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

what?

[underworld]

does someone want to explain this in layman's terms?

Re:what?

[cperciva]

256 + 652 = 908
908 + 809 = 1717
1717 + 7171 = 8888, which is a palindrome.

However,
196 + 691 = 887
887 + 788 = 1675
1675 + 5761 = 7436
7436 + 6347 = 13783
and contining on for a few million digits still doesn't end up at a palindrome.

Re:what?

[h2odragon]

good explaination.

Now, can you explain why anyone would spend time doing this? :)

Re:what?

[danalien]

Some find this sort of activity as "fun".

Re:what?

[norwoodites]

There is one misprint:
256 + 652 is not 908 but 808.

Re:what?

[EvanED]

Try again; 908 is correct.

Re:what?

[KittyTheCat]

5+5=10

Re:what?

[lahi]

I tend to make silly mistakes in addition as well, but if I were to post any corrections, I'd make damn sure I was right. You, on the other hand, obviously didn't.

Try with a calculator next time.

-Lasse

Re:what?

[norwoodites]

I just noticed that, I cannot add.

Re:what?

[(startx)]

actually, the original poster is correct, 256+652 IS 908, not quite sure how you got 808

Re:what?

[sloveless]

Maybe it's your Mac? Time to get a real computer.

Re:what?

[billbaggins]

Digitwise addition, a technique well-known to first-grader starting out on the path of adding multi-digit numbers, before they master the art of remembering the 'carry' digit every time...

Re:what?

[gmenhorn]

The description is in layman's terms. Well, in sub-layman's terms and it is confusing. His description is : "...by reversing and adding their digits". That's it. Hmm, okay. Does this mean for 196: reversed = 691 and adding their digits 6+9+1 = 16? It wasn't until I went to the site he links to until I figured out what was going on. And then I wondered, why did I waste my time?

Re:what?

[Anonymous Coward]

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

one plus one is two ! so fucking what ?

Re:what?

[Waffle Iron]

Now, can you explain why anyone would spend time doing this?:)

They do it to try to find out if repeating these steps for 196 ever ends up with a palindrome.

Re:what?

[GoogolPlexPlex]

Surely it's not so complicated to find more of these numbers? This guy has got 100 million of them, some over 41 millions digits long - all the ones that stem from 196, anyway.

Re:what?

[Tokerat]

He already has one! Didn't you say he had a Mac... :-P

Re:what?

[ComaVN]

That's silly, if it doesn't, this is NOT the way to prove that. It can only prove if a number is not in the sequence.

Then again, there are plenty of people who consider it fun to learn 10000+ digits of pi. Now THAT will impress the chicks.

Re:what?

[Grizzlysmit]

Then again, there are plenty of people who consider it fun to learn 10000+ digits of pi. Now THAT will impress the chicks.

so thats my problem with girls, I need to memorize pi.

Re:what?

[jcoy42]

I just noticed that, I cannot add.

It kills me that your email is from physics.uc.edu..

A Geek who gets off on numbers ...

[pgrote]

That pretty much sums it up.

His website is a great treasure chest of information for folks looking to do this in their spare time. He seems to be pretty level headed, but just gets off on this.

Re:A Geek who gets off on numbers ...

[Warped-Reality]

1-800-HOT-GRIT

And the reason is...?

[garf]

So why search for these numbers?

Confused...I am...

Re:And the reason is...?

[garf]

Takes all sorts I suppose..:)

Re:And the reason is...?

[garf]

Oh I have...mathematics is a beautiful art, don't get me wrong. I just can't understand the use of these numbers other than their single propertie.

Now control theory, p and s plane...oh joy :))

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!!!

Re:Simple Example

[cperciva]

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 /.

You aren't talking about this by any chance, are you?

Re:Simple Example

[teetam]

Nice to see /. posting that story, even if mine was rejected. Thanks for pointing me to the story.

Re:Simple Example

[SuperguyA1]

Actually what the article says is that all numbers under 5000 or something like that fall into one of 3 'seeds'. for example 196 would be a seed 887.

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

In a nutshell....

[moniker_21]

Pick a number, any number. Reverse the digits in the number, add those reversed digits to the original number. Does this sum create a palindrome? If not, repeat the process with the new sum. By example:

87+78 = 165
165+651 = 726
726+627 = 1353
1353+3531 = 4884, a palindrome!

This article is saying that for the thousands of numbers tested, every one except 196 has exhibited this property.

Re:In a nutshell....

[Andrew Allan]

I've found another one!!!

Try doing it with 691! ;-)

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: In a nutshell....

[Black Parrot]

> This article is saying that for the thousands of numbers tested, every one except 196 has exhibited this property.

Now try it in base 195.

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:In a nutshell....

[Weird Dave]

Oh, man! Do you actually call that a perl program? Here's how it is supposed to look (this is the exact same program with the syntax fixed ;-):

#!/usr/bin/env perl
use strict;
use warnings;

for my $i (1 .. 100) {
&paltest($i,$i,0);
} # for

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

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

Re:In a nutshell....

[Weird Dave]

And, here's the program with its algorithm fixed, but admittedly with a few warnings showing up because it passes 196:

#!/usr/bin/env perl
use strict;
use warnings;

my $reclimit = 100;

for my $i (1 .. 200) {
my $r = &paltest($i,$i,0);
if ($r == -1) {
print "$i yields no palindrome in $reclimit recursion levels\n";
} else {
print "$i yields a palindrome at recursion level $r\n";
} # else
} # for

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

if($number eq reverse($number)) {
return $reclevel;
} elsif ($reclevel > $reclimit) {
return -1;
} else {
my $rev = reverse ($number);
return &paltest ($rev+$number,$orig,$reclevel+1);
} # else
} # paltest

Re:In a nutshell....

[tesmako]

I hate you. I spent an hour writing a program to calculate these damn numbers and crunching on 691 before I got the joke :P Oh well, I learnt a bit of GMP in the process, guess it was not all wasted.

Re:In a nutshell....

[tesmako]

Can't let perl have all the fun, a quickly thrown together and badly tested C solution: lychrel.c

Requires GMP, build it with something along the lines of

gcc -o lychrel -O2 -lgmp lychrel.c

and run with a number as argument to try it for the lychrel property. Makefiles are for wimps ;)

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: In a nutshell....

[protonman]

Or base 1!

Re:In a nutshell....

[plaa]

This article is saying that for the thousands of numbers tested, every one except 196 has exhibited this property.

Wrong, 196 is though to be the smallest integer with this property. Check the integer sequence referenced. It gives 45 integers which are thought to have this property, starting with 196, 295, 394, 493, 592, 689, 691,...

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

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.

Re:In a nutshell....

[N1KO]

Does your code support integers larger than 16, 32, 64 bits?

Re:In a nutshell....

[tesmako]

No, took an hour to both implement it and then run it for long enough to notice that it wasnt likely to finish anytime soon. As it happens I have since slept and discovered that the code is somewhat moronic too but that's what you get for trying to code in the middle of the night, crappy code to solve an unsolvable problem :)

Also could be noted that I implemented it in Matlab at first, but noticed that it ran off the range of a 64 bit integer quickly, so I reimplemented it (badly) in C with GMP to support arbritrarily sized numbers and get some speed out of it (at this point I still thought 691 was just hard, not that hard :), seeing I had never used GMP before it involved a bit of manual reading and such.

And just to note, one should not use GMP for this problem, was too sleepy to see that yesterday though, better off maintaining and array of actual base 10 digits and doing the adds "by hand".

Re:In a nutshell....

[Kredal]

Moderators! Attack! The previous poster didn't give credit for his sig, which all self-respecting geek will know comes from Office Space!

Re:In a nutshell....

[Tom Womack]

No. As far as I can tell (I admit I only used long-longs, so interesting things might happen after 18 digits), there are four essentially-different series starting with numbers less than 10^4: the ones generated by 196, 879, 1997 and 7059.

The code to check this is trivial, and at http://www.chiark.greenend.org.uk/~twomack/lychrel .cpp

I've extended the search to starts up to 10^7, and found a little under 2000 routes; there are many more route-starts beginning with the digit 1 than with any other digit. The program is fast enough that it's not worth downloading the tables of route-starts.

http://www.xs4all.nl/~itsme/projects/math/196/ba se 2.html

has the observations required to show that 10110_bin *never* forms a palindrome in base 2.

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

Re:Real world applications?

[Tablizer]

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

What else are unemployed trivia-loving geeks gonna do?

Re:Real world applications?

[feed_me_cereal]

OK, I just read about these numbers for the first time just now, but I'd guess that there isn't a lot of significance to them. I mean, if you're looking at the digits, then it becomes relevant that you're using base-10. Usually in these sorts of cases, you only find important stuff in dealing with base-2. But then again, this is just my ever-so-slightly educated guess. If anyone really does know how these numbers may be significant, I'd like to hear it!

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.

Re:Real world applications?

[grytpype]

You know, I sensed some bogosity here, but I couldnt' quite tell what it was. I think you've put your finger on it. Are there any interesting number-theoretical properties that depend on what base you express the number in?

Re:Real world applications?

[evilquaker]

Are there any interesting number-theoretical properties that depend on what base you express the number in?

Normality (all digits in decimal expansion occur with the same frequency) is one. In particular, the question of whether numbers like pi and e are normal is still unsolved. The interesting thing is that it's relatively easy to prove that almost every number is normal (i.e. those that aren't form a set of measure zero). But proving that any specific number is normal is difficult (unless it's chosen to have such properties, such as the number 0.12345678910111213141516...).

Of course, there is the property of being "absolutely normal" (i.e. normal in every base). Again, it's trivial to prove that almost every number is absolutely normal, but no one has proved that any "naturally occurring" number is absolutely normal.

Re:Real world applications?

[DustMagnet]

People seem to have a very hard time seporating numbers from their representations. I used to teach a course on computer representation (int, float, machine language). I used to start one lecture by writing the numeral five on the board and explaining it wasn't five. Just a representation of five. Maybe I should have started with "cat". People don't that the letters C-A-T aren't a cat.

It took most people a while to understand (long office hours). Some people never understood the difference. They got really lost when I got to 1's and 2's complement.

The title of the article states that it's an amateur search. I can't see why a professional would bother. It's entirely unique to beings with 10 fingers and doesn't exsist in Plato's world of numbers.

Kind of like when the date and time was 20/02/2002 20:02. It's kind of cool, but just an artifact of how Americas represent dates and times.

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?

[CresentCityRon]

Yes I thought of that. Sort of like being "English Centric". Regardless of base if there is only one number that doesn't obey this rule then that is interesting. I'm not sure what for but interesting.

A solution waiting for the problem.

Re:Arbitrary definition of a palindrome?

[Papineau]

Not just the test. The digits reversing part is also based on base-10. Could be interesting to mix the bases for the test and the reversing.

As for the use of base-10 rather than another base... probably because we have 10 fingers? :P

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:Arbitrary definition of a palindrome?

[treat]

Since when does pure mathematics need to have an obvious application?

This is not pure mathematics, as it relies on the numbers being represented in a certain base.

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."

Re:Arbitrary definition of a palindrome?

[grammar fascist]

Try base 196. It works in that: 10 + 01 = 11! Wow!

You can do that to every number in at least one base. What definitely would be interesting is to find some number it only works for in its own base. Or to find some special properties of numbers that are defined by what bases it works in, how many steps it takes in each base...

In any case - and I'm sure you've heard this a lot - it doesn't matter that there's no apparent useful application. Stop being so practical.

Re:Arbitrary definition of a palindrome?

[feed_me_cereal]

yeah, but having 10 fingers isn't important to general mathematics. In this context, it's arbitrary.

Re:Arbitrary definition of a palindrome?

[Misanthropic Lycanth]

It does not have to be "base-10 Centric," but it has to have a lower limit. Any number n is a palindrome in base-(n-1) and any base greater than n.

Before any of you start posting that 196 is not a palindrome in base-197, think about what 10 through 15 are in hex.

Why?

[SagSaw]

Can someone out there explain what, if anything, is the mathematical significance of Lychrel numbers? I understand the basic definition, but I'm not sure what is gained by showing a particular number is or isn't a Lychrel number.

How about compression

[oliverthered]

Could you use this process for compression? numbers that are palindromes could be expressed as the smallest root number.

Re:How about compression

[oliverthered]

Why not?

if you find a palindrome! in say an image that can be produced from a far smaller number you could use it as a compression method.

Re:How about compression

[juggleboy]

I'd love to see the decompression algorithm for this...

Re:How about compression

[oliverthered]

Well here goes,
Using an example from the thread.

Say you have a channel in an image where some consecutive bytes run like this.
53351221 (in base 10) they could be compressed to
398 and 456

398 and 456 could be represented by there order (e.g. 398 is the nth root number) which would be 398, 456

the compressed result may be somthing like
35 , 39

you would decompress by finding the 35'th and 39'th root numbers and doing a reverse + add until a palidrome is reached.

There are probably far too few numbers that this method would work for to make compression praticle but it's still an interesting possiblilty.

Interesting, Inspiring and math script kiddie

[CresentCityRon]

I found the website to be rather interesting - I remember discussing this in a finite math course many years ago but we spend only like 10 mins on it.

I thought it was great that someone without a lot of math background but a hell of a lot of energy could jump right in and make a difference. I have a great education and background and haven't done as much. I feel ashamed. I gotta start rockin!

It was funny that he doesn't fully understand the math, the programs and still gets things done. He just gets programs that others have created and puts them to use. A math research script kiddie. There should be a website for this. Dump off your interesting math code and people can download and run those that are interesting.

Wrong approach?

[sfraggle]

It seems to me that this is the wrong kind of approach to this problem. They would be much better trying to find a generalised proof. You can search as high as you want but you can never prove that the next iteration will not yield a palindromic number otherwise. The difficult part to me seems to be to describe the problem in a simple mathematical form that can be analysed.

Re:Wrong approach?

[sfraggle]

> Can you prove that Pi does not repeat at some
> point not yet calculated?

Yes, it was proved over a century ago by Ferdinand Lindemann.

Re:Wrong approach?

[sfraggle]

Whoops, my mistake. Lindemann proved Pi is transcendental but Johann Lambert proved earlier (1768) that Pi is irrational, enough to show that the digits will never repeat

Re:who cares?

[GigsVT]

Well, I guess this falls into one of those, "I couldn't hurt anything, and if they enjoy doing it, then so what?"

Who knows, maybe someone will think of an application for it. Base two wasn't particularly relevant before binary computers came about, stuff like that.

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: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".

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:All the apathy here...

[RickHunter]

Yes, that's a good point. I believe whats special is that 196 is the start of this chain, and only numbers this is true for are the ones in this chain. The question is: do they have a special property, or is it just some kind of base 10 fluke?

Re:All the apathy here...

[NoMoreNicksLeft]

It's obvious to me anyway, on an intuitive level, that if an iteration goes beyond a few digits in size, the likelyhood of it palindroming plummets. I bet number of non-Lychrels that require 50+ iterations is unbelievably low, or even zero.

Of course, I too wonder what in the hell this is good for.

Re:All the apathy here...

[Zaak]

In fact, it's impossible to prove that there are any Lychrel numbers.

Actually, it might be possible to find a non-constructive proof of the existence of Lychrel numbers, that gives you no insight to whether 196 is one, or why they happen.

TTFN

Re:All the apathy here...

[Sivar]

That's true. I need to stop posting right after I get up. :)

Re:All the apathy here...

[pediddle]

I bet number of non-Lychrels that require 50+ iterations is unbelievably low

But remember that there are an infinite number of numbers, so if there is *any* possibility of this happening, there will be at least one number that does it.

And why would the probability be zero? What if you find a number that requires 49 iterations. What then is mathematically significant about 50?

You can't prove stuff based on your intuitions, especially when those intuitions don't make sense.

Re:All the apathy here...

[NoMoreNicksLeft]

Well, the fact that it's limited to integers, and that the possibility of a string of n decimal digits matching such a pattern, when n grows with each iteration like it does.

Sort of like the chance of finding the string "333" in decimal Pi, versus finding "3333333333333333333333333". Both are in there, multiple times even, but its obvious one is much more likely than the other. 50 was arbitrary though, it might be 20 or 100. You get the idea.

Re:All the apathy here...

[blonde rser]

Numbers that resolve into the series also have the same characteristic... for example 295 + 592 = 887 394 + 493 = 887 689 + 986 = 1675 Prove it for 196 and all these numbers fall with it.

Re:All the apathy here...

[NoMoreNicksLeft]

Go to hell, worthless slashdot troll.

Re:All the apathy here...

[DrSkwid]

Come on, you can do better than that.

tbh, life is boring up @ the karma kap. I try and lose a few points just to have the fun of getting them back again.

so again your intuition of me trolling is wrong.

Re:All the apathy here...

[protonman]

> Sort of like the chance of finding the
> string "333" in decimal Pi, versus
> finding "3333333333333333333333333".
> but its obvious one is much more likely than
> the other.

Thank you for taking the time to explain why statistics sucks so much.

Get a clue on real math.

Re:All the apathy here...

[NoMoreNicksLeft]

I troll on usenet, breaks up the monotony. Also, it lets me see how the old-timer trolls used to do it back in the day...

Damn, kinda makes me wish it were 1986 again.

Re:All the apathy here...

[NoMoreNicksLeft]

Fuck you. No, really. Fuck yourself and drop dead. Am I supposed to be rainman or something?

I must have forgot how Pi devolves into a string of 20 "3" digits right after 3.14...

Re:All the apathy here...

[martyn s]

Select "code" in the pulldown menu next to the preview button.

Re:All the apathy here...

[protonman]

> Am I supposed to be rainman or something?

Yeah, definitely, yeah.

Re:All the apathy here...

[DrSkwid]

I looked up my first posts with google.
cringeworthy
I took a flamebait hit btw. tnx.
Karma: Excellent bah, not even a score any more
This place is getting dull without the rpg elements.

you'd think they were trying to get rid of people !

Re:All the apathy here...

[NoMoreNicksLeft]

Karma: Somewhere between -20 and 50.

Gotta love that...

Re:All the apathy here...

[dragons_flight]

Given a 2*n digit number, it suffices to generate a palindrome if the sum of the i-th and (2n-i+1)-th digit is less than 10 for all i between 1 and n. It follows that at least (2n)/(2^n) numbers of length 2*n will immediately form palindromes. While less obvious, it is also true that if the sum of the i-th and (2n-i+1)-th digit is greater than 10 then the next iteration can only generate a palindrome if the sum of every digit and it's counterpart is greater than 10 (e.g. 9292 -> 12221). Not all numbers with this property will immediately form palindromes (e.g. 9393 -> 13332), but it is a requirement. This property holds for an additional (2n)/(2^n) numbers.

Hence the probability that a number of length 2*n will immediately form a palidrome is 1/(2^(n-1)) for each iteration.

On average, the number gains 0.5 digits per iteration of the algorithm. Consequently for a number with 2*n digits, after infinite iterations, you expect to have encountered a number of palindromes approximately equal to Sum(1/2^(n-1+k/2)), k=0 to infinity => ~6.8*2^(-n).

A straight forward density argument shows that there have to be some Lychrel numbers and that most numbers with a large number of digits are Lychrel numbers, but of course it doesn't tell you which particular numbers have this property.

Obviously I haven't been entirely rigorous, but afterall this is slashdot.

Re:All the apathy here...

[Prune]

>> 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.

Read the article. These numbers don't count exactly because they follow in that chain. Only the seed of a chain counts.

Re:All the apathy here...

[evilquaker]

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

Actually, I think it's the opposite: when there's a proof either way, it will probably just be a mathematical curiosity (or it could turn out to be interesting, but I doubt it...). Until then, it's an unsolved problem. If you find the proof, you will most likely be the first person in the history of the Earth to know the answer. The fact that it's a relatively obscure problem, and that AFAIK no one has even come close to finding a method to attack the problem make it one of the better problems to work on if you like this feeling of adventure. Other problems (odd perfect number, Goldbach's, Twin Primes, Collatz) that fail one of the above two tests don't have this kind of promise.

Re:All the apathy here...

[underworld]

I don't have a full theory, but I would like to suggest a general direction for a theory. Since being introduced to this concept, I find it a bit intriguing (or perhaps I just have too much free time) - anyhow...

the number 196 yields 887 in its quest for Lychel-ness. When computing the solution for 887, there are some interesting patterns that emerge. For instance, the 8th through the 13th sequence look like this:

18211171
17111271
--------
35322452
25422343
--------
40744805
50844704
--------
91589509
90598519
--------
182188028

The interesting aspect is the middle pattern of 11, 22, 44, 89/98, 188. In the 15th, 17th, 18th, and 21st, there's another pattern of sorts:

15: 1004 266 2310
17: 2273 556 2622
18: 4536 211 6344
21: 8739 2221 9467

But the really really interesting sequence is the 23rd:

23: 71272 788 37206

The 788 in the middle will be added to an 887 reversed, and thus might have the affect of an infinite loop. It's seems similar to dividing 10 by 3 - you always end up with a remainder and keep dividing.

Creating a theory to find these might be quite difficult. Imagine a formula to determine all of the irrational numbers -

While trying to "guess" at a likely Lychel candidate using 887 as a middle pattern basis, I tried a few such as 58872 (pall. at 3 iteration) and 58876 (pall. at 27th iteration). What's interesting abough 58876 is that 25th and 26th sequences show a similar pattern as 887's 8th and 9th sequence by itself:

25: 1115552 11 1455111
26: 2231093 22 4010222

Which of course was quelched by the 27th sequence:

27: 44511974 47911544

I still don't fully understand why 887 doesn't resolve but it seems to have something to do with the numbers being higher than 5 and/or causing a carrying 1 when adding.

my $.05 worth...

It may not generate anything useful right now

[gatesh8r]

But that doesn't mean it can't be useful later. A great example of this is the logarithm. Always nice now that it's used in seismology and understanding computer performance, huh? Yet all it was useful for until the 20th century was slide rules.

Re:It may not generate anything useful right now

[Cryptnotic]

Hey man, slide rules helped end WWII. Bombadiers used slide rules to calculate when to drop a M pound bomb from a plane moving at V mph at an altitude Y to hit a target at some distance D in front of the plane. Those aren't simple slide rules, they're often two or three dimensional things custom made for certain types of bombs or aircraft or whatever. Anyway, back when computers took up the entire wing of a building, they didn't have automatic targetting computers in warplanes.

Anyway, ordinary slide rules were commonly used by engineers up until pocket calculators became available (which I guess would be the 1960's or so...). It is the only way to quickly multiply large numbers by hand (unless you're Rainman and can do it in your head).

In case you don't know, logarithms are rather simple.

Say you want to find A*B, where A and B are rather large numbers and you don't want to multiply them by hand. log(A*B) = log(A) + log(B), a useful property. So find log(A) and log(B) using your slide rule. Add them. Then do the inverse log. That's A*B. Multiplying by using addition and tables of logarithms. Fun stuff, huh?

And yes, it works with any "base" for the logarithm, base e (the "natural" log), base 2, base 10, whatever.

Re:It may not generate anything useful right now

[nelsonal]

Similar mechanisms were also used by submarines to hit moving targets with torpedos. Also pilots used them well into the 1980s to aid plotting a route.

Re:It may not generate anything useful right now

[dillon_rinker]

Yet all it was useful for until the 20th century was slide rules.

?????

You're heart's in the right place but your facts are way wrong on this one. Logarithms were developed for the purpose of changing a then-hard problem (multiplication) into an easy one (addition). They were useful for centuries before the 20th. Read any standard text on the history of mathematics.

I was going to post the solution

[sulli]

but the lameness filter blocked me. Dammit!

Re:I was going to post the solution

[falzer]

I have also discovered a truly marvelous proof for this statement, which, unfortunately, this margin is too small to contain.

Is this really maths?

[happy monday]

I don't think this is a mathematical problem... reversing the digits in a number isn't a mathematical procedure, is it? I don't think there is a mathematical relationship between one particular number and another which has the same digits in reverse. Why should there be? That's like thinking that spelling a word backwards will give you a word with the opposite meaning. I think this situation is just a coincidence, and the only meaning is related to the appearance of the numerals themselves rather than any property of palindromic numbers.

Re:Is this really maths?

[jaavaaguru]

Then doesn't it seem a huge coincidence that nearly ALL number that have this procedure applied to them come out as palindromes? That has got to be significant. We just don't know why yet.

Re:Is this really maths?

[Anonymous Coward]

Oh I suppose you could make it into math if you really wanted to. but you'd have one procedure for base10 and another for other bases. An example of doing it with 4 digit numbers (you could do a Summation of you wanted to do it for n-digit numbers of course). I will be using ascii-ized syntax (% is remainder after division, ^ is raised to the power of) since most browsers do not support the HTML extensions for mathematical expressions.

Let w = input number
Let v = resulting palindrome

v = (((w/(10^0))%10)*10^3) + (((w/(10^1))%10)*10^2) + (((w/(10^2))%10)*10^1) + (((w/(10^3))%10)*10^0);

I believe that is correct for all cases. if you want to reverse the digits of something mathematically. And I didn't test it on negative values so don't try it.

Representing a digit-flip as an equations means you could apply the calculus to it and come up with interesting results.

I don't like the method used in the article because it seems like brute force, where as manipulating some equations seems more elegant.

Re:Is this really maths?

[ziriyab]

Just because there is no symbol associated with a mathematical manipulation doesn't mean it's not a math problem.

Proof for 196

[dave_mcmillen]

The number 196 NEVER becomes a palindrome, no matter how many iterations you do. I have assuredly found an admirable proof of this, but the Post Comment box is too narrow to contain it.

(Apologies to Fermat.)

Reversing numbers

[ergo98]

This seems like an absurdly irrelevant number study, but it is intriguing from a purely number theory perspective.

In any case, what is the mathematical function for inverting the digits of a decimal value? (i.e. 192 -> 291). I'm not including textual "move the digits around". I hardly am concerned with number theory anymore, but this does intrigue me.

No I just have to go count the tiles...

Re:Reversing numbers

[mariube]

In C code, this is how to do it binary, according to /usr/share/computers/fortune

n = ((n >> 1) & 0x55555555) | ((n << 1) & 0xaaaaaaaa);
n = ((n >> 2) & 0x33333333) | ((n << 2) & 0xcccccccc);
n = ((n >> 4) & 0x0f0f0f0f) | ((n << 4) & 0xf0f0f0f0);
n = ((n >> 8) & 0x00ff00ff) | ((n << 8) & 0xff00ff00);
n = ((n >> 16) & 0x0000ffff) | ((n << 16) & 0xffff0000);

-- C code which reverses the bits in a word.

Of course, all of this can be represented with division and modulus, although it would grow bigger and uglier. :)

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.

so long, and thanks for all the fish

[HD Webdev]

Thank goodness it's not the number 42...I'd be on my way to the local pub, towel in hand, if it was!

slow news day?

[mother_superius]

This site doesn't seem to be much. He seems to have found that the numbers will generally come to make a palindrome over time. This is something cute but nowhere on the site does he even try to explain relevance to anything. In addition, this guy doesn't seem to know much of anything with math. He considers infiniti a number, he is suprised by obvious things, etc. He reminds me of the Greek philosophers before science as we know it came to be. He thinks whether something makes sense rather than taking it step by step in a proof (or for the simplicity of explaining it, a pseudoproof).

158

[matula]

The number 196 happens to be the first of them. How about 158? 158 + 851 = 1009, which isn't a palindrome.

Re:Bah

[Kwikymart]

yah, we could use base 36 (digits + 26 letters) and try to get "racecar" out of it.

Re:who cares?

[stipe42]

care to elaborate? That sounds interesting.
stipe42

It's not true, though.

[dark-nl]

879, 1997, and 7059 also have this property, whatever it is. The guy even explains this on his site. I wonder who he is, and why he doesn't put his name anywhere.

Re:It's not true, though.

[Misanthropic Lycanth]

He has put his name on the website. Several times. It's Wade VanLandingham. Check the milestones page. He also searches for the numeric expression for his name in the numbers that result from the additions.

Palindrome?!

[mkrist]

What IS a palindrome? I saw that it is something with the same number? Can someone please make it clear to me? Thanks.

Re:Palindrome?!

[Andy_R]

If you reverse the digits of a number and get the original number again, it is palindromic.

The term is more usually used for words or sentences that have the same property of reversibility (spaces are generally ignored), such as "madam, I'm adam".

Re:Palindrome?!

[eric6]

the longest one i've heard of (and the title of a book, no less), is "go hang a salami, i'm a lasagna hog"

Fractal Drawing

[DennisZeMenace]

If anyone has some time to waste, it'd be fun to create a fractal drawing of it. Well it's not really a fractal, since it's integer, but anyway. It'd be only a single line of pixels (or a symmetric 2d reflection), and the color code represents how many iterations it took to reach a palindrome. I'm guessing it may not look real spectacular, but who knows...

DZM

Distributed system impossible

[MoogMan]

Excuse me for possible retardation, but surely distributed Lychrel number algorithms *cant* exist? The equation is recursive and therefore each calculation requires the knowledge of the number before it... And so, only one thread even can be used to process this with one instruction (add).

Re:Distributed system impossible

[Misanthropic Lycanth]

This confused me the first time I read it too. They aren't proposing a distributed "help us find the end of the 196 problem" program. They are proposing a "let's all find more numbers like 196" program.

Although I am interested in knowing whether or not a distributed 196p program could exist. CS and mathematicians, is it possible?

Re:Distributed system impossible

[MoogMan]

After some contemplation I have realised that, to some extent it is possible... but not in the way that you and I orignally thought. Finding the solution for one number is (as far as I can tell), non-distributable. However, a system where each client would look for one number can be split up into some parts... The small routine that figures out the numbers, another system that cross references "thread numbers", where a number like 196 will end up as another number (887?) and so if the solution for 196 is found, the solution for 887 is implicitly found. There could be a main database storing this correlation, and whenever a client solves a number, it checks against the database for other non-solved numbers and eliminates from the database...
Any thoughts on that process?

Re:Distributed system impossible

[Misanthropic Lycanth]

It's still one computer, one number. It cuts down the list of total numbers, but 200+ iterations will do that anyway.

The only thing I can think of is breaking the number down in chunks. Take the first m digits and the last m digits and produce a new set of digits. The ends of the sequences are "fuzzy" because a lesser significant digit could carry. A central computer would assemble the digits from different sources and check.

This is clearly not a solution. It takes too much effort to organize this.

Re:who cares?

[ergo98]

It gets to be more questionable when they talk about a "SETI@Home" type of distributed application : For something so thoroughly useless, that is nothing more than an absurd waste of electricity (and it is a waste of electricity: Computers are left on when they would have been turned off, and barring that the CPU is actively consuming power rather than the dramatically lower power usage halt instruction which it would be running normally).

Of the 90 that start as palindromes...

[NoMoreNicksLeft]

Did he ever bother to check if they're the palindromic result of a smaller non-Lychrel number?

Time Cube

[Myco]

I'm almost sure the answer to all of this has to do with Nature's Harmonious Time Cube.

It hurts to understand

[JustShowMeTheFives]

So the iteration, when seeded with 196 or whatever, can run endlessly without becoming a palindrome (in base 10)...and this is the only one found under 10,000. Isn't that because after 10,000, the numbers are just too large to realistically luck out like that? After all the iteration practically doubles every time. That would get your needle in a haystack quick enough. Are we supposed to be helping or hurting the terrorists with this?

Re:who cares?

[GigsVT]

Yeah, that is a little more questionable when you start talking about opportunity cost, but when you get down to the basics, there are a lot of useless pursuits that waste resources but make people happy in some way.

Car audio springs to mind. Think of all the gasoline wasted by burning an extra 500 watts all the time. Unless you are going to go facist on people and tell them they can't waste their time and money with things that don't really benefit society, then you really can't say much about something like this.

196 is very definitely palindromic

[mike_lynn]

... in binary.

(196) 11000100 + 00100011 = 11100111

Re:Base ten

[Blakflag]

Well if your gonna check for palindromes in binary then you should be adding and flipping digits in binary also.

But scarily enough, if we invert that bastardized conversion and add in binary while checking in base 10...

196 d
11000100.b
11000100 + 00100011
101010010101111111001111.b
1010100101011111110011 11.b
11100111 d

A Freakin decimal palindrome on the first addition. WiErD mathematics.

Re:Base ten

[Blakflag]

oops the microsoft calculator did an evil thing to me.. Guess I dont know how to add any more.

But actually its
11000100.b + 00100011.b
11100111.b

which is a binary palindrome. Like most binary addition probly ends up. :P

How is it NOT pure math?

[yerricde]

This is not pure mathematics, as it relies on the numbers being represented in a certain base.

Number theory per se is considered part of pure mathematics, the N in NIGGERS. When my number theory book defined some basic concepts in terms of first principles, it defined "base" soon after it defined "multiplication". It roughly went like this: The base b representation of the positive integer n is a finite sequence of non-negative integers less than b such that the last element x > 0 and the sum[i = 0..n-1](x[i]*b^i) = n.

This statement about 196 and base 10 is a statement about sequences of numbers.

Re:How is it NOT pure math?

[martyn s]

Watch this post get modded down because it says "the N in NIGGERS".

Go to either -1 or GeoCities

[yerricde]

I have assuredly found an admirable proof of this, but the Post Comment box is too narrow to contain it.

I suspect that you're either bluffing or just making a bad joke analogous to the DMCA jokes, the Beowulf jokes, the "All Your Base" jokes, the "2. ???; 3. PROFIT!" jokes, the "Priceless" jokes, etc. that get moderated to -1 on Slashdot.

On the other hand, if you really do have a proof or disproof of the existence of Lychrel numbers, then fire up Emacs, make a web page outlining your proof, and post it on a public server. Go to GeoShitties if you have to. Leave the finest details to the reader if you have to. Pure mathematicians want to see it. Bad.

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...

hack value

[bp33]

This effort is interesting for its hack value, but it seems to me that 196 isn't special so much as a "Lychrel" Number but as an input to the formula. That is, a prime number clearly has a unique and specific mathematical property, but 196 is just .. 196.

I propose a different formula. Take a number, multiply it by the reverse of itself (instead of adding it) and then, oh, subtract the original number plus its reverse, and repeat until you have, hmm ... the length of any substring of the resulting digits that are numerically ascending or adjacently equivalent is a prime number. Bonus points if there is more than once sequence in the result that is a prime number.

This is a made-up-while-typing-in-the-slashdot-comment-box problem, yet you could have whole web sites dedicated to the search for these 'special' numbers. But are they really 'special' numbers?

BTW, I searched the site and Google but didn't find any indication of *why* the name Lychrel was chosen, only the blackboard entry where they were first named.

196 doesn't work in binary.

[UserAlreadyExists]

It is indeed an arbitrary definition of palindrome. Try doing this in binary:
196 (Dec) = 1100100 (Bin)
1100100 + 0010011 = 1110111
However, there may be different sets of Lychrel numbers in different bases. (I don't know, this is the first time I've heard of Lychrel numbers.)

The first question that comes into mind...

[Wolfier]

Is the set of all palindromic numbers of the same 'size' as all natural numbers?

Re:The first question that comes into mind...

[Eharley]

So you want to know if there is an infinite number of palindromic numbers? Probably.

There can't be more palindromic numbers than natural numbers since each palindromic number is a natural number. And since we measure infinities by putting sequences in correspondence with the natural numbers, the size of the set of palindromic numbers cannot exceed the size of the set of natural numbers. Consult W. Rudin's "Real Analysis" for a good proof of this.

If there is only a finite number of palindromic numbers than a computer could exhaustively search and enumerate all of them. Then again, how would we know that we've reached the end? They could be very spaced out. What we need is a mathematical proof.

However, number theory is a very hard nut to crack. The algebraic structure of numbers is quite elusive. Showing how many palindromic numbers there are one way or the other is probably going to be exceedingly difficult. If not because number theory is hard but because there is no practical interest in this subject.

Okay, the facts...

[pla]

As the primary author of the distributed Lychrel search, I would like to point out a few things (I have paraphrased all "quotes" below based on a perusal of comments so far)...


Does this have any use?

Absolutely none. If you have a use, let us know. Otherwise, it doesn't really have any application. We *have* discovered that iteration of reverse-and-add has fractal nature, does that at least get us a few "coolness" points?


This won't work, the algorithm has too much serial dependancy.

Wrong on two counts. First, the core concept of reverse-and-add *does* yield to parallel implementation (Jason Doucette worked through this one, and has come up with a *really* elegant solution). Second, searching for Lychrels does not involve "deep" iteration of reverse-and-add. It only requires taking a *lot* of numbers to a certain arbitrary depth (10000 digits more than suffices at the starting value range we currently can deal with). So, although deep iteration takes some work to efficiently parallelize, Lychrel searching gives a nearly linear speedup with the CPU count.


Why limit yourselves to base-10?

For the simple answer, all lower bases have trivial proofs of either an infinite number of non-terminating sequences, or no known non-terminating sequences. This makes base-10 the lowest "interesting" base to work in. Of course, the question strikes me as odd... Why not ask why we use base 10 for counting? Why not base 2, or 7, or 60? Just as meaningful of a question.


What does "Lychrel" Mean?

196 exists as the lowest (base-10) number that does not seem to terminate on iteration of reverse-and-add, but not the only one. Obviously, any consequent of 196 (such as 887) will also never terminate. Other numbers also never terminate, such as 879, and they never converge with the series generated by earlier numbers either. So, needing a name for these numbers, Wade VanLandingham picked the word "Lychrel" (pronounced la-shrel), and the active 196 community accepted it into common use.


And, the big question...

Can I run your distributed client?

Yes and no. First of all, it will take me about another 3 weeks of development to finish the client (the server program already works adequately, for now). Second, our currently available server machine can only handle around 100 clients, and just within the 196 community we have offers to run about half of those already.

So, although we will gladly consider offers to run the client, as our single biggest need for help we require a nice server with a fat pipe. The actual bandwidth needed won't go that high, but the total could (our next data set, 1E10, we predict will take 20-40MB of traffic. The 1E11 set will take up to 17 times that, and so on... It grows quickly).

Additionally, most Slashdot readers run Linux. Although I plan to write a Linux client, at the moment my optimized reverse-and-add routine only builds under Windows (mostly because I hate AT&T syntax assembler, I actually prefer coding for Linux otherwise). So, if anyone wants to volunteer to convert a 250+ line inline assembler function (with MMX) from intel format to AT&T, drop me a line (you would of course get full credit for your contribution).


Umm... That seems to cover most questions.

Re:Okay, the facts...

[bp33]

> Additionally, most Slashdot readers run Linux.

Really? I don't want an OS flame war but wonder how you knew this.

Re:Okay, the facts...

[gloth]

For the simple answer, all lower bases have trivial proofs of either an infinite number of non-terminating sequences, or no known non-terminating sequences. This makes base-10 the lowest "interesting" base to work in. Of course, the question strikes me as odd... Why not ask why we use base 10 for counting? Why not base 2, or 7, or 60? Just as meaningful of a question.

I'd be very interested to see any of the proofs that you're talking about. As a math geek myself, I firmly believe that proofs/theory are they way to go here. Number crunching can never show that there are any Lycrel Numbers at all!

Re:Okay, the facts...

[pla]

From http://www.math.niu.edu/ ~rusin/known-math/96/palindrome...

First, there is a regular family which can be shown to extend to any power of 2:
Base 2: 10(n 1s)1101(n 0s)00

After 4 iterations, becomes same thing with n increased by 1.

Base 4: 10(n 3s)3323(n 0s)00
After 6 iterations, becomes same thing with n increased by 1.

Base 8: 10(n 7s)7767(n 0s)00
After 8 iterations, becomes same thing with n increased by 1.

Base 16: 10(n Fs)FFEF(n 0s)00
After 10 iterations, becomes same thing with n increased by 1.
...

Base 11: 1246277(n As)A170352495681825A5026571A506181864A5143171(n 0s)0872542
After 6 iterations, becomes same thing with n increased by 1.

Base 17: 10023AB83E3B983CFGEC556G4G010(n 0s)0FGCG10FG505GF020CGF(n Gs)GG11G4F655DDGGB299B3D38BB320G
After 6 iterations, becomes same thing with n increased by 1.

I will look for the "missing" proofs under 10, and post again later if I find them.


Incidentally, I agree with you that iteration will not "prove" anything, and we can't even say with certainty that the numbers we find count as Lychrels (since we can't even prove that any non-terminating base-10 numbers exist under repeated reverse-and-add, without which Lychrels simply do not exist).

However, our search *has* illustrated quite a few properties of reverse-and-add, and we hope that will lead us in the direction of a proof.

For example, at our current "solved" size, 1E9, the set of Lychrels grows by a factor of 17 for each new decimal order of magnitude. Simply extrapolating that shows that this trend cannot continue forever (in fact, it *must* end by 1E27), so why such a high rate of growth at lower numbers?

Basically, although I have a reasonably good (though certainly non-expert) ability to create formal proofs, neither I nor anyone else (including a few "real" mathematicians, not just hobbyist hacks such as myself) really know where to begin in proving the existance of Lychrels. If you have any ideas on this, feel free to let me know. ;-)

Re:Okay, the facts...

[spongman]

okay, i gott ask, how are you parallelizing this? it seems like a serial operation to me.

Re:Okay, the facts...

[pla]

okay, i gott ask, how are you parallelizing this

For deep iteration, I won't say (since I didn't do most of the work on it, I don't consider it mine to reveal). Suffice it to say that I have written a decent amount of parallel code (MPI, for my 8-node Beowulf cluster), and IMO Jason's solution would work very well.

For the distributed Lychrel search, I uniformly divide the search range up into a number of large chunks (currently 9000, giving each client between one third and one half of a billion numbers to check). The client programs work on one chunk at a time, finding a set of "relative" Lychrels to an arbitrary depth (currently 10k digits). The client then returns its set of relative Lychrels, and by running what amounts to a standalone version of the client on the amassed results of all clients over the entire range, I end up with an accurate listing of all Lychrels in the target search range (or, more accurately, a set of "relative" Lychrels over an input set consisting of all positive base-10 integers).

Perhaps the non-obvious part here involves the idea of "relative" Lychrels... Basically, Lychrels only exist relative to earlier known Lychrels (ie, they don't converge with earlier ones). Merging those lists into one larger list takes little effort (compared to the task of identifying the Lychrels relative to each list in the first place).

As an example, consider the lists {0, ..., 499} and {500, ..., 999}. From the first list, 196 starts the only unique series that does not terminate. From the second, both 879 and 887 form unique nonterminating sequences. This gives us a combined result (such as the distributed clients would return) of {196, 879, 887}. Doing the *exact* same processing with this new list as the input, we find 887 results directly from 196, and 879 does not. So, as a final result we get {196, 879}, correctly identifying all Lychrels (over the set of all integers) under 1000. And, the post processing took only 0.333% of the CPU time of the "real" work. Implementation-wise, on a fast enough machine this post-processing could occur in real-time, merging a client's returned list of results with a master list of all results known so far.

Re:Okay, the facts...

[spongman]

ah, so I assume this is for finding lychrels and not for finding the palindrome for 196?

I can't think of any way to parallelize the iterative reverse/add process itself...

Re:Okay, the facts...

[Yunzil]

Why not ask why we use base 10 for counting? Why not base 2, or 7, or 60? Just as meaningful of a question.

No really. Unless you have 2 or 7 or 60 fingers. :)

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 ...

Re:palidrome vs number bases

[cperciva]

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

No. In any base, the number of palindromes less than n is O(n^(1/2)).

Re:who cares?

[DrSkwid]

Binary computers actually arrived on the scene over 200 years after Boolean algebra was invented and refined by George Boole and first presented in a paper by him in 1854.

"Boole's system of logic is but one of many proofs of genius and patience combined." was how De Morgan commented. It is not recorded how many whiny teens said "so what".

It's first real practical use was for telephone switching.

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:Three Years Of Computing

[bp33]

The accounts are also interesting when you note the times it has taken to crunch a million numbers. John Walker took three years on his Sun 3. 4-5 years later it took two months on the hardware available at the time. According to the milestone reports on the 196 page, it takes about a week now (on a uniprocessor P4 @ 1.8Ghz).

The code may be different but its the order of magnitude difference in the run times that's significant.

OT: Stack Overflow in Perl

[bentini]

(Do NOT run this on numbers that don't have known palindromes since it will cause a stack overflow. :)

Would this, actually? If so, it's a shame. That's an obvious case for tail-recursion elimination. I guess perl doesn't demand this like scheme does?

Will parrot support the ability to do stack-based tail recursion elimination? I know that this has been one of the big pains of java-based scheme implementations: For security reasons, it's hard to mess with the stack in the apppropriate ways. Right? Cause that code needs only constant stack space, right?

It'd be a shame if this new technology everyone is investing so much in... OK, I meant parrot, that apparetnly perl 6 will be based on... Is not going to have hooks to support that type of optimization that doens't just improve coefficients, but takes you from O(n) to O(1)...

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.

A Perl Script to Find Lychrel Numbers :-)

[@madeus]

While I've been wating for the film to start I came up with this....

The only problem is doesn't work when the numbers are _really_ large (like hundreds of characters long), I think that's an issue with one of the functions :/.

#!/usr/bin/perl
# Stupid Perl script to find Lychrel numbers.
# Yeah yeah inefficent, but very quick and easy to write :-)
# Iain Collins, iain_collins@mac.com

use Math::BigInt;
use Strict;

my $start_number = $ARGV[0];

print "Searching For Lychrel Number, starting at: $start_number....\n";

my $forward_number = $start_number;
my $reversed_number = reverse($forward_number);
my $result;
my $result_tmp;
my $i = 0;
my $count = 0;

while ($i == 0) {
my $n = Math::BigInt->new($forward_number);
$count++;
$result = $n->badd($reversed_number);
$result =~ s/\+//g;
$forward_number =~ s/\+//g;
$reversed_number =~ s/\+//g;
print "$count: $forward_number + $reversed_number = $result\n";
$result_tmp = reverse($result);
if ($result_tmp == $result) {
print "Palendrome Found?\n";
$i++;
}
$forward_number = $result;
$reversed_number = reverse($forward_number);
}

Similar problem: Home Primes

[akruppa]

There is a similar problem, the "Home Primes". A definition is on Eric Weisstein's MathWorld page.

In a nutshell, you start with a composite number, write down the prime factors in decimal in ascending order of size and so get a new number, which may either be prime or composite. If prime, then this is the Home Prime of the number you started with. If composite, repeat.

All the numbers below 100 reach their Home Primes rather quickly - except for 49 (and thus, 77). This one is expanded to 95 steps by now and has grown into a 194 digit number which is becoming increasingly difficult to factor. The last 15 steps of this sequence were done by Paul Leyland of Microsoft Research, Cambridge, and myself. The 95th step is factored down to a 153 digit composite which we are struggling with right now. See Patrick De Geest's page for the current status of this problem.

A difference between Home Primes and the 196 problem is that Home Primes can be shown to exist for every number - the probability that no Home Prime is found in as the number of expansion steps goes to infinity converges to zero. It is, however, quite possible that the number gets too large to factor with today's resources before the Home Prime is found. (That seems to be happening in the 49 case right now.) AFAIK no guarantee of a terminating sequence exists for the Palindrome problem, it is possible (and likely) that the 196 sequence spins off into infinity without ever becoming palindromic.

As far as I can tell, there is no practical use for either Palindromic numbers of Home Primes. It's just recreational - a way to spend spare time no better or worse than board games or sports on the TV. Except it involves massive amounts of cpu time and pretty advanced algorithms (i.e. ECM and GNFS) - the study of which I find extremely intriguing. It's probably one of the geekiest ways to spent your time (not that I were proud of that, I merely can't help it).

Alex

Another odd thing about 196

[Archie Binnie]

Okay, so it's late and I was tired... but the first thing I tried when I reached for a calculator was reversing 196 (=691) and adding the sum of the digits (1+9+6). Adding them both together gives 707 (a palindrome).

Spooky or what?

But seriously, could this have anything to do with why it may have this odd property?

Re:Another odd thing about 196

[ghastard]

Okay, so it's late and I was tired... but the first thing I tried when I reached for a calculator was reversing 196 (=691) and adding the sum of the digits (1+9+6). Adding them both together gives 707 (a palindrome).

Take this even further. By taking 196, and adding 16 (1+9+6) to it you get 212, another palindrome!

This is pretty cool.

BUGFIX: A Perl Script to Find Lychrel Numbers

[@madeus]

Of couse, was using "==" to check for matches (Perl wiggs out at this). Testing for matches by treating the values as a string works fine. (i.e. by using "eq" instead of "==").

This versions works, even with ReallyBigNumbers...

#!/usr/bin/perl
# Stupid Perl script to find Lychrel numbers.
# Yeah yeah inefficent, but very quick and easy to write :-)
# Iain Collins, iain_collins@mac.com

use Math::BigInt;
use Strict;

my $start_number = $ARGV[0];

print "Searching For Lychrel Number, starting at: $start_number....\n";

my $forward_number = $start_number;
my $reversed_number = reverse($forward_number);
my $result;
my $result_tmp;
my $i = 0;
my $count = 0;

while ($i == 0) {
my $n = Math::BigInt->new($forward_number);
$count++;
$result = $n->badd($reversed_number);
$result =~ s/\+//g;
$forward_number =~ s/\+//g;
$reversed_number =~ s/\+//g;
print "$count: $forward_number + $reversed_number = $result\n";
$result_tmp = reverse($result);
if ($result_tmp eq $result) {
print "Palendrome Found?\n";
$i++;
}
$forward_number = $result;
$reversed_number = reverse($forward_number);
}

Why?

[Snaller]

.... erm ... yeah... but why? Do they expect to find a message from God... or?

So the meaning of life....

[Snaller]

...is actually 196!

Damn! I've been worshipping the wrong number all this time!

lychrel-seeker for hexadecimal

[k-zed]

If anyone's even marginally interested, here's a quick'n'dirty C program to seek these 'magic numbers' - of the hexadecimal sort. Should compile on practically anything - just type 'make' ...

the sources

set is the same size as naturals

[honkycat]

As you've pointed out, every palindromic natural is a natural therefore set of all palindromic naturals is smaller or equal in size to the set of all naturals.

For any natural, reverse its digits and concatenate into a palindromic natural. This doesn't give you every palindromic natural, but it shows that the set of all palindromic naturals is larger or equal in size to the set of naturals.

Thus, the set of palindromic naturals is either smaller or equal in size to the set of all naturals and it's either larger or equal in size to the set of all naturals. It follows that it's equal in size.

Re:set is the same size as naturals

[Eharley]

But I think that the idea is that you have to reverse and add the digits of the number at least once before it is a palindrome.

So we could run into a problem if there is a palindromic number that when multiplied by 2 is no longer a palindrome. Any thoughts?

Re:set is the same size as naturals

[Wolfier]

Okay. The fact that the size of the sets are equal is actually really trivial. I was just posting my first thought if I were to attack the problem - I wanted to know what you guys think FIRST if you were to try solving this. :)

So God works in decimals?

[Second_Derivative]

Yeah, I'm sure 196 is meaningful in base 10. And maybe something else is meaningful in base 14 or whatever. Unless god works in decimal I don't see much significance in this other than interesting artefact of our number system. Still, if someone could create a proof...

False assertion in Fourmilab paper

[blonde rser]

In order for addition of a digits-reversed number to yield a palindrome, there must be no carries in the addition and hence each pair of digits must sum to 9 or less.

This is an assertion from John Walker's Three Years Of Computing . Several other sites reference this statement and it appears to be over generalized. Certainly for any addition of this form the sum is a palindrome but not all digit-reversed sums that are palindromes are of this form. For example conside
74 + 47 = 121
or
7744 + 4477 =12221
These are only a few counter examples. There may be some general rule on how to generate all such numbers.

I hope this statement isn't fundemental in any greater works.

Fairly Simple Perl Script

[suwain_2]

Being the insane geek I am, I quickly whipped up a Perl script that I *think* works. I tested it with an example given (87), and got 4884, which is correct.

The program will print out the number of time's it's looped (the number of numbers it's tried), and then what the number is. Every million numbers (I wanted it to be big enough that it wasn't printing out miles of crap, but small enough that I got occasional outputs to know what it was up to), it prints out the time elapsed, the number of repititions, and the current number.

#!/usr/bin/perl

$in = $ARGV[0];
for ($x = 1; $x <= $in; $x++) {
$reverse = reverse($in);
$in = ($in + $reverse);
$back = reverse($in);

if ($in == $back) {
print "$x\t$in\n";
exit;
}

if ($x % 1_000_000 == 0) {
$time = (time - $^T);
print "\tat $x... $time sec elapsed... $in\n";
}
}
# the end...

Constructive criticism and whatnot is welcomed. (And yes, I know, my variable names suck...)

Re:Fairly Simple Perl Script

[suwain_2]

Sorry for the entire comment being fixed-width... I put my code in tags, and that didn't seem to work so well... So I changed the whole thing to code and hit submit. BTW, 12 million repititions in 176 seconds, and nothing found... But I'm beginning to wonder... Answers very quickly get expressed as integers like "1.40025556644157e+15"... Does a reverse() of this return 51+e7...", or does Perl have the "real" number internally? My script doesn't work for long if the former is the case.

Re:Fairly Simple Perl Script

[jareds]

But I'm beginning to wonder... Answers very quickly get expressed as integers like "1.40025556644157e+15"... Does a reverse() of this return 51+e7...", or does Perl have the "real" number internally? My script doesn't work for long if the former is the case.

No, Perl does not perform arbitrary precision integer arithmetic.

Try perl -e 'print scalar reverse (2**60);', for example.

Also, I don't understand why you're using $x <= $in as the test case for your loop. $in will grow much faster than $x.

Re:Fairly Simple Perl Script

[suwain_2]

Ugh... It was kind of written in five minutes, without a whole lot of thought.

Now that I think of it, I'm not sure what's with

$x <= $in

either, but it seems to work for smaller numbers at least. :)

I hate when I write code, and then later don't understand why something works... Thanks for your comments, though.

what about other bases?

[GPS]

What's so special about base 10, other than that most of us have 10 digits on our hands? (Devil's advocate)

Numbers patterns become interesting when they can be mapped to real-world phenomena where the phenomena closely "follow" the patterns.

Has anyone tried to find "Lychrel Numbers" on other bases?

Longest delayed palindromes. World Record!

[flowerp]

This webpage has some of the longest delayed palindromes ever found. It's fun reading, too. This guy has programmed a lot of interesting stuff in his CS "career" so far ;)

http://www.jasondoucette.com/worldrecords.html

Re:who cares?

[GigsVT]

It is not recorded how many whiny teens said "so what".

Haha, now with the modern marvel of Slashdot, the "so what" score of any new idea can be qutified and archived for history.

"Slashdot's system of comments is but one of many proofs of ignorance and impatience combined." :)

What a waste of time...

[hal9k]

Assuming that there is any significance to such numbers is wasteful. As if the base 10 numbering system is the root of the universe....

useless comment on the Lychrel page

[Kwantus]

well, duh, yes, OBVIOUSLY if N is Lychral so is its reversal R and iteration N+R :p how could anyone slip into thinking (or writing) 196 was the only one less than 10000 ??

Re:useless comment on the Lychrel page

[Kwantus]

oh shoot

i'd meant also to mention this iteration was something i'd played with quite a bit as a kid reading Creative Computing. Until I figured out it wasn't the only way to make palindromes. I had mental blocks of my own I guess :)

This is old news

[Anonymous QWord]

I remember reading about this in a magazine from 1984. 'Twas The Transactor, volume 4 issue 6. It includes an implementation for the 6502, though it gives up after 255 digits. And it's less than 196 bytes. Get with the times. ;)

Anyone want source?

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!

How about 158? How about NO

[decaying]

1009 + 9001 = 10010
10010 + 01001 = 11011 Which is a palindrome

Well, except for...

[UberQwerty]

...all of its children. After each iteration beginning with 196, you get another number that will never become a palindrome.
196 + 691 = 887
887 + 788 = 1675

Obviously, if 196 doesn't work, then 691 won't, and neither will 887, or 788, or 1675 etc
Or did the article mention that another condition for Lychrel numbers was that the number can't be part of another one's sequence?

The sets are of equal size

[jareds]

Then again, how would we know that we've reached the end? They could be very spaced out. What we need is a mathematical proof.

You can't reach the end. Any number whose digits are all less than 5 is obviously palindromic, and the set of such numbers is countably infinite. Thus, the set of palindromic numbers is not smaller than the set of natural numbers.

Since, as you pointed out, the size of the set of palindromic numbers cannot exceed that the set of natural numbers, the sets are of equal size.

I KNEW it was a Fermat joke

[yerricde]

the joke you so obviously missed

No, I did not miss the Fermat joke. Note that I accused dave_mcmillen of making a "bad joke". I've just noticed that Fermat jokes in Slashdot pure-math articles are quickly becoming as tired as Beowulf cluster jokes in hardware articles.

A Perl script for Buddha

[The Panther!]

Sorry, I wrote this on my windows machine at work, so I don't have a proper sh-bang at the top, but this'll tell you whether or not a number converges to a palindrome in less than 10,000,000 iterations, and if so, where and how. A preliminary run took a long time. :-) Drop the iterations down to about 1,000 and it's pretty quick, and gives you an idea what might be interesting to explore.

(ps-yes, it was nicely formatted and commented, but the LAMENESS filter rejects code like that. ;-)

# Do some perl madness
sub FindPalindrome { my ($value) = @_; for (my $j=1; $j=10000000; $j++) { $value += reverse($value); if ($value==reverse($value)) { print("$_[0] found in $j steps, palindrome of $value.\n"); return $j; } } print("$_[0] might be a Lychrel number!\n"); return $j; } { for (my $i=0; $i1000; $i++) { FindPalindrome($i); } }

Re:A Perl script for Buddha

[jareds]

Your program will flag anything as a possible Lychrel number once it gets to the point where Perl represents it in scientific notation, which will always happen in way less than 10,000,000 iterations. In fact, it will always happen in less than 70 iterations. Stick $value in the string printed if it might be a Lychrel number to see what I'm talking about.

Re:A Perl script for Buddha

[The Panther!]

Quite right. I realized that when I tried to compute 1,000,000 iterations of 196. I had (mistakenly) thought that Perl operated on strings for integers, which would ultimately mean it internally supported BigNums. A minor change to the script to use Math::BigNum and it works fine, though quite a bit slower. :-)

Direct Link

[Grimwiz]

If you want to link directly to the sequence without using the search box use the following URL:
http://www.research.att.com/cgi-bin/access.c gi/as/ njas/sequences/eisA.cgi?Anum=023108

My daughter recently covered this at school, but they didn't discuss any exceptions to the rule. However, they chose random 3 digit numbers to check, so there was only a 13/900 chance each child would have a LOT of homework that night :-)

math research script kiddie

[epine]

Couldn't have said it better myself. I guess a lot of people haven't paid any attention to what Wolfram is trying to say in his new book: this kind of weird algorithmic pattern shit is as common as sand on a beach. Around 1983 people were playing around with wondrous numbers, which at least has the replayability advantage of not producing a monotonic sequence.

My first curse upon the world: this problem has a proof that involves analyzing 100,000 special cases producing 10,000 pages of dense results, and none of these cases can be reduced.

My second curse upon the world: some idiot bothers to find it.

The suggestion that no pure mathematician has any clue about where to begin is not equivalent to saying no pure mathematician has any clue about whether to begin. A proof is hardly worth the paper it gets published on if it doesn't reflect back on other branches of theory.

Thank goodness!

[flacco]

I, for one, am glad they haven't thrown away that distributed processing power on something trivial like curing cancer instead.

hmm new encryption formulas!

[linuxislandsucks]

hmm we need to add those formulas to gnu pgp makes a nice set of new encryption formulas.....

Encryption keeping governmental power from ging absolute!

No, more like distributed.net, not SETI@Home...

[FlyerFanNC]

...cuz there may actually be a solution. SETI@Home *has* no solution.

Re:who cares?

[jandrese]

Huh? Maybe you've changed the base on me without telling, but isn't 1854 + 200 == 2054?

Besides, much like this theory, boolean algebra was all but ignored by the mathematical community at large until the 1940s, when the introduction of computers made the field suddenly relevant.

Re:who cares?

[DrSkwid]

hehe those darn century numbers

well I was approximately correct.

my point was that sometimes an idea can arrive ahead of it's time

Mind you, the palindromic properties of numbers will probably have to wait for quite some time.

All no. 1..10,000 become pal. besides 196 = FALSE

[Juge]

Sivar said:

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

This is NOT true. This has been quoted incorrectly for a long time. There are about 250 numbers under 10,000 that never appear to resolve into a palindrome.

If 196 will never solve, then it is obvious that 691 will never solve. Also, it is obvious that 295, 394, 493, 592, and again 691 will never solve (what are normally called the 'consequences of 196').

The first iteration after 196, 887, will thus never solve, as well as 788. From these two number, it is obvious that 986, 887, 788, 689 will never solve (the consequences of 887).

The other lychrel sequences under 10,000 are 879, 1997, and 7059. Each of these numbers, and their sequences, and all consequences of these numbers and their sequences will also never solve.

If you are interested in Lychrel numbers, my page here: http://www.jasondoucette.com/worldrecords.html

It shows the numbers that take the longest length of time before they solve. So, they are the closest numbers to Lychrels that aren't Lychrels. It also details my section of the 196 Quest.

Jason Doucette
www.jasondoucette.com

Bruce Dickinson, Mathematician?

[ohboy-sleep]

So the best we've been able to come up with is that 196 is... weird. But why? Let's analyze it. Hmmm, it appears to be a cubed number. What? No!... It's... It's... 6 cubed or "6... 6... 6... the Lychel of the Beast!" (apologies to Iron Maiden)

Re:Bruce Dickinson, Mathematician?

[ohboy-sleep]

D'oh!! I pooched the 4th grade math on that one. My bad.

Math has too much free time

[kenp2002]

I can see that there are plenty of cryptic number patterns that math people seek. But this one seems a bit, well, irrelivant. I can see the need to seek out primes and such but this "new" pattern looks like it was invented for the sake of inventing. I have a super-hard to calculate number sequence I called Slashdom. It is any integer when multiplied by 642867150 results in a number prefectly divisible by the first digit of the result. There I am a genius, now let's devote time and effort into seeking these Slashdoms so we can use them in encryption.

Come on, I tried to find one of the non-palindrom number and my first try succeeded, 3421965. Purely from a math stand point you increase you odds of a non-palidrom number by having this sequence odd-prime-even somewhere in your number.

I find this a novelty, but god help us of some twit gets a noble prize for something related to this.

Re:Math has too much free time

[kenp2002]

As a follow up just as an FYI like in logic we can NEVER test a number under these circumstances for success we can only test for failure. Because the iterations are infinite we can NEVER assure that an interation like this is a success unless we competed the interations (we can't because they are infinite) but we can test for failure (as the first failure stops the process) But come on is this process useful? Primes seem so much more useful than playing tag with palidroms...

P.S - Yes I am cranky again today, sob... It was my birthday yesterday and I only got 1 happy birthday :( I need a hug...