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Écoutez et entrez les chiffres que vous entendez Envoyer Supprimer cliff Afficher le profil Autres options 30 avr 1994, 23:25 Groupes de discussion : sci.math De : cl...@watson.ibm.com (cliff) Date : 29 Apr 1994 19:42:27 GMT Local : Ven 29 avr 1994 20:42 Objet : Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur Title: Cliff Puzzle 22: Factorions From: cl...@watson.ibm.com If you respond to this puzzle, if possible please include your name, state, and e-mail address. If you like, tell me a little bit about yourself. You might also directly mail me a copy of your response in addition to any responding you do in the newsgroup. I will assume it is OK to describe your answer in any article or publication I may write in the future, with attribution to you, unless you state otherwise. Thanks, Cliff Pickover * * * Factorions are numbers that are the sum of the factorial values for each of their digits. For example, 145 is a factorion because is can be expressed as 145 = 1! + 4! + 5! The largest known factorion is 40,585 = 4! + 0! + 5! + 8! + 5! Some mathematicians believe that this is the largest possible factorion. (In fact, 145 and 40,585 are the only multi-digit factorions known to humanity.) Can you end the loneliness of the factorions? I would be interested in hearing from any of you who can find a larger factorion for an article I am writing. Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. John Scholes Afficher le profil Autres options 1 mai 1994, 15:16 Groupes de discussion : sci.math De : jscho...@kalva.demon.co.uk (John Scholes) Date : Sun, 1 May 1994 13:55:21 +0000 Local : Dim 1 mai 1994 14:55 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article <2prnv3$1...@watnews1.watson.ibm.com> cl...@watson.ibm.com "cliff" writes: > Factorions are numbers that are the sum of the factorial values for > each of their digits. For example, 145 is a factorion because > is can be expressed as 145 = 1! + 4! + 5! > The largest known factorion is 40,585 = 4! + 0! + 5! + 8! + 5! > Some mathematicians believe that this is the largest possible factorion. > (In fact, 145 and 40,585 are the only multi-digit factorions known to > humanity.) > Can you end the loneliness of the factorions? > I would be interested in hearing from any of you who can find a larger > factorion for an article I am writing. Unless I am missing something, this puzzle is pretty silly. A factorion must have less than 8 digits, because 8 * 9! < 10^7. A simple C program to check all such digits (time to write <5 minutes, time to run < 1 minute) gives just 1,2,145 and 40585. No doubt one can debate whether 1 and 2 qualify. -- John Scholes Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. Kevin S. Brown Afficher le profil Autres options 2 mai 1994, 00:58 Groupes de discussion : sci.math De : kevin2...@delphi.com (Kevin S. Brown) Date : 1 May 1994 21:23:39 GMT Local : Dim 1 mai 1994 22:23 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur CW> Factorions are numbers that are the sum of the factorial values for CW> each of their digits. For example, 145 is a factorion because is CW> can be expressed as 145 = 1! + 4! + 5! The largest known factorion CW> is 40,585 = 4! + 0! + 5! + 8! + 5! Some mathematicians believe that CW> this is the largest possible factorion. (In fact, 145 and 40,585 are CW> the only multi-digit factorions known to humanity.) Can you end the CW> loneliness of the factorions? I would be interested in hearing from CW> any of you who can find a larger factorion for an article I'm writing. JS> Unless I am missing something, this puzzle is pretty silly. JS> A factorion must have less than 8 digits, because 8 * 9! < 10^7. JS> A simple C program...gives just 1,2,145 and 40585. I don't think you are missing anything. It is known that these are the only four integers equal to the sum of the factorials of their base 10 digits. This is discussed in Joe Roberts' book "The Lure of the Integers" on page 35. He mentions a general theorem due to B. L. Schwartz stating that the sum of a function f(n,d) of the base b digits of N is equal to N for only a finite number of cases if n [max f(n,d)] lim sup ---------------- < 1 where d = 0 to b-1 n b^n In addition to the simple example f(n,d) = d! for all n, he gives the more interesting example f(n,d) = d^n. In this case the theorem tells us there are only finitely many cases where an n-digit number is equal to the sum of the nth powers of its digits. The known occurrences of this include 153, 1634, 8208, 9474, .... up to 4679307774. The theorem says there are only finitely many of these, but it isn't known if 4679307774 is the largest. (It's known that an upper bound for such a number is 10^60.) Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. John Scholes Afficher le profil Autres options 3 mai 1994, 04:36 Groupes de discussion : sci.math De : jscho...@kalva.demon.co.uk (John Scholes) Date : Mon, 2 May 1994 23:04:11 +0000 Local : Mar 3 mai 1994 00:04 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article <9405011722592.kevin2003.DL...@delphi.com> kevin2...@delphi.com "Kevin S. Brown" writes: > ... only finitely many cases where an n-digit number is equal to the sum of > the nth powers of its digits. The known occurrences of this include 153, > 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774 > is the largest ... It doesn't seem too hard to find some more ... 32,164,049,650; 32,164,049,651; 40,028,394,225; 42,678,290,603; 44,708,635,679; 49,388,550,606; 82,693,916,578; 94,204,591,914 Any improvements, anyone? -- John Scholes Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. John Scholes Afficher le profil Autres options 3 mai 1994, 10:52 Groupes de discussion : sci.math De : jscho...@kalva.demon.co.uk (John Scholes) Date : Tue, 3 May 1994 09:28:42 +0000 Local : Mar 3 mai 1994 10:28 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article <9405011722592.kevin2003.DL...@delphi.com> kevin2...@delphi.com "Kevin S. Brown" writes: > ... only finitely many cases where an n-digit number is equal to the sum of > the nth powers of its digits. The known occurrences of this include 153, > 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774 > is the largest ... Another one: 28,116,440,335,967 (none with 12 or 13 digits). -- John Scholes Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. John Scholes Afficher le profil Autres options 3 mai 1994, 17:42 Groupes de discussion : sci.math De : jscho...@kalva.demon.co.uk (John Scholes) Date : Tue, 3 May 1994 16:25:26 +0000 Local : Mar 3 mai 1994 17:25 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article <9405011722592.kevin2003.DL...@delphi.com> kevin2...@delphi.com "Kevin S. Brown" writes: > ... only finitely many cases where an n-digit number is equal to the sum of > the nth powers of its digits. The known occurrences of this include 153, > 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774 > is the largest ... A few more: 16-digit 4,338,281,769,391,370 (& +1) 17-digit 21,897,142,587,612,075 35,641,594,208,964,132 35,875,699,062,250,035 These seem to take about half-an hour each on an elderly PC running interpreted basic. So it looks as though it should be feasible with a little more trouble to go all the way and find the largest. -- John Scholes Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. Sujet remplacé par "Digital Invariants (was Factorions)" par Dan Hoey Dan Hoey Afficher le profil Autres options 3 mai 1994, 22:00 Groupes de discussion : rec.puzzles, sci.math Suivi : rec.puzzles De : h...@AIC.NRL.Navy.Mil (Dan Hoey) Date : Tue, 3 May 1994 20:38:13 GMT Local : Mar 3 mai 1994 21:38 Objet : Digital Invariants (was Factorions) Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur kevin2...@delphi.com "Kevin S. Brown" writes that there are > ... only finitely many cases where an n-digit number is equal to the sum > of the nth powers of its digits. The known occurrences of this include > 153, 1634, 8208, 9474, .... up to 4679307774... but it isn't known if > 4679307774 is the largest ... According to David Wells's book _The Penguin Dictionary of Curious and Interesting Numbers_ these numbers are called digital invariants. They were mentioned on rec.puzzles in 1992, and Dik Winter provided a list from 1985 that I confirmed with a program I wrote. The following is a list of all 89 base-ten digital invariants. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208, 9474, 54748, 92727, 93084, 548834, 1741725, 4210818, 9800817, 9926315, 24678050, 24678051, 88593477, 146511208, 472335975, 534494836, 912985153, 4679307774, 32164049650, 32164049651, 40028394225, 42678290603, 44708635679, 49388550606, 82693916578, 94204591914, 28116440335967, 4338281769391370, 4338281769391371, 21897142587612075, 35641594208964132, 35875699062250035, 1517841543307505039, 3289582984443187032, 4498128791164624869, 4929273885928088826, 63105425988599693916, 128468643043731391252, 449177399146038697307, 21887696841122916288858, 27879694893054074471405, 27907865009977052567814, 28361281321319229463398, 35452590104031691935943, 174088005938065293023722, 188451485447897896036875, 239313664430041569350093, 1550475334214501539088894, 1553242162893771850669378, 3706907995955475988644380, 3706907995955475988644381, 4422095118095899619457938, 121204998563613372405438066, 121270696006801314328439376, 128851796696487777842012787, 174650464499531377631639254, 177265453171792792366489765, 14607640612971980372614873089, 19008174136254279995012734740, 19008174136254279995012734741, 23866716435523975980390369295, 1145037275765491025924292050346, 1927890457142960697580636236639, 2309092682616190307509695338915, 17333509997782249308725103962772, 186709961001538790100634132976990, 186709961001538790100634132976991, 1122763285329372541592822900204593, 12639369517103790328947807201478392, 12679937780272278566303885594196922, 1219167219625434121569735803609966019, 12815792078366059955099770545296129367, 115132219018763992565095597973971522400, and 115132219018763992565095597973971522401. My program checked that there are no more up to 60 digits, and it is easy to prove that there can be none larger. As a devout zerophilist, I of course include zero on the list. While I believe that zero is, properly speaking, not a one-digit number but a zero-digit number (customarily written with a leading zero), it is a digital invariant in either case. Dan Hoey H...@AIC.NRL.Navy.Mil Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. Sujet remplacé par "Loneliness of the Factorions" par Dik T. Winter Dik T. Winter Afficher le profil Autres options 4 mai 1994, 00:58 Groupes de discussion : sci.math De : d...@cwi.nl (Dik T. Winter) Date : Tue, 3 May 1994 23:30:45 GMT Local : Mer 4 mai 1994 00:30 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article <767982326...@kalva.demon.co.uk> jscho...@kalva.demon.co.uk writes: > In article <9405011722592.kevin2003.DL...@delphi.com> > kevin2...@delphi.com "Kevin S. Brown" writes: > > > ... only finitely many cases where an n-digit number is equal to the sum of > > the nth powers of its digits. The known occurrences of this include 153, > > 1634, 8208, 9474, .... up to 4679307774... but it isn't known if 4679307774 > > is the largest ... > > A few more: 16-digit 4,338,281,769,391,370 (& +1) > 17-digit 21,897,142,587,612,075 > 35,641,594,208,964,132 > 35,875,699,062,250,035 > 1 digit (of course) 1; 2; 3; 4; 5; 6; 7; 8; 9 3 digits 153; 370; 371; 407 4 digits 1634; 8208; 9474 5 digits 54748; 92727; 93084 6 digits 548834 7 digits 1741725; 4210818; 9800817; 9926315 8 digits 24678050; 24678051; 88593477 9 digits 146511208; 472335975; 534494836; 912985153 10 digits 4679307774 11 digits 32164049650; 32164049651; 40028394225; 42678290603; 44708635679; 49388550606 82693916578; 94204591914 14 digits 28116440335967 16 digits 4338281769391370; 4338281769391371 17 digits 21897142587612075; 35641594208964132; 35875699062250035 19 digits 1517841543307505039; 3289582984443187032; 4498128791164624869 4929273885928088826 20 digits 63105425988599693916 21 digits 128468643043731391252; 449177399146038697307 23 digits 21887696841122916288858; 27879694893054074471405; 27907865009977052567814 28361281321319229463398; 35452590104031691935943 24 digits 174088005938065293023722; 188451485447897896036875; 239313664430041569350093 25 digits 1550475334214501539088894; 1553242162893771850669378; 3706907995955475988644380 3706907995955475988644381; 4422095118095899619457938 27 digits 121204998563613372405438066; 121270696006801314328439376 128851796696487777842012787; 174650464499531377631639254 177265453171792792366489765 29 digits 14607640612971980372614873089; 19008174136254279995012734740 19008174136254279995012734741; 23866716435523975980390369295 31 digits 1145037275765491025924292050346; 1927890457142960697580636236639 2309092682616190307509695338915 32 digits 17333509997782249308725103962772 33 digits 186709961001538790100634132976990; 186709961001538790100634132976991 34 digits 1122763285329372541592822900204593 35 digits 12639369517103790328947807201478392; 12679937780272278566303885594196922 37 digits 1219167219625434121569735803609966019 38 digits 12815792078366059955099770545296129367 39 digits 115132219018763992565095597973971522400 115132219018763992565095597973971522401 If I am right this took about 2000 seconds on a Vax back in 1985. Doing the same base 12 took 60 times as much. (I think those were seconds indeed, otherwise I would never have done base 12.) Here follows the total number of solutions (including one digit solutions) for the different bases: base total# largest (preceded by # digits in parenthesis) 2 1 ( 1) 1 3 5 ( 3) 122 4 11 ( 4) 3303 5 17 (14) 14421440424444 6 30 (18) 105144341423554535 7 59 (23) 12616604301406016036306 8 62 (29) 11254613377540170731271074472 9 58 (30) 104836124432728001478001038311 10 88 (39) 115132219018763992565095597973971522401 11 134 (45) 12344AA12A721803422912A8AA4963568083A268456A4 12 87 (51) 15079346A6B3B14BB56B395898B96629A8B01515344B4B0714B Now that computers are so much faster I might try to expand the list. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098 home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: d...@cwi.nl Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. Sujet remplacé par "Digital Invariants" par Erich Friedman Erich Friedman Afficher le profil Autres options 5 mai 1994, 05:52 Groupes de discussion : rec.puzzles, sci.math De : Erich Friedman Date : Wed, 4 May 1994 14:46:33 GMT Local : Mer 4 mai 1994 15:46 Objet : Re: Digital Invariants Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur In article Dan Hoey writes: > According to David Wells's book _The Penguin Dictionary of Curious and > Interesting Numbers_ these numbers are called digital invariants. Is this book still in print? Erich Friedman fried...@macs.stetson.edu Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. davids8782 Afficher le profil Autres options 8 mai 1994, 01:42 Groupes de discussion : sci.math De : davids8...@delphi.com Date : Sat, 7 May 94 20:36:00 -0500 Objet : Re: Digital Invariants Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur Erich Friedman writes: >In article Dan Hoey writes: >> According to David Wells's book _The Penguin Dictionary of Curious and >> Interesting Numbers_ these numbers are called digital invariants. >Is this book still in print? >Erich Friedman >fried...@macs.stetson.edu Yes it is. I got a copy about a year ago. About ten dollars. The US address is Penguin Books, 375 Hudson Street, NY,NY, 10014. Or you can call the 800 operator and ask for the 800 number for Penguin Books. Répondre à l'auteur Transférer Vous devez vous connecter pour pouvoir envoyer des messages. Pour envoyer un message, vous devez dans un premier temps rejoindre ce groupe. Veuillez mettre à jour votre pseudonyme dans la page Paramètres d'abonnement avant de publier des messages. Vous ne disposez pas de l'autorisation nécessaire pour publier un message. Sujet remplacé par "Loneliness of the Factorions" par Tomas Antonio Mendes Oliveira e Silva Tomas Antonio Mendes Oliveira e Silva Afficher le profil Autres options 9 mai 1994, 23:13 Groupes de discussion : sci.math De : t...@ci.ua.pt (Tomas Antonio Mendes Oliveira e Silva) Date : Mon, 9 May 1994 15:32:54 GMT Local : Lun 9 mai 1994 16:32 Objet : Re: Loneliness of the Factorions Répondre à l'auteur | Transférer | Imprimer | Message individuel | Afficher l'original | Signaler ce message | Rechercher les messages de cet auteur Kevin S. Brown (kevin2...@delphi.com) wrote: : CW> Factorions are numbers that are the sum of the factorial values for : CW> each of their digits. For example, 145 is a factorion because is : CW> can be expressed as 145 = 1! + 4! + 5! The largest known factorion : CW> is 40,585 = 4! + 0! + 5! + 8! + 5! Some mathematicians believe that : CW> this is the largest possible factorion. (In fact, 145 and 40,585 are : CW> the only multi-digit factorions known to humanity.) Can you end the : CW> loneliness of the factorions? I would be interested in hearing from : CW> any of you who can find a larger factorion for an article I'm writing. : : JS> Unless I am missing something, this puzzle is pretty silly. : JS> A factorion must have less than 8 digits, because 8 * 9! < 10^7. : JS> A simple C program...gives just 1,2,145 and 40585. : I don't think you are missing anything. It is known that these are the : only four integers equal to the sum of the factorials of their base 10 : digits. This is discussed in Joe Roberts' book "The Lure of the Integers" : on page 35. He mentions a general theorem due to B. L. Schwartz stating : that the sum of a function f(n,d) of the base b digits of N is equal to N : for only a finite number of cases if : n [max f(n,d)] : lim sup ---------------- < 1 where d = 0 to b-1 : n b^n : In addition to the simple example f(n,d) = d! for all n, he gives the more : interesting example f(n,d) = d^n. In this case the theorem tells us there : are only finitely many cases where an n-digit number is equal to the sum of : the nth powers of its digits. The known occurrences of this include 153, : 1634, 8208, 9474, .... up to 4679307774. The theorem says there are only : finitely many of these, but it isn't known if 4679307774 is the largest. : (It's known that an upper bound for such a number is 10^60.) Here goes the complete list of integers for this last case [ f(n,d) = d^n ]. Incidently, they are usually called Plus Perfect Numbers, and can be computed for bases other that 10 (for any given base there is always a finite number of them). 0 1 2 3 4 5 6 7 8 9 153 370 371 407 1634 8208 9474 54748 92727 93084 548834 1741725 4210818 9800817 9926315 24678050 24678051 88593477 146511208 472335975 534494836 912985153 4679307774 32164049650 32164049651 40028394225 42678290603 44708635679 49388550606 82693916578 94204591914 28116440335967 4338281769391370 4338281769391371 21897142587612075 35641594208964132 35875699062250035 1517841543307505039 3289582984443187032 4498128791164624869 4929273885928088826 63105425988599693916 128468643043731391252 449177399146038697307 21887696841122916288858 27879694893054074471405 27907865009977052567814 28361281321319229463398 35452590104031691935943 174088005938065293023722 188451485447897896036875 239313664430041569350093 1550475334214501539088894 1553242162893771850669378 3706907995955475988644380 3706907995955475988644381 4422095118095899619457938 121204998563613372405438066 121270696006801314328439376 128851796696487777842012787 174650464499531377631639254 177265453171792792366489765 14607640612971980372614873089 19008174136254279995012734740 19008174136254279995012734741 23866716435523975980390369295 1145037275765491025924292050346 1927890457142960697580636236639 2309092682616190307509695338915 17333509997782249308725103962772 186709961001538790100634132976990 186709961001538790100634132976991 1122763285329372541592822900204593 12639369517103790328947807201478392 12679937780272278566303885594196922 1219167219625434121569735803609966019 12815792078366059955099770545296129367 115132219018763992565095597973971522400 115132219018763992565095597973971522401 I can send the C source code of the program that generated this list to anyone interested. Since it was one of the first programs that I wrote in this language don't expect too much in terms of readability. ***************************************************** * Tomas Oliveira e Silva * * INESC Aveiro/DETUA * * Universidade de Aveiro * * 3800 AVEIRO PORTUGAL Email: t...@inesca.pt * *****************************************************