Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Posting-Version: version B 2.10.1 6/24/83 (MC840302); site turing.UUCP Path: utzoo!linus!philabs!cmcl2!seismo!mcvax!turing!dik From: dik@turing.UUCP Newsgroups: net.games,net.math,net.puzzle Subject: Re: Solution to DSR Contest Announced Message-ID: <224@turing.UUCP> Date: Fri, 16-Nov-84 18:02:21 EST Article-I.D.: turing.224 Posted: Fri Nov 16 18:02:21 1984 Date-Received: Sat, 17-Nov-84 20:19:24 EST References: <130@ncvet.UUCP> Organization: CWI, Amsterdam Lines: 163 Apparently-To: rnews@mcvax.LOCAL Well some time ago the referenced article appeared, and many among us will have pondered about the generation of the complete set of solutions to the problem. Here it follows. The solutions were generated in slightly less than 2000 seconds on a CDC Cyber 170-750 (about 15.5 times as fast as a DEC Vax 11/750). (In summary the problem is to find n digit numbers were the number is the sum of the n-th powers of its digits.) First a table displaying for a number of digits the number of solutions. There are no solutions of 40 or more digits. # digs # sols # digs #sols # digs # sols 1 9 14 1 27 5 2 - 15 - 28 - 3 4 16 2 29 4 4 3 17 3 30 - 5 3 18 - 31 3 6 1 19 4 32 1 7 4 20 1 33 2 8 3 21 2 34 1 9 4 22 - 35 2 10 1 23 5 36 - 11 8 24 3 37 1 12 - 25 5 38 1 13 - 26 - 39 2 So there are 88 solutions. It may be seen that the majority of the solutions has an odd number of digits (70 odd, 18 even) even if we ignore the (trivial) one-digit solutions. This appears not to be accidental, because the same holds if we pose the problem for other bases than base 10. Is there someone who has an explanation for this? The largest solution is the 39-digit number: 115132219018763992565095597973971522401 The complete list of solutions: n = 1 1 2 3 4 5 6 7 8 9 n = 3 153 370 371 407 n = 4 1634 8208 9474 n = 5 54748 92727 93084 n = 6 548834 n = 7 1741725 4210818 9800817 9926315 n = 8 24678050 24678051 88593477 n = 9 146511208 472335975 534494836 912985153 n = 10 4679307774 n = 11 32164049650 32164049651 40028394225 42678290603 44708635679 49388550606 82693916578 94204591914 n = 14 28116440335967 n = 16 4338281769391370 4338281769391371 n = 17 21897142587612075 35641594208964132 35875699062250035 n = 19 1517841543307505039 3289582984443187032 4498128791164624869 4929273885928088826 n = 20 63105425988599693916 n = 21 128468643043731391252 449177399146038697307 n = 23 21887696841122916288858 27879694893054074471405 27907865009977052567814 28361281321319229463398 35452590104031691935943 n = 24 174088005938065293023722 188451485447897896036875 239313664430041569350093 n = 25 1550475334214501539088894 1553242162893771850669378 3706907995955475988644380 3706907995955475988644381 4422095118095899619457938 n = 27 121204998563613372405438066 121270696006801314328439376 128851796696487777842012787 174650464499531377631639254 177265453171792792366489765 n = 29 14607640612971980372614873089 19008174136254279995012734740 19008174136254279995012734741 23866716435523975980390369295 n = 31 1145037275765491025924292050346 1927890457142960697580636236639 2309092682616190307509695338915 n = 32 17333509997782249308725103962772 n = 33 186709961001538790100634132976990 186709961001538790100634132976991 n = 34 1122763285329372541592822900204593 n = 35 12639369517103790328947807201478392 12679937780272278566303885594196922 n = 37 1219167219625434121569735803609966019 n = 38 12815792078366059955099770545296129367 n = 39 115132219018763992565095597973971522400 115132219018763992565095597973971522401 -- dik t. winter centrum voor wiskunde en informatica postbus 4079 1009 AB amsterdam nederland +31 20 592 4102 (polish your dutch) UUCP: decvax!mcvax!turing!dik