Factorial Base and Base 10

Date: 11/02/2001 at 05:00:34
From: Leonard Oltshoorn
Subject: Factorial base and base 10

Hello,

Let n be a number written in base 10, which has an interpretation in 
factorial base as well. Let m be the value of its interpretation in 
factorial base. Which is the greatest n for which m <= n ?

Thanks.
Leonard

Date: 11/05/2001 at 03:52:06
From: Doctor Floor
Subject: Re: Factorial base and base 10

Hi, Leonard,

Thanks for writing.

Your question will give a quite large number as answer, so let me show 
as an example how to solve the question with base 4 in stead of base 
10. Of course your base 10 question can be solved in a similar way.

First I make a table of the place values in base 4 as well as in 
factorial base. In base 4 the nth digit (from behind) has a place 
value of 4^(n-1), while in factorial base it has place value n! In the 
table I also indicate the difference, as well as the maximum value the 
digit can have to be interpretable in both bases (in factorial base 
the maximum is n, in base 4 the maximum is 3).

digit#    base 4    fact base    difference    maximum value
  1         1           1            0              1
  2         4           2            2              2
  3        16           6           10              3
  4        64          24           40              3
  5       256         120          136              3
  6      1024         720          304              3
  7      4096        5040         -944              3

Now we see that among the numbers 1, 10, 100, .... the number 
1000000 is the first for which the factorial base interpretation 
has a larger value than the base 4 interpretation. We can also see 
that the largest possible difference that can be made with a 
six-digit number is found with 333321 and 333320 and brings a 
difference of 2*2+3*10+3*40+3*136+3*304 = 1474. This 1474 is more 
than 944, but less than 2*944. That shows that

  1333321

is the largest number (written in base 4) for which the base 4 
interpretation has a larger value than the factorial base 
interpretation, since clearly, there is no such number greater than 
2000000.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/