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### 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/
```
Associated Topics:
High School Number Theory

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