Newsgroups: rec.puzzles, sci.math
Followup-To: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: Tue, 3 May 1994 20:38:13 GMT
Subject: Digital Invariants (was Factorions)

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
Hoey@AIC.NRL.Navy.Mil