AD, followed by the 11 year differences (22, 33, 44, etc.) up until 99 and 101AD when we had 2 again, then a whole century of 10 year differences.  I'm not sure all this counts however as no one was actually counting the years like this at that point, as it wasn't customary to start counting from the birth of Christ until around 700AD, so the first palindromic year actually recognized as such was probably either 686, 696, 707, 717 or 727.  Note that throughout the first 1000 years, most palindromic years were separated by only 10 years.  The last time we had a separation smaller than 11 years was the last turn of the millenium, with the two years between 999AD and 1001AD.  For the next ten-thousand years, we'll have to deal with 110 years spearations with a few 11-year abberrations at each millennium. 2992-3003, 3993-4004, etc.

And yes, I am extremely bored here at work today.  :)


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