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/
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