General forum: Is 8795 a boring number?

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Is 8795 a boring number? 2008-03-14 Ingo Althofer Mathematicians con prove that there are no boring natural numbers. (If there were some, take the smallest one - and voila, this one would be interesting at least because it were the smallest boring one.) In real life things are a bit differently (and being "the smallest boring one" is sort of second class-interesting). Well known is "The On-Line Encyclopedia of Integer Sequences", directed by Neil Sloane. Its web address is http://www.research.att.com/~njas/sequences/ Visitors of the site may enter any string in the search line, and as a result all possible hits in the data base are shown. Skimming the encyclopedia shows that the smallest natural number not present is 8795 Does anybody here has an idea why 8795 might be interesting (for instance as member of some "interesting" sequence). Of course, it would be best not only to have an idea but also to achieve that 8795 comes in the encyclopedia. Ingo Althofer (who did something rather interesting onthe 8795-th day of his life). 2008-03-14 wccanard "Mathematicians con prove that there are no boring natural numbers." I don't buy that. Isn't this basically the same line of reasoning as Russell's Paradox etc etc? If you try and formalise what a proof is, then you can define boring as the smallest number N without a proof that it's interesting, but the problem is that this will surely *not* be a proof that N is interesting, because the definition of N will be beyond the scope of what a proof is---it's kind of a "meta-proof" only. 8795 looks like a very boring number to me, though. In fact the most interesting thing about it that springs to mind is that it's so _large_! I'm sure there are plenty of smaller boring numbers. Ingo: can you explain more carefully what you did? Are you saying that 8795 is the smallest number you found that is not mentioned in the encyclopaedia at all? You have to be careful with arguments like this: there are plenty of very "dense" sequences in the encyclopaedia: for example the encyclopaedia will contain the sequence of positive odd integers, but the entry in the encyclopaedia for sequence won't mention 8795 because the encyclopaedia only carries the first 40 or so terms of the sequence, and then will give a general formula for the rest of the terms. If you have a few hours to spare you can try and find the first 40 numbers not mentioned at all in the encyclopaedia and then submit them as a new entry. I think Sloane might have strong reservations about this sequence though :-) This might also already have been done---did you search for "boring"? wcc 2008-03-14 Ingo Althofer Hi wcc, > "Mathematicians con prove that there are no boring natural numbers." > > I don't buy that. I meant "proving" in the sense of having social agreement on this statement. > 8795 looks like a very boring number to me, though... > I'm sure there are plenty of smaller boring numbers. The answer is "no", when you accept that a natural number is not boring when it is explicitly mentioned in Sloane's encyclopedia. > .. can you explain more carefully what you did? Are you > saying that 8795 is the smallest number you found that > is not mentioned in the encyclopaedia at all? Right. Lisa Schreiber, one of my math students, made a complete automatic scan of the database (on January 11, 2008). As a result, amongst other things she got a count how often each number was present (as a string) in the database. 8795 was the only natural number below 10,000 which was not present. > ... the encyclopaedia only carries the first 40 or so terms > of the sequence, and then will give a general formula for the > rest of the terms. Right. We had searched only for numbers mentioned explicitly as a string. > If you have a few hours to spare you can try and find the > first 40 numbers not mentioned at all in the > encyclopaedia We have the first 50 (from date January 11, 2008). > and then submit them as a new entry. I think Sloane > might have strong reservations about this sequence though :-) Me, too. > This might also already have been done---did you search > for "boring"? Today, "boring" gives five hits - none of them in relation to "small boring numbers". Ingo. PS: The smallest negative integer not in the database on January 11, 2008 was -634 (meaning absolute value). 2008-03-14 MichaeI X I do not understand your proof: In a first step you setup a list of potentially boring numbers, then you remove the smallest one, because it is that "second class interesting" one. Then you continue with that method, until it becomes boring: Voila, the rest is a list of really boring numbers. In practice, you should run both parts of that procedure in parallel or take a non-procedural approach: Assumed there were an infinite amount of natural numbers and some really, really boring ones among them. Every eliminating method becomes boring sooner or later, then there are still some boring numbers left. You noticed well, my counter-proof has the assumption that something can "become boring". That's what I remember from university: Mathematic is extremely hard when Dynamics is involved. 2008-03-14 furbolero The proof is supposed to be the next one: *Suppose* there is a set of "boring numbers" ("those with no special property", take it loosely). However the lowest number IS the "first lowest number" and then it has a special property, and thus is not boring. Silly reductio ab absurdum :) 2008-03-14 furbolero sorry the typo: ...the lowest number in that set IS the "first boring number", and then... 2008-03-14 slaapgraag Well, another proof of hw boring this is: LG player nr 8795 is zargamox, he never played one single game here.... 2008-03-14 MarleysGhost > Then you continue with that method, until it becomes boring: LOL! Michael X, you have the right insight. I'm bored already. 2008-03-14 wccanard Here's my favourite example of this "boring" paradox, which actually fits in very nicely in the context of a games site. We don't really want to play games that go on forever! The nice thing about games like hex and dots and boxes is that they are guaranteed to finish in a finite time, and if we make little modifications to games like chess (say, we demand that if a position repeats 3 times then the game _is_ a draw, rather than saying just that either player may claim a draw) then these games become finite too. It's obvious what I mean by "finite", right? A game is _finite_ if every play of the game finishes in a finite time. OK, so let's say that on a games site not too far from here, all the games on it are finite (let's say we have dots and boxes, hex and four in a row). In fact let's say that one of the guidelines for implementing new games onto the site is that they must be finite games. Now it's time to introduce a new finite game onto the site. Here's a fun finite game! It's called "hypergame". The rules of it are as follows. Player 1, on his first move, _names_ one of the games on the site; that is his move. And then the two players start playing the game that player 1 has named, with player 2 having the first move. So for example, in a game of hypergame, player 1 can say "dots and boxes", and then a game of dots and boxes starts with player 2 having the first move. Hypergame is a neat game! You can challenge someone to hypergame and then consider your opponent, and try and find a game that you're better at than he/she is, and play that game as your first move. Hypergame in fact encourages people to get better at every game on the site! So Hypergame is a Good Thing. Moreover, Hypergame is also *clearly* a finite game, because on Player 1's first move he has to name a finite game, and that game will clearly finish in a finite time. So Hypergame is finite. The site admin likes Hypergame so much that he implements it as one of the games on the site. A few weeks later he is bewildered by a complaint from a user that it appears that an infinite game is going on on the site, which is contrary to the guidelines. And of course the game is: P1 "hypergame" P2 "hypergame" P1 "hypergame" ... :-) 2008-03-14 Gregorlo wccanard, i'm pretty sure i explained that paradox here in the forums a few years ago ;) 2008-03-14 slaapgraag Gregorio, link please? 2008-03-14 Gregorlo i'm looking for it! unfortunately, i have just discovered google doesn't throw results of LG other than empathy games :( so i'm handlooking in all pages in all forums... be patient... i will shallow my words if not ^_^ it's been my all-time favourite paradox! i can hardly believe i did not post it here!! :D 2008-03-14 Gregorlo Damn! I surfed in all forums pages since i am registered here, and didn't found my post. I just read the titles, so maybe i missed the right post. :( i'll check more tomorrow... btw, i laughed out loud again reading all those posts of Andrew-Villasante, who copypasted a lot of sentences from http://www.7freedom.com/, but changing the order, and sometimes changing some words. Chaotic form of art! I laughed again, how nostalgic! returning to topic,,, nobody remembers me posting about hypergame?? :( 2008-03-14 Robin Gregorio, I remember that someone posted that paradox a few years ago. If you say it was you, I believe you :) 2008-03-14 wccanard Gregorio: I am well aware that the paradox is well-known :-) I certainly didn't invent it! 2008-03-14 Ray Garrison It would seem that 8795 is not boring at all. Just look at all the excitement it is causing in this forum! 2008-03-14 Gregorlo wccanard,,, of course i didn't want to imply that!!! :D i just wantd to point out that it was "discussed" before in the forums ;) and i agree with Ray!!! 2008-03-14 wccanard 8795 _is_ boring, because if Sloane had decided to store a different number of numbers by default in his encyclopaedia then we would have got another number. 2008-03-14 wccanard To amplify my point, search the encyclopaedia for the sequence 95,195,295,395,495 You get a few very dull sequences, including things like the fascinating "Numbers n such that string 9,5 occurs in the base 10 representation of n but not of n-1" etc etc, and 8795 would be in a sequence like this were the first 100 or so terms of each sequence to be listed rather than just the first 30 or so. If Sloane thinks that a sequence like this is worth including in the encyclopadeia then why not do the same thing with the string "7,9,5" and voila! 8795 is officially interesting! Here's an idea which sounds much more intrinsically interesting (to me at least): instead of doing what you did, which relies crucially on how far people could be bothered to write down their fascinating sequences, why not find the smallest positive integer f(n) which is not mentioned as the m'th term of any sequence in the database for 1<=m<=n? This gets around the problem of 8795 being the 8795'th term in the sequence 1,2,3,4,5,6,... . I might be much more persuded by an argument that f(1) is interesting-because-it-is-so-boring, although of course there's a danger that 57 is the first term in the fascinating sequence of numbers that have got a 5 and a 7 in, and not the first term in any other sequence... 2008-03-14 Ingo Althofer Thanks for all the feedback. I'll try to give a whole bunch of answers in this posting. Michael X wrote: > Then you continue with that method, until it becomes boring: > ... You noticed well, my counter-proof has the assumption that > something can "become boring". That's what I remember from > university: Mathematic is extremely hard when Dynamics is involved. Good point to mention the aspect of dynamics. ************************************************ wccanard wrote about "hypergame": > ... Player 1, on his first move, _names_ one of the games > on the site; that is his move. And then the two players > start playing the game that player 1 has named, with > player 2 having the first move... The paradox might be simply repaired by formulating: The first move of hypergame is to propose any LittleGolem game EXCEPT hypergame. ********************************** Ray Garrison: > It would seem that 8795 is not boring at all. > Just look at all the excitement it is causing in this forum! Unfortunately, most of the excitement is not about 8795 itself, but about my introducing paragraph and about weaknesses of Sloane's dictionary. Had I forseen this, I would have simply asked "Who knows GOOD reasons why 8795 might be interesting." On the ohter hand, I like the discussion. It is not boring for me, yet. ******************************************** wcc made several interesting remarks. One of his central points is: 8795 is not in the data base mainly by bad luck, because it comes too late in certain sequences which are in the data base. I like wcc's proposal: > ... why not find the smallest positive integer f(n) > which is not mentioned as the m'th term of any sequence > in the database for 1<=m<=n? When Lisa Schreiber would not have other (more interesting) duties I would ask her to work out this sequences f(n), for instance for all n, 1 <= n <= 40. **************************************** In one of his answers css wrote: > ... and voila! 8795 is OFFICIALLY [capital letters > by me] interesting! Although Sloane's encyclopedia is the first address for many research question, it is of course not OFFICIAL. And, as not only Sloane himself is feeding the monster, but a whole team (more than 20 people in the editorial board), it is quite natural that soft or even nonserious entries will occur in the data base. Perhaps Sloane should set up a second team of helpers who only had to try cleaning up?! (But to state it clearly: The percentage of garbage in Sloane's eancyclopedia is far far below the percentage of garbage in google and co.) ******************************************* Finally, a little exercise on one of wcc's "extreme" examples: > To amplify my point, search the encyclopaedia for the > sequence 95,195,295,395,495 > > You get a few very dull sequences, including things like the > fascinating "Numbers n such that string 9,5 occurs in the > base 10 representation of n but not of n-1" Question: What is the asymptotic density of this set within the natural numbers? Ingo. 2008-03-14 wccanard Last question: I say zero! 2008-03-15 Gregorlo @ingo: you don't repair the paradox, you just spoiled it!! ;) :P 2008-03-16 Dustin 8795 is semi-prime. That makes it semi-interesting... 2008-03-16 wccanard Just because a number is prime or semi-prime (a product of two primes) doesn't make it interesting, does it? Doesn't it have to be something like "the smallest semiprime which is the sum of 7 cubes in 2 different ways" or something? 2008-04-01 berghildur 8795 is related to the unlucky number 13 and the number of the Beast 666: 8795 = 13*666 + 666/(6+6+6) + 666/6.66 ------------------------------------ Let G be a generalised Fibonacci sequence, whose terms G(n) are defined as follows: G(1) = G(2) = G(3) = 0 G(4) = 1 G(n) = G(n-1) + G(n-2) + 10*G(n-3) + 5*G(n-4) {n>4} The first few terms of this sequence are: 0, 0, 0, 1, 1, 2, 13, 30, 68, 238, 671, 1739, 5130. This sequence has the following property: Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795. This sequence happens not to be unique, since - strangely enough - there exists also a generalised Fibonacci sequence U (for "unlucky") and another one, B (for "Beast"), such that Sum(U(k)/13^k) = 1/8795, and Sum(B(k)/666^k) = 1/8795. The discovery of these sequences is left to the reader. ------------------------------------ 8795 is also related to the number pi, as follows: pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/22) + arctan(1/633) + arctan(1/2985) + arctan(1/7614) + arctan(1/8795) 2008-04-01 wccanard Heh :-) Regarding the first comment, what is the smallest number *not* "related to 13 and 666"? ;-) The second comment is indeed nice. I've seen this trick before: it was explained to me in the following way: "explain why 1/9899=0.0001010203050813213455..." (the point being that 1,1,2,3,5,8,13,21,34,55 are the first few terms in the usual Fibonacci sequence). The 3rd is also nice---again a "standard trick" if you like, but it somehow looks more spectacular than it is! You could try and argue for 2,5,13,22,633,2985,7614,8795 to be added to Sloane's database ;-) 2008-04-07 Ingo Althofer Hi Berghildur, greetings from Shanghai, where I just read your contribution on 8795. > Let G be a generalised Fibonacci sequence, whose terms G(n) > are defined as follows: > > G(1) = G(2) = G(3) = 0 > G(4) = 1 > G(n) = G(n-1) + G(n-2) + 10*G(n-3) + 5*G(n-4) {n>4} So, the generating coefficients for this sequence are a rather simple finite sequence: (0, 0, 0, 1; 1, 1, 10, 5) > The first few terms of this sequence are: > 0, 0, 0, 1, 1, 2, 13, 30, 68, 238, 671, 1739, 5130. > > This sequence has the following property: > > Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795. Did someone else check if this result is reallz true? > This sequence happens not to be unique, since - strangely enough - there > exists also a generalised Fibonacci sequence U (for "unlucky") and another > one, B (for "Beast"), such that Sum(U(k)/13^k) = 1/8795, and > Sum(B(k)/666^k) = 1/8795. The discovery of these sequences is left to > the reader. Are the generating coefficients for these sequences similarily simple, compared with the first sequence? Ingo. PS: I always thought Berghildur was a volcano hunter from Iceland - and not from Kyrgysistan. 2008-04-07 wccanard > > Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795. > Did someone else check if this result is reallz true? It's easy to check. Consider the generating function associated to the recurrence relation; it's F(x)=sum G(k).x^k. Now use the initial conditions and the definition of the recurrence to write F(x) as a rational function of x [we get F(x)=x^4/(1-x-x^2-10x^3-5x^4) ] and finally substitute in x=1/10 to compute the infinite sum. If you do this exercise you'll easily see how Berghildur reverse-engineered the process and obtained the recurrence relation from the number. The clever observation is not obtaining the recurrence relation, it's observing that you can rig it so that the coefficients are so small: this boils down to the fact that 8795=10000-1000-100-10*10-5*1. Now you can answer your other questions yourself, armed with the trick. 2009-02-04 Ingo Althofer Finally, 8795 has finished its life as a boring number. Thanks to Omar E. Pol, it is now present in the Online Encyclopedia of Integer Sequences. Namely, it has the form p(n)*p(n+2) - 2*p(n+1), where p(n) is the n-th smallest prime number. Set p(n)=89, p(n+1)=97, p(n+2)=101. Cheers, Ingo. A152532 2009-02-05 Carroll Isn't p(n) simply the nth prime number? ("http://primes.utm.edu/nthprime/index.php#nth")? Here for n=24,25,26. So what is the next "Boring Number" so we may laugh a little more? 2009-02-05 wccanard 8796 ;-) 2009-02-05 wccanard This whole thread is based on what I believe is a false premise. The original claim was that 8795 was the smallest positive integer not in Sloane's tables. But 8795 is in sequence A000027, which is the sequence 1,2,3,4,5,6,7,8,9,... . 2009-02-05 christian freeling Very perceptive! 2009-02-05 Hjallti Is this the list of all numbers, or the list of all numbers without double digit? 2009-02-05 wccanard In some sense, Hjallti, you've put your finger on the problem. Sloane doesn't list the first 1,000,000 terms of every sequence, he just stops when he gets bored (like I did; A000027 is "all positive integers"). So 8795 is probably in many of the sequences in the database (because there are all manner of sequences in there, e.g. "positive odd numbers", "numbers with a 5 in" etc etc), it's just that people got bored before 8795 every time. Probably there are other ways of getting 8795 in---you can "contribute new terms" to Sloane's database! Perhaps someone wants to contribute a few more numbers that have a 5 in? ;-) 2009-02-05 Ingo Althofer What I meant: Before December 06, 2008 8795 was the smallest integer not mentioned EXPLICITLY in the data base. @Carroll: I will try to look it up and tell you. Ingo. 2009-02-06 Ingo Althofer @Carroll and all other numerologists: After 8795 has come in, the smallest numbers not explicitly in the Encyclopedia are: 11630 12067 (special: -12067 is in !) 12407 Cheers, Ingo 2009-02-06 MichaeI X Now, that's really interesting :P 2009-02-06 Tim From a visual point of view, 11630 certainly is one of the most boring numbers I've ever seen, although its octal equivalent 26556 looks promising ...

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