Interesting.
On Feb 11, 2015, at 11:57 AM, rcs@xmission.com wrote:
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.
Hmm, e-2 = 1/2! + 1/3! + . . . . Also, 1/e = 1 - 1 + (1/2! - 1/3!) + (1/4! - 1/5!) + . . . = 2/3! + 4/5! + 6/7! + . . ., not a representation I've seen before. QUESTION: There seems to be a natural definition of "weakly normal to factorial base": Definitions: Given any interval [a,b) in [0,1): * For an integer N >= 0: limit as k -> oo of (1/k) Sum_{1 <= j <= k} X_[a,b)(a_j / j) = b-a. ** For a fraction f in [0,1): limit as k -> oo of (1/k) Sum_{1 <= j <= k} X_[a,b)(c_j / (j+1)) = b-a. where X_S denotes the characteristic function of the set S. Definitions: (Fully) "normal to factorial base" is defined analogously but involves *joint* distributions and the characteristic functions of all subsets of [0,1)^n of form [a,b)^n, for all n = 1,2,3,.... Hoping that the meaning is clear, I'm too lazy to write that all out here. It's obvious that almost all numbers (pos. ints. and fractions) are normal to factorial base. QUESTION: What does normality to factorial base imply about normality to ordinary "geometric" bases like our decimal system? Or vice versa? --Dan
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
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