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# factorial base

A positional base in which each place value instead of being a power of the base is a factorial. For example, the integer which is represented in base 10 as 47 (because $4\cdot 10^{1}+7\cdot 10^{0}$) is represented in factorial base as 1321, or $1\cdot 4!+3\cdot 3!+2\cdot 2!+1\cdot 1!$.

Factorial base representation has applications in combinatorics and cryptography.

The factorial base representations are unambiguous as long as the maximum allowed digit for a given place value is not exceeded (e.g., the least significant digit $d_{1}$ can only be 0 or 1, while the most significant digit in an 7-digit factorial base number $d_{7}$ has to be in the range 0 to 7).

With this limitation placed in the definition, and the observation that

$n!-1=\sum_{{i=1}}^{{n-1}}i!i$ |

it is obvious that factorial base is unambiguous, though it has the potential to use an infinite amount of distinct digits even as the less significant place values are limited in what values they can contain.

Though this is true of fractions, though in the opposite direction (the most significant fractional place values are more limited in the range of digits they can contain), factorial base has the advantage that the representation of a rational number always terminates. This is not always the case in a fixed base where the representation of a rational number could be repeating when the denominator is coprime to the base (see: factorial base representation of fractions).

The Lucas-Lehmer code maps unique factorial base representations of an integer $n$ to the permutation of $n$ elements in lexicographical order.

A007623 of Sloane’s OEIS lists the first few integers written in factorial base, A046807 lists palindromic numbers in factorial base, A118363 lists factorial base Harshad numbers, etc.

## Mathematics Subject Classification

11A63*no label found*

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## Comments

## clarification of unambiguous claim for factorial base?

Can you offer either a theorem and proof, or examples

that make this claim clear?

"The factorial base representations are unambiguous as long as the maximum allowed digit for a given place value is not exceeded (e.g., the least significant digit can only be 0 or 1, while the most significant digit in an 7-digit factorial base number has to be in the range 0 to 7). Thus factorial base has the potential to use an infinite amount of distinct digits even as the less significant place values are limited in what values they can contain."

## Re: clarification of unambiguous claim for factorial base?

For 7 you can write 101 ( = 1*3!+0*2!+1*1!) or 31 ( = 3*2!+1*1!), but with the second representation you've exceeded the maximum allowed digit in the two-factorial place-value. Doesn't get any clearer than that.

P.S. What's the longest proof that 1 + 1 = 2?

## Re: clarification of unambiguous claim for factorial base?

I think you would be wiser to listen to "jac"'s suggestion and rewrite that paragraph. It is not clear to me either and many other users will find it difficult to read. On the other hand, the fact that any number can be represented uniquely in a factorial base is no trivial fact, and a proof of this fact would be a nice addition to the entry.

T

## Re: clarification of unambiguous claim for factorial base?

Sniff... Torquemada: could you please pay attention to who wrote what?

Lando: Could you not write in that way that makes people like Torquemada think I said something I didn't actually say?

Its all gonna have to wait until after Memorial Day anyway.

## Re: clarification of unambiguous claim for factorial base?

> Sniff... Torquemada: could you please pay attention to who

> wrote what?

My mistake, sorry. However, "jac" did rise a legitimate point. The paragraph could be clearer in exposition, and a proof of that fact would be extremely helpful to the reader.

T

## Re: clarification of unambiguous claim for factorial base?

Lisa, I'm sorry that my remarks caused others to lash out at you.

I still think that the article is clear enough, but then again, I understand written English better than native speakers. My suggestion: clarify by obfuscating. Find some way to put x amount of Greek letters into the page. That should do the trick.

Will