Are there infinitely many N-digit numbers equal to the sum of the N-th powers of their digits (Armstrong...
arithmetic/digits/armstrong.numbers New
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Problem
Are there infinitely many N-digit numbers equal to the sum of the N-th powers of their digits (Armstrong numbers)? Did anyone find the other pairs of numbers with the properties mentioned in the previous posting? These are the pairs (well actually the first of each pair) of base 10 numbers which both end in the digit 0, and are equal to the sum of the Nth powers of their digits, where N is the number of digits. -- Michael Jones Just so everyone is in the know, N-digit numbers that are equal to the sum of the N-th powers of their digits are called Armstrong numbers, so the above question is asking for Armstrong numbers that end in the digit 0. Hopeless questions: Are there infinitely many Armstrong numbers? Are there infinitely many Armstrong numbers that (begin,end) in a given string of digits? Somebody had to ask them. -- Chris Long, Rutgers University
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> Are there infinitely many Armstrong numbers? No. For an N-digit number, the sum of the N-th powers of the digits is at most N*9^N, while the number is at least 10^(N-1). Since 61*9^61 ~ 9.87E59 < 10^(61-1), there are no 61-digit Armstrong numbers, or larger ones for that matter. -- Joe [ Editor: the list of Armstrong numbers ending in 0 was deleted in the quoted posting, and no answer to the other questions was found in the collected postings. ] > ... 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 ... -- Kevin S. Brown <kevin2003@delphi.com> 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 <Hoey@AIC.NRL.Navy.Mil>, May 1994