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

who cares?

[stipe42]

I'm rather partial to odd occurences, patterns and facts about numbers and number theory. But I could not find anything on any of the linked pages that could explain why this is interesting. All it seems to be is a variation on: 'well if you take this really convoluted and arbitrary iterating test, every number will work with enough iterations. Except for this one number.' It seems to me that just about any arbitrary iterating test will work for all numbers except for a handful. In order to differentiate the test there must be something unique about it. Are the numbers useful? Do the numbers correspond with numbers found by another test, like every other prime number or something? If not it's just very complicated numerical eeny-meenie-minie-moe.

stipe42

Arbitrary definition of a palindrome?

[PseudoThink]

Seems to me their palindrome test is a bit limited, since they only appear to be testing base-10 numbers. What's the use in that? Why not test base-2 or base-16 or whatever? Probably because there is no useful application to this arithmetic curiosity?

Re:Arbitrary definition of a palindrome?

[Uruk]

Since when does pure mathematics need to have an obvious application? Some people study math just because it's interesting. Sometimes, people come up with areas of number theory that don't immediately look promising, but that later get developed into something very useful, like optimum golomb rulers, or the mathematics that goes into public key crypto.

To get into the mind of a mathematician, you must understand the cardinal rule of math - that there is no such thing as an uninteresting number. All numbers have interesting aspects about them (strange prime factorizations, that they're palindromes, that they're the smallest sum of three consecutive cubes, whatever) but here's the real kicker - there's no such thing as an uninteresting number because if anybody was to ever find an unintereting number that had absolutely nothing special about it, it would be interesting purely for the reason that it doesn't have anything interesting about that.

Grasp that, and you can grasp why people do things like this. It's an intellectual exercise that some happen to like quite a bit.

Re:In a nutshell....

[tealover]

If you're going to copy stuff, you should at least give credit or show a link to the site that you're stealing from.

This link comes from a link on the www.p196.org page.

Moderators: Please mod the parent poster down for dishonesty.

Thanks.

Re:All the apathy here...

[pomakis]

What would be interesting is coming up with a proof of why 196 exhibits this peculiar property. Until then, it's actually impossible to prove that a number is a Lychrel number. In fact, it's impossible to prove that there are any Lychrel numbers. Whose to say that the 20 billionth iteration of 196 isn't going to result in a palindrome?

Until and unless there's a proof of why Lychrel numbers exist, the whole concept is quite uninteresting beyond a passing "neat".

Cpu Cycles

[Mister Transistor]

I think that my "extra" CPU cycles would be much better put toward distributed AIDS or cancer research. SETI seems somewhat of a waste of time, pedantic stuff like this even more.

Re:All the apathy here...

[xanth]

196 _can't_ be the only number < 10,000 with this property of never generating a palindrome. In fact, 887, 1675, and 7436 must also have this property:

196 + 691 = 887
887 + 788 = 1675
1675 + 5761 = 7436

Clearly, if any of 887, 1675, or 7436 eventually lead to a palindrome, then so will 196. So why do people just keep talking about the special 196? It might be the first, but certainly not the only one < 10,000.

Re:All the apathy here...

[ddstreet]

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

By definition the numbers 691, 887, 788, 1675, 5761, 7436, and 6347 must also have the same problem, since they're in the chain following 196.
196 + 691 = 887
887 + 788 = 1675
1675 + 5761 = 7436
7436 + 6347 = 13783

Re:what?

[Anonymous Coward]

What the fucking hell is slashdot on ?
Too much sun lately ?

one plus one is two ! so fucking what ?

Re:Real world applications?

[wurp]

I don't understand why this is interesting at all. These properties only matter for numbers expressed in base 10 (I mean, for other bases other numbers might exhibit the property, but the property is inherent to a standard base expression of the number).

A particular base expression of a number is not the number, it's a representation of the number. There are plenty of ways to express a number that don't involve any base, much less base 10. To me, interesting mathematical properties are independent of the expression of the number, like primality, arithmetic properties, whether it's algebraic or trancendental, etc.

The notion of 'palindrome' doesn't apply to numbers at all. It may apply to your representation of the number, but I can come up with a representation that is or is not a palindrome for any number you like. I just don't get the interest.

A little logic

[SiliconEntity]

I'm not familiar with this problem, so what I'm going to say is probably well known to students in the field.

It seems like the best way to produce a palindrome on the next step is for the sum of the kth digit and the kth-from-the-end digit to be less than 10. Then there will be no carries and we get a nice palindrome.

For random numbers, the chance of this being true is 1/2 for each digit in the first half of the number. Therefore with a number of length 2n digits, the chance that it will be palindromic on the next step is 1 in 2^n. (That' s one in two-to-the-nth power.)

If a number is not a palindrome on one step, it will become about one digit longer from the reverse-and-add. So at each step that it is not palindromic, the chance that it ever will become palindromic decreases.

From this perspective, it's not surprising that most small numbers become palindromes after a few steps, but that as we get to larger numbers we will find more and more that seem to never become palindromic. After some length the chance of ever again getting a palindrome is so remote that there is no point in continuing - your computer is more likely to make a mistake than for the number to happen to have the special form that can create a palindrome.

196 just happens to be a number which "gets lucky", it escapes out of the small-number region where most form palindromes. Once you get past a dozen or so steps you'll probably never get a palindrome.

There doesn't have to be anything special about 196, it's all a matter of chance and odds.

That's how I see it, anyway.

Re:In a nutshell....

[plaa]

(I'd bet there are either infinitely many such numbers or none...)

Actually, given a little thought, that's quite trivial to prove:

Suppose there is an integer N that doesn't become palindrome. Then every integer in its calculation sequence is also an integer that doesn't become palindrome. So either there are no such integers, or there are infinitely many. Duh!

But the question forms out whether there are infinitely many base numbers: I'd bet that there are either no Lychrel numbers, or there are infinitely many "base" Lychrel numbers.