Amateur Quest For Lychrel Numbers

Habberhead writes "Some people are aware of the quest for a palindromic solution for the number 196. Basically any number that doesn't form a palindrome by reversing and adding its digits is known as a Lychrel Number. (Sequence Number A023108 of Sloan's On-Line Encyclopedia of Integer Sequences) The number 196 happens to be the first of them. In over a year's worth of time, and more than 2 quadrillion calculations, this guy at www.p196.org has reversed and added the number over 100 MILLION times. His current answer is over 41 million digits long! Apparently he and a few others are also working on a distributed computing program for finding larger and larger Lychrel Numbers. It looks like they have in mind a Seti@Home style program with visible results."

Comments

Simple Example

[teetam]

Consider 196:
196+691 = 887 (which is not a palindrome)
Apply the same for 887, 887+788 = 1675 (not a palindrome)

Apparently, you can go on forever like this without ever reaching a palindrome!

152, on the other hand, which I picked randomly, quickly reaches 707 which is a palindrome.

Personally, I don't find this interesting at all. I posted a story a week ago about the prime number problem being solved for the first time with a deterministic algorithm and it was rejected by /. OOPS! Did I just go offtopic? Sorry, mods!!!

Real world applications?

[MattC413]

What are some real-world applications that this process generates?

Maybe some psuedo-random number generation with the huge strings of numbers that this comes up with?

Any way that this could be used in some sort of encryption?

There HAS to be some useful purpose to this.. There must be, or it wouldn't be the way it is! *twitch, twitch*

-Matt

All the apathy here...

[Sivar]

This may seem like a trivial and silly waste of time, and it probably is, but the number 196 is interesting. Why? Read this quote:

Whether all numbers eventually become palindromic under this process is unproved, but all numbers less than 10,000 have been tested. Every one becomes a palindrome in a relatively small number of steps (of the 900 3-digit numbers, 90 are palindromes to start with and 735 of the remainder take less than 5 reversals and additions to yield a palindrome). Except, that is, for 196. This number had been carried through 50,000 reversals and additions by P. C. Leyland, yielding a number of more than 26,000 digits without producing a palindrome. Later, P. Anderton continued the process up to 70,928 digits without encountering a palindrome.

ALL numbers up to 10,000 become palindromes very quickly... except for the number 196?

Re:In a nutshell....

[Uruk]

Here's simple code to check this property for all numbers from 0 to 100 - adjust to test it for arbitrary numbers: (Do NOT run this on numbers that don't have known palindromes since it will cause a stack overflow. :)

#!/usr/bin/perl -w
use strict;

for(my $x=1; $x < 100; $x++) {
paltest($x, $x, 0);
} # End for

sub paltest {
my($number, $orig, $reclevel) = @_;

if($number eq reverse($number)) {
print "$orig yields a palindrome at recursion level $reclevel.\n";
return 1;
} else {
my $rev = reverse($number);
return paltest(($rev + $number), $orig, ($reclevel + 1));
} # End else
} # End paltest

Re:Arbitrary definition of a palindrome?

[xenocide2]

Of course, this leads to such quote by Cramer or Cauchy or someother such famous mathematician: "When I suddenly find anything useful concerning my work, I stop."

Generalization to arbitrary bases

[Raiford]

Check the sites on this. There are generalizations of the phenomenon to arbitrary bases

http://www.mathpages.com/home/kmath312.htm

palidrome vs number bases

[Alien54]

and contining on for a few million digits still doesn't end up at a palindrome.

Of course this is only relevant depending on base ten numbers. You milage will vary depending on the base.

It is a quirk of numbers based on the nuances of the notation system you are using, and as such is amusing for some.

I imagine there may be more palidromes in a base two system, vs, say, a base 666 system. (to choose an arbitrary base).

Oddities of this sort of thing might have some usefulness in offbeat cryto systems, but beuyond that ...

Three Years Of Computing

[Eharley]

There is a very nice account of one famous
computer geek's battle with this number.

http://www.fourmilab.ch/documents/threeyears/thr ee years.html

The account reminds me that computers are more
for just word processing and surfing the web. We
can explore interesting and amusing phenomenon
with them. I wish I weren't so jaded.

Re:In a nutshell....

[Anonymous Coward]

If 196 truly isn't a lychrel number, then neither is 887, 788, 1675, 5761, or any number after that in the sequence for that matter -- as soon as you hit a palindrome, every number used in a sum/reversal up to that point is a lychrel number. Could it be that the numbers in that sequence are the only ones that exhibit this property?

Re:Go to either -1 or GeoCities

[jbrw]

You need to go read "Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem" which appears to be the name for the US edition of Simon Singh's excellent book, published in the UK under the name of "Fermat's Last Theorem".

Besides explaining the joke you so obviously missed, it is an excellent book about mathmatics generally - and this is from someone who detests maths. I only wish this was around when I was doing maths in high school and i'd been forced to read it. Oh well...

The "magic" of 196

[floateyedumpi]

I was interested in the magic of the number 196, so I computed the "palindrome yield" for all numbers up to 3000. I defined as a "Lychrel number" any for which no palindromic sum was found after 1000 iterations. Remember that the probability for a subsequent number in the series to yield a palindrome when summed with its reverse is .45^(n) where n is the number of digits in that number. At a depth of 1000, n~400, and the probability is ~1e-140!!! Of course, it's not really random, for why else could the number 89 succeed with a palindrome of length 13 (probabilitly 3e-5). However, as you'll see, we could have chosen a cutoff of 100 or even 30.

The plot showing the sequence of iteration-to-palindrome depths for each integer is available here.

The Lychrel numbers (iteration depth>10000) are colored red. Interesting, the maximum non-Lychral depth (number of iterations until palindrome) was 24, which occurs right away at 89 (try it, its a fun one). After that, the depths recur in similarly patterned blocks, with a typical spacing of about ~100 (or occasionally a very close spacing of only 2), and some interesting gaps. The first few Lychrel numbers:

196, 295, 394, 493, 592, 689, 691, 788, 790, 879, 887, 978, 986, 1495, 1497, 1585, 1587, 1675, 1677, 1765, 1767, 1855, 1857, 1945, 1947, 1997, 2494, 2496, 2584, 2586, 2674, 2676, 2764, 2766, 2854, 2856, 2944, 2946, 2996

Can you spot the patterns in this sequence? The only thing special I can see about 196 is it is the first Lychral number!