The only theorem I know of is that a rational number always has a finite representation in factorial base; therefore e is irrational. Back in the day, I was looking through the file of paper tapes in the PDP1 room (this served as the software library), and I came across one labeled simply "e". The handwriting might have been either Eric Jensen or Bill Ackerman. It was only a couple of folds, a few hundred characters. Curious, I printed it out, and then ran the assembler on it. Indeed, it promptly printed out a few thousand digits of e. I examined the code more closely, and discovered it created a factorial base representation of (the fractional part of) e, one digit per machine word, and then did a simple decimal conversion, multiplying the factorial base representation by decimal 10, and printing the integer "carry" that falls off the front end; rinse, repeat. The program, in assembly language, easily fit on one page. A gem. Rich -------- Quoting Daniel Asimov <asimov@msri.org>:
I've always liked factorial base -- which uses integer coefficients --
the version for nonnegative integers:
(*) N = a_1 1! + a_2 2! + . . . + a_k k!, with 0 <= a_j < j for all j
and the one for fractions in [0,1):
(**) f = c_1 / 2! + c_2 / 3! + . . . + c_k / k! + . . . with 0 <= c_j <= j for all j .
If the c_j's are all = j, then the series sums to 1.
The nice thing is, this doesn't depend on a specific choice of base, so the factorial representation of a number might be of number-theoretic interest.
But I don't know of theorems linking number-theoretic properties of a number to factorial representations.
E.g., can one say something about the representation (*) of a prime number? About the representation (**) of an algebraic number as compared to a transcendental one? Etc.
--Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun