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For my latest post I’ll move away from my usual topics of astronomy and space travel and I’ll talk about an interesting unsolved problem in mathematics which is: ‘Is the number 196 a Lychrel number?’.

What is a Lychrel number?

There is a simple mathematical operator called reverse and add where you take a number and add to it the number with its digits reversed. For example:

if we reverse and add 15 we get 66, (because 15 + 51 =66)

we reverse and add 236 we get 868 (because 236 +632=868)

For nearly all numbers, if you carry out the reverse and add process multiple times you will eventually arrive at a number which is the same written backwards, known as a palindrome. Some examples are shown below.

The vast majority of numbers lower than 10,000 form a palindrome in 4 or fewer steps, although there are some notable exceptions. For instance, as illustrated in the notes at the bottom of the post, the number 89 requires 24 reverse and add steps before we get to the 13 digit number 8 813 200 023 188, which is a palindrome.

However, when we repeat the the reverse and add process with the number 196 then no matter how many times it has been applied a palindrome has not been found. A Lychrel number is a defined as:

a number which will never result in a palindrome no matter how many times the reverse and add process is carried out.

The term Lychrel was coined by the computer scientist Wade Van Landingham, who has spent a lot of effort in the search to determine whether or not 196 is a Lychrel number and created a website dedicated to the search http://www.p196.org/. On this site he states.

“Lychrel” was simply a word that was not in the dictionary, not on a Google search, and not in any math sites that I could find. If there is any “hidden meaning” to the word, it would simply be that it is a rough anagram of my girlfriend’s name Cheryl. It was a word that hit me while driving and thinking about this. I liked the sound of it, and it stuck. There is no secret to the word.

Is 196 a Lychrel number?

No one knows for certain if 196 is a Lychrel number, but the question has been investigated  by many computer scientists over the years. In 1990, a programmer named John Walker applied 2,415,836 reverse and add steps to 196, no palindrome was found and the final number was a million digits in length. Since then as computers have advanced, the reverse and add process has been applied more and more times. In February 2015 a computer programmer called Romain Deb carried out the reverse and add process 1 billion times. The final number had 413,930,770 digits and is so large that, if we were to print it out,  it would be over 1,000 km long. During these 1 billion steps process no palindrome was found.

Even so, this does not prove that 196 is a Lychrel number. If a computer is used to apply the reverse and add process billions of times, the only thing this can possibly prove is that 196 isn’t a Lychrel number. This would happen if after one of these steps the result turned out to be a palindrome. Nobody has been able to mathematically prove that no matter how many times we carry out the reverse and add process for the number 196, we will NEVER find a palindrome. So it is more correct to call 196 a candidate Lychrel number

Other Lychrel numbers

196 is not the only candidate Lychrel number. If we consider all the numbers under 1000 then 196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986 are also candidate Lychrel number. The interesting thing is that, if we carry out the reverse and add process for all of these numbers with the sole exception of 879 and 978 (which is 879 its digits reversed) then we arrive at the same sequence of numbers as if we had started at 196. Some examples are shown below.

For the reason 196 is referred to as the ‘seed’ of all these other candidate Lychrel numbers.

Other bases

Everything I have written so far relates to Lychrel numbers in base 10. If we consider other bases then some numbers can be shown to be Lychrel numbers. For example in base 2, the number 10110 (which is equivalent to the number in 22 base 10), can be mathematically proven to be a Lychrel number. The proof is a little too detailed to be included in this post, but if you’re interested the details are in the following reference:

http://www2.math.ou.edu/~jalbert/courses/byrnes_glascock.pdf

On that note I’ll sign off for now. I hope you’ve enjoyed the change of subject :-).  My next post will be on the more familiar topic of astronomy.

The Science Geek

Notes

 

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

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