Newsgroups: rec.puzzles, sci.math
Followup-To: rec.puzzles
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: 25 Jul 92 22:13:33 GMT
Subject: Digital Invariants (was Number Theory)

uunet!eliot!andyc (Andy Panda Collins) writes:

> I don't know much about this news-group, but I thought
> that I might post this list of number-theoretic numbers.
> Let N be an n-digit number which is separated into it's
> decimal digits, raise each digit to the nth power and sum
> them.  Let N be a unique number if the sum is equal to N.

Well, this topic is more the province of recreational math than of
number theory, so I've redirected it to rec.puzzles such topics are
more customarily treated.

According to David Wells's book _The_Penguin_Dictionary_of_Curious_
_and_Interesting_Numbers_ these numbers are called digital invariants.
Wells notes that

  When G. H. Hardy wished, in his book _A_Mathematician's_Apology_,
  to give examples of mathematical theorems that were not `serious',
  he chose two examples, `almost at random', from Rouse Ball's
  _Mathematical_Recreations_.

  The first was the fact that 8712 and 9801 are the only 4-digit
  numbers that are multiples of their reversals.

  The second was the fact that, apart from 1, there are just 4
  numbers that are the sums of the cubes of their digits.

I found all the numbers that are multiples of their reversals last
year (and coined the term `palintiples' to refer to them) so I suppose
it is not so surprising that I should be interested in this problem as
well.

Andy lists all the digital invariants up to 14 digits, and I wrote a
program that confirms his list.  The next few are:

15 digits: none
16 digits: 4338281769391371
           4338281769391370
17 digits: 35875699062250035
           35641594208964132
           21897142587612075
18 digits: none
19 digits: 4929273885928088826
           4498128791164624869
           3289582984443187032
           1517841543307505039
20 digits: 63105425988599693916
21 digits: 449177399146038697307
           128468643043731391252
22 digits: none
23 digits: 35452590104031691935943
           28361281321319229463398
           27907865009977052567814
           27879694893054074471405
           21887696841122916288858
24 digits: 239313664430041569350093
           188451485447897896036875
           174088005938065293023722
25 digits: 4422095118095899619457938
           3706907995955475988644381
           3706907995955475988644380
           1553242162893771850669378
           1550475334214501539088894.

I have also verified that there are no digital invariants of 46 or
more digits.

David Wells lists one more digital invariant, though, of 39 digits:

  115132219018763992565095597973971522401

and he says it is the largest one known.  However, I am not sure but
what this may have been proven to be the largest one that can exist
by whoever discovered it.  I do wish Wells were a little more
conscientious with his attributions.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil