Newsgroups: sci.math
From: hoey@AIC.NRL.Navy.Mil (Dan Hoey)
Date: Thu, 30 Jun 1994 22:05:47 GMT
Subject: Multiplicative Persistence (was: Solve this funny problem...)

fra...@davina.inria.fr (Franck ) writes:

> Take one number, 47 for example, and multiply its digits [together],
> until you have only one digit :
> 4*7 = 28
> 2*8 = 16
> 1*6 = 6
> This number is said to have a peridicity of 3, because we have
> processed 3 multiplications before having one digit.
..
> A researcher tells me that i can't find a number with periodicity
> 12, 13 ....

As p...@eurocontrol.fr (Philip Gibbs) noticed, Franck used both
`peridicity' and `periodicity' in his message.  I have no idea how the
first might have been derived, and the second is a rank misnomer.
Fortunately, the term `multiplicative persistence' is already
established in the literature for this concept.  David Wells's _The
Penguin Dictionary of Curious and Interesting Numbers_ cites

  N.J.A. Sloane, `Multiplicative Persistence', _Journal of
  Recreational Mathematics_, vol. 6.

Wells [presumably summarizing Sloane] states, ``No number less than
10^50 has a higher multiplicative persistence [than 11] and it is
conjectured that there is an upper limit to the multiplicative
persistence of any number.''

Philip noted that the only nonzero numbers that can arise from
multiplying the digits of a number have all their prime factors less
than ten.  I will call such numbers `low-pf numbers.'  I also wrote a
program to find the low-pf numbers of high persistence.  Up to 10^100,
the only ones with persistence greater than 6 are the following:

          MP(338688) = MP(2^8 3^3 7^2) = 7
          MP(826686) = MP(2 3^10 7) = 7
         MP(2239488) = MP(2^10 3^7) = 7
         MP(3188646) = MP(2 3^13) = 7
         MP(6613488) = MP(2^4 3^10 7) = 7
        MP(14224896) = MP(2^9 3^4 7^3) = 7
MP(3416267673274176) = MP(2^6 3^27 7) = 7
MP(6499837226778624) = MP(2^24 3^18) = 7

         MP(4478976) = MP(2^11 3^7) = 8
       MP(784147392) = MP(2^6 3^6 7^5) = 8
     MP(19421724672) = MP(2^21 3^3 7^3) = 8
    MP(249143169618) = MP(2 3^2 7^12) = 8
    MP(717233481216) = MP(2^9 3^5 7^8) = 8

       MP(438939648) = MP(2^12 3^7 7^2) = 9
    MP(231928233984) = MP(2^33 3^3) = 9

   MP(4996238671872) = MP(2^19 3^4 7^6) = 10
 MP(937638166841712) = MP(2^4 3^20 7^5) = 10

These include several numbers missing from Philip Gibbs's table, and
omit those numbers in his table that contained the digit zero.  [I
don't know why Philip included the latter.  He said something about
allowing the digit zero, but if the digit zero is ignored, then the
conjecture is false: for instance MP0(2^270)=12, MP0(2^872)=13.]

The smallest numbers of large persistence are then MP(2677889)=8,
MP(26888999)=9, MP(3778888999)=10, and MP(277777788888899)=11.  The
first two improve Philip's results--I suspect he didn't consider using
the digit 6.

I have a stronger conjecture than the one Wells provided, though I
suspect Sloane had it.  Note that the numbers I listed are all much
less than 10^100.  I noticed that the low-pf numbers of persistence 3,
4, 5, and 6 also seem to dwindle well before 10^100.  If no more
appear, then the following are the maximum low-pf numbers for their
persistence.

                          MP(1438916737499136) = MP(2^24 3^6 7^6) = 6
                         MP(36381499733311488) = MP(2^35 3^2 7^6) = 5
                    MP(6863918177676863471616) = MP(2^59 3^5 7^2) = 4
MP(268476843674491723183742239912919243314992) = MP(2^4 3^17 7^38) = 3

The low-pf numbers of persistence 2 continue past 10^100, but appear
to be dwindling as well.  This is not surprising: The number of
N-digit low-pf numbers increases quadratically with N, but (assuming
the digit "5" appears with positive density) the probability that the
product of the digits is nonzero decreases exponentially.  Thus the
expected number of low-pf numbers with persistence greater than one
converges.  So I conjecture:

  Of the positive integers whose prime factors are less than ten,
  only finitely many have multiplicative persistence greater than
  one.

or equivalently

  Of the positive integers whose decimal representation lacks the
  digit "1", only finitely many have multiplicative persistence
  greater than two.

If this conjecture is true, then it is may be one of those that is
theoretically, but not practically, provable.

wieck...@deca.cs.umn.edu (Zbigniew Wieckowski) said that Banach spaces
are somehow related here.  I would be interested to hear an
explanation of the connection, unless it is as superficial as his
remarks about Kaprekar's process.

Dan Hoey
Hoey@AIC.NRL.Navy.Mil