Talk:Happy number

old comment

I removed the text:

It can be shown mathematically that no matter what the initial number t is, the sequence t₀,t₁,t₂,.. will eventually settle between 1 and 163. What happens to this sequence after it is inside this interval can be estimated by a simple computation.

It turns out that the only alternative to ending up at 1 is to be stuck in the cycle

4, 16, 37, 58, 89, 145, 42, 20, 4, ...

as I couldn't find these claims in either reference, and I strongly doubt they've been proved. dbenbenn | talk 21:23, 28 Jan 2005 (UTC)

The second part is in the references. The first one I proved. I did not know on Wikpedia you must submit the proofs. What is a good way of proceeding about that? Assuming that the text fragment is accurate, and since it adds value to the article, maybe it should be left. What do you think? Oleg Alexandrov | talk 21:39, 28 Jan 2005 (UTC)

I should have given you more time to explain yourself. :) I got your message. I think, in the fragment above I said that it can be proved that things will settle between 1 to 163, and only from there on one can do some computerized checking. So, you want me to include the proof? It is not that hard.

Which reference do these facts appear in? Note that the second claim implies the first. I agree that if you can show any number eventually goes below 163, then it's simply a matter of checking those cases. dbenbenn | talk 21:48, 28 Jan 2005 (UTC)

OK, I made a mistake. In those references it was mentioned only that we either are in the cycle 4, 16, .... 4, or otherwise end up at 1.

You are right, it was not mentioned that you get stuck in 1 to 163, and what happens next.

So, I can prove that you get stuck in 1 to 163, and I had written a code to show what happens next, but that one was voted for deletion (which I think was appropriate).

So, we have 2 options. (a) Keep things the way they are now. (b) Put back the statement with the proof. What to do about the "fact" of what happens after you are between 1 to 163 I don't know. That can be checked only computationally.

Suggestions welcome. Oleg Alexandrov | talk 21:54, 28 Jan 2005 (UTC)

Are you referring to the MathWorld article? It says

"Unhappy numbers have eventually periodic sequences ... which do not reach 1 (e.g., 4, 16, 37, 58, 89, 145, 42, 20, 4, ...)."

E.g. means for example. The article does not say that unhappy numbers always end up in the 4, 16, ... cycle.

If you have a proof of this fact, I suggest that you should publish it. You're a postdoc, you need to publish papers! And once it's been peer reviewed, we can refer to it here no problem.

Additionally, I invite you to write your proof here on this talk page. I would be interested in seeing it. But I don't think it would be appropriate to include in the article until it's been peer reviewed. dbenbenn | talk 22:06, 28 Jan 2005 (UTC)

Got it! I did not know you guys are so strict about things, but that makes sense. So, let's drop it.

About the proof. The idea is quite simple. If you start with a large number, and sum the squares of its digits, then, as you might guess, what you get is much smaller than the original. If you do it several times, you keep on getting much smaller number each time, until you are between 1 and 1000. From there on, a bit more care is needed with calculations (but is still simple) to see how much further you can drop. For example, for 999 you end up after one step with 81+81+81 obviously, which is 243. From the range 1-243 the number with the largest sum of squares of digits is 199, for which you get 1+81+81=163. And from here on, a little program can check where you end up after several iterations.

This could make a nice math competition problem. But I agree with you, not worth the trouble puttin on Wikipedia. Oleg Alexandrov | talk 23:07, 28 Jan 2005 (UTC)

Er, um, let me back-pedal please. I hadn't actually thought about this sequence at all when I wrote the above. I had just assumed that things were difficult. Now having thought about it for a few minutes, I agree it's quite simple. Any number above 999 goes to a number with fewer digits, so you eventually get to something 999 or below. Then it's just a matter of checking. This level of reasoning can certainly go in the article. I'll start adding it right now. dbenbenn | talk 23:18, 28 Jan 2005 (UTC)

Done. For what it's worth, it was never about strictness. The problem was simply that no justification for the claims at the top of this thread were given. dbenbenn | talk 23:54, 28 Jan 2005 (UTC)

There is a problem with proof. You see, you focus too much on what happens over 100. If you are at 99, you jump up to 162. And then, if you drop below 100 again with your reasoning, you can come back later. So it is not clear if you ever get stuck under 100. So I think you should say we are stuck between 1 and 162 and leave it that way. What do you think?Oleg Alexandrov | talk 05:44, 29 Jan 2005 (UTC)

I disagree. The computer program proves (presumably; I haven't actually checked it) that every number 1 to 99 is either happy or goes to 4. And every number above 99 eventually goes into the range 1–99. So (I think) the proof as I modified it is still correct.

The interesting fact, which I was trying to get at in my edits, is that 99 is the last number that gets bigger, which is why it's "special" with respect to this process. 162 isn't special in that sense. If you're going to stop at 162, I think you might as well stop at 1000. But the extra analysis to get down to 99 reveals some interesting structure in the sequences. dbenbenn | talk 06:24, 29 Jan 2005 (UTC)

If you put it that way you are right. And emphasizing 99 is good too. What I got confused about was the following: I would have really liked to not even talk about what happens under 163 (over 100 versus under 100), as there things jump around quite a bit. For comparison, while once we are in 1 to 163, we are dead stuck there. Your proof was not wrong, I misread it expecting something else. Oleg Alexandrov | talk 16:36, 29 Jan 2005 (UTC)

Checking up to 163

The following program by User:Oleg Alexandrov can be used to check the claim in the article about 1 to 163:

% Find the happy numbers, and the cycles which do not lead to happy numbers.
% Assume that we start at some integer between 1 and 163.
% It can be proved that the sequence t_0, t_1, ... as in the article,
% always stabilizes in this interval.

function main(m)

   A=zeros(163, 20); % row i will store the cyclical sequence starting at i
   
   for i=1:163
      N=i; % current term
      for j=1:20 % can prove that at most 18 iterations are necessary to start repeating the cycle
	 A(i, j)=N;
	 N=sum_digits(N);
      end
      
      if  (N ~= 1 & N ~= 4  & N~= 16  & N ~=  37 & N ~=    58 & N ~=    89 & N ~=   145   & N ~=  42   & N ~=  20   & N ~=   4)
	 disp('We have a problem! We neither got a happy number nor are we');
	 disp('in the cycle 4    16    37    58    89   145    42    20     4 ');
      end
   end

   
   A(1:163, 1:20) % display a table showing in each row the with all the cycles (please ignore the trailing zeros)

function sum=sum_digits(m)

   sum=0;
   p=floor(log(m)/log(10)+0.1)+1; % number of digits
   for i=1:p
      d=rem(m, 10);
      sum=sum+d^2;
      m=(m-d)/10;
   end

Dear dbenbenn, I took so much of your time. I am flattered. Thanks. Oleg Alexandrov | talk 00:45, 29 Jan 2005 (UTC)

That's what Wikipedia's all about. Thanks for explaining it to me, and I'm sorry I misunderstood at first. dbenbenn | talk 03:17, 29 Jan 2005 (UTC)

Checking up to "anywhere"

The following tiny C program checks up to "any" number (< MAXINT), which can be given as optional argument (default=99), to see that no number gives an infinite loop, but always ends up at 1 (happy) or 4 (unhappy)).

It prints a message "xxx is (un)happy" for each number, unless an additional second argument is given (e.g. "silent"), in which case only the currently checked number is displayed (without linefeed).

In view of the highly non-optimized code (KISS principle), it takes about 20 sec on my PC to check up to 10⁷ (ten million) (but maybe most time is spent in printf calls).

/* happy.c */

 sum(int i){ /* (recursively) calculate sum of squares of digits */
  return(( i<10 ) ? i*i : sum( i/10 ) + sum( i % 10 ));
 }

 happy(int i){ /* count iterations needed to reach 1 or 4 */
  int c=1;
  for ( ; i > 1; c++ ) if ( i==4 ) return( -c ); else i = sum(i);
  return( c );
 }

 main(int c,char**v){
  int i, h, max = ( c>1 ? atoi( v[1] ) : 99);
  for( i=1; i<=max; i++ ){
   printf( "%15d\r", i); h = happy(i);
   if( c<3 ) printf( "\t\t is found to be %s after %d iterations.\n",
               (h>0 ? "happy":"unhappy"), abs(h)-1 );
  }
 }

Enjoy... — MFH 23:34, 9 Mar 2005 (UTC)

A thought about proofs using computers

The proof on the main page has some quite developed (and interesting) reasoning concerning numbers below 1000, and terminates by "a computer program can easily verify that in the range 1 to 99...". Now,

• first, the computer program checks as easily (in less than a millisecond) up to 999, thus making the longest part of the proof superfluous. (Understand me well, I find this part nevertheless most interesting!)
• secondly, it cannot be checked by reading the proof whether this statement (about 1..99) is true or false. In order to have a "complete" proof, one would need the explicit list of the sequences for the remaining numbers (where the sequences could be truncated whenever a number less than the "starting point" is attained).

Of course, for the latter (explicitly displayed list) the difference between 99 and 999 is crucial. Thus, in some sense, the "lack" or "necessity" of what is missing in the end (in order to have a complete proof) justifies what is (without it) "superfluous" at the beginning. Quite remarkable! — MFH: Talk 17:20, 29 September 2005 (UTC)

unhappy 42

I made a program that prints the sequences obtained for the numbers 1..99, but only up to the point where a number less than its predecessor is obtained (i.e. the point from which on the sequence is no more increasing). I noticed that most often this point was the number 42. More precisely, I counted for each number how many times it plays the role of such a "breakpoint" (for 1..99). Here are the results:

   1: 5.    2: 1.    4: 1.    5: 2.    8: 1.    9: 1.   10: 7.   11: 2.   13: 2.   16: 3.   17: 1.
  18: 1.   20: 2.   25: 5.   26: 1.   29: 2.   32: 1.   34: 3.   36: 1.   37: 4.   40: 1.   41: 5.
 42: 14.   45: 1.   49: 1.   50: 2.   51: 2.   52: 2.   53: 2.   58: 2.   61: 3.   64: 2.   65: 5.
  68: 2.   69: 1.   73: 1.   74: 1.   80: 1.   81: 2.   82: 1.   85: 1.   90: 1.    All others: 0.

I.e., while 42 is the breakpoint for 14 numbers, all other numbers are breakpoints for at most 7 numbers.

Of course this result is related to the choice of the first 100 numbers, but I don't think this is very important. (In fact, 42 remains the favourite taking into account all numbers up to 999, but the distribution becomes more homogenious for the others.) Strange, this 42.... — MFH: Talk 22:21, 29 September 2005 (UTC)

It's the meaning of life :-) --HappyCamper 02:42, 5 November 2005 (UTC)

I wonder if that's why they chose to reference these numbers in the Doctor Who episode "42"? :-) 81.178.232.81 15:52, 20 May 2007 (UTC)

History?

So who first defined happy numbers? How did they come up with the name? What significance (if any) has the research of happy numbers had?--AlexSpurling 20:21, 5 October 2006 (UTC)

According to Unsolved problems in Nymber Theory by Richard Guy, Reg. Allenby's daughter thought of them. They have little significance, just are interesting. Bubba73 (talk), 21:05, 5 October 2006 (UTC)

Can you add that to the article, with the reference? (I don't have the book) That's an interesting tidbit. - DavidWBrooks 15:09, 6 October 2006 (UTC)

done. I wish I knew her actual name though. Bubba73 (talk), 15:25, 6 October 2006 (UTC)

unhappy 23

23 is NOT happy: 23²=529
5²+2²+9²=110
1²+1²+0²=2
2²=4
4²=16
1²+6²=37
3²+7²=58
5²+8²=89
8²+9²=145
1²+4²+5²=42
4²+2²=20
2²+0²=4
Please feedback if i'm wrong or not!!! wiki@dbserver.nu (March 30, 2007)

Wrong. 23 -> 2²+3² = 13, not 23². Sum of the square of the digits. Bubba73 (talk), 14:54, 30 March 2007 (UTC)

Sorry!

Trivia

Note: this conversation has been moved from the editors' Talk pages, to bring it to wider attention. It concerns a Dr. Who episode mention and whether it belongs in the article. - DavidWBrooks 21:37, 20 May 2007 (UTC)

Hi. I've taken the Doctor Who reference back out of the Happy number article. Wikipedia's trivia policy explicitly mentions "popular culture" sections as a prime example of what not to do in an article. It's very cool that happy numbers got mentioned on Doctor Who, I agree, but it's a fact about a Doctor Who episode, really - it doesn't help anyone who came to Wikipedia to find out about happy numbers. --HughCharlesParker (talk - contribs) 19:15, 20 May 2007 (UTC)

True, but you're overlooking another sector of readership: those who didn't come to wikipedia to find out about happy numbers, who probably don't know they exist, but who encounter them via a Dr. Who search - and therefore learn about them. Which is exactly why the reference should be in the article. If there were lots of such references then you wouldn't want them - e.g., you wouldn't include every TV-show reference to "pi" - but this is so unusual and unexpected that it's just the kind of thing wikipedia does well. Over-literal trivia-stomping in this case cuts off a route of spreading a little recreational mathematics to a world that needs all the math help it can get. - DavidWBrooks 19:24, 20 May 2007 (UTC)

Hi again. I've moved the conversation here to keep it together. You're right about people coming to the article because of the Doctor Who episode - that's why I read it. People like me, though, who came to it that way, already know it was in the doctor who episode. I'm not sure what you mean about people who didn't come to wikipedia to find out about happy numbers - if you didn't want to find out about them, why would you be reading the article? I think I must have misunderstood you - what have I missed? --HughCharlesParker (talk - contribs) 19:54, 20 May 2007 (UTC)

PS - I don't think I've pointed you to the most helpful policy page Handling Trivia is more helpful. The paragraph about pink-eye and Mir sums up what I'm on about. --HughCharlesParker (talk - contribs) 19:58, 20 May 2007 (UTC)

Here's what I have clumsily tried to convey: "trivia" mentions are an entry into many serious articles that might draw in people and enlarge their knowledge. A Dr. Who fan might stumble on this article, be intrigued about Happy Numbers, learn more about them, become a mathematician and solve all of Hilbert's Problems! OK, maybe that's a stretch ... but I love serendipity and think it should be encouraged, even if it litters some articles. - DavidWBrooks 21:37, 20 May 2007 (UTC)

I think it definitely merits some mention in this article - one or two sentences, at least. I'm not sure about having much more than that, but a lot of articles have a section "XYZ in popular culture", etc. There definitely should be some mention about it here. Bubba73 (talk), 23:22, 20 May 2007 (UTC)

I certainly think it's worth mentioning. Instead of a popular culture section which had one entry (and might never get a second), I have placed it in the tiny happy prime section where it belongs naturally in agreement with WP:TRIVIA. Note also that WP:TRIVIA is not policy but a guideline. And WP:HTRIV is neither; it's an essay. I don't think the paragraph there about pink-eye and Mir is a good analogy to happy numbers. Mir is an extremely notable topic with huge amounts of notable information "competing" for space. Happy numbers are a rather obscure topic which have a small article and may never have been mentioned in popular culture until they suddenly got this high-profile reference which probably greatly increases the number of people who have ever heard about happy primes. By the way, I did not know about the Doctor Who reference until an editor inserted mention in Happy number which I watch, and I enjoyed hearing about it. I look forward to see the episode when it airs on a channel I can see. PrimeHunter 23:51, 20 May 2007 (UTC)

I think the line about Dr. Who is hard to see in the article. If it shouldn't go in its own section, then I suggest making it a subsection of Happy Primes, so it appears in the table of contents and is set apart a little from the rest of the text. Bubba73 (talk), 00:25, 21 May 2007 (UTC)

The happy primes section is only five lines and the only place primes are mentioned. I don't think the last two lines should be the only subsection in the article, and I don't think it's necessary to show this reference in the table of contents. PrimeHunter 00:53, 21 May 2007 (UTC)

Bubba73: you're right, a lot of articles have a section "XYZ in popular culture" - but they shouldn't. Check out what wikipedia isn't and the trivia guideline. An increasing number of the "popular culture" sections have the trivia template at the top. --HughCharlesParker (talk - contribs) 18:18, 21 May 2007 (UTC)

Not everybody agres with this point of view, however; it's a guideline not a requirement. Many (most?) "pop culture" sections or their equivalent are, indeed, pointless collections of distracting fancruft. But not all. The template shouldn't be indiscriminatly slapped down on every section. - DavidWBrooks 19:27, 21 May 2007 (UTC)

It is so unusual for something like this to be mentioned (other than in the TV show Numb3rs), that I think it should certainly be in this article. I've known about happy numbers 20 years or so, and I think they are fairly well known in math circles. Bubba73 (talk), 19:45, 21 May 2007 (UTC)

:) Happy numbers: As Seen On TV!! Oh well. I guess this isn't going to bring down the project. --HughCharlesParker (talk - contribs) 19:48, 21 May 2007 (UTC)

Currently it is only one sentence. i think there is a consensus to let it stand as it is, right? Bubba73 (talk), 19:51, 21 May 2007 (UTC)

If we'd reached a consensus we'd all agree. I'm not agreeing (would Encyclopedia Brittanica include this line? Why not?) - I'm giving up. I just don't think that one short sentence is worth the effort when wikipedia is, as you point out, riddled with trivia. --HughCharlesParker (talk - contribs) 20:02, 21 May 2007 (UTC)

Wikipedia:Consensus says "Consensus does not mean that everyone agrees with the outcome; instead, it means that everyone agrees to abide by the outcome." It certainly looks like we have reached consensus with this short mention in the happy prime section and no trivia-like section. PrimeHunter 20:41, 21 May 2007 (UTC)

(unindent) I'm sorry, I misinterpreted your comments. We've written dozens of sentences about whether or not one sentence should remain. There is no "trivia" section now. Someone added a link to it in the article about Dr. Who episode. Bubba73 (talk), 20:32, 21 May 2007 (UTC)

definition issue

User:UKPhoenix79 seems to be taking issue with the definition (the first sentence of the article). I think it is clear: the process ends in 1, or in an infinite loop; those numbers that end in 1 are happy, other numbers are not. User:UKPhoenix79 has twice removed the correct explanation of the process. I would like User:UKPhoenix79 to explain any intended changes here before making any further changes, so we can fix this problem. Others are welcome to chime in, of course. -- Doctormatt 23:42, 21 May 2007 (UTC)

Perhaps the confusion comes in because if you continued the process on 1, it would be an infinite loop to itself. However, the definition states that you stop the stop the process once you hit 1. Maybe it would be clearer if it stated that the infinite loop doesn't include 1. Bubba73 (talk), 23:50, 21 May 2007 (UTC)

Yes, I think that certainly doesn't help. Would anyone mind if I broke the first sentence into a couple of sentences, so the iterative process is clearly defined first, and then happy numbers are defined based on that? Something like this:

Starting with any positive whole number, the following process can be applied: replace the number by the sum of the squares its digits, and repeat until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers and those that do not are unhappy numbers.

How does that sound? Doctormatt 00:07, 22 May 2007 (UTC)

I think that is good and makes it clearer, except I'd probably say that the process stops or terminates if you reach 1. Or you could state it that it goes to an infinite loop on 1 if it is happy, and an infinite loop not including 1 is not happy. Bubba73 (talk), 00:52, 22 May 2007 (UTC)

Mathematical Proof of fallacy of current statement on Happy Numbers

Take the sum of the squares of its digits continue iterating this process until it yields 1, or produces an infinite loop.

According to this statement a number is happy once it produces reduces to a 1 or produces an infinite loop. If that is so and making the assumption that the current sentence is correct we will prove that this statement is incorrect this by testing two known unhappy numbers and check the results.

Lets try 2
2² = 4 ← infinite loop start
4² = 16
1² + 6² = 37
3² + 7² = 58
5² + 8² = 89
8² + 9² = 145
1² + 4² + 5² = 42
4² + 2² = 20
2² + 0² = 4 ← infinite loop end

Lets Try 1979
1² + 9² + 7² + 9² = 212
2² + 1² + 2² = 9
9² = 81
8² + 1² = 65
6² + 5² = 61
6² + 1² = 37 ← infinite loop start
3² + 7² = 58
5² + 8² = 89
8² + 9² = 145
1² + 4² + 5² = 42
4² + 2² = 20
2² + 0² = 4
2² = 4
4² = 16
1² + 6² = 37 ← infinite loop end

Note that neither number is listed as a happy number here and in fact the smallest happy number aside from 1 is 7. This statement is then false and needs to be corrected. As it is blatantly false as it currently stands and actually stating (unintentionally) that EVERY number is happy. Actually an infinite loop is what defines an unhappy number. -- UKPhoenix79 08:08, 22 May 2007 (UTC)

Your initial quote omitted the first sentence which is what defined happy numbers in the article. What the article really said before each of your 3 changes [1][2][3] was:

A happy number is any number that eventually reduces to 1 when the following process is used: take the sum of the squares of its digits, and continue iterating this process until it yields 1, or produces an infinite loop. Numbers that are not happy are called unhappy numbers. [4]

I haven't seen anybody claim that you get a happy number if the process gives an infinite loop. Hopefully you will not revert the new extended definition which looks fine to me. PrimeHunter 17:21, 22 May 2007 (UTC)

I can see how that original statment can be ambiguous. Bubba73 (talk), 18:26, 22 May 2007 (UTC)

Question

I've heard a few names for the numbers 4, 16, 37, 58, 89, 145, 42, and 20. The most common being "miserable numbers". I can't find any source for these numbers have a special name as they relate to the whole Happy number thing. Has anyone else seen a source that has these numbers named specifically?

This is all I know. Bubba73 (talk), 02:26, 15 June 2007 (UTC)