So what is the sequence giving the factorial base representation of (the fractional part of) e? Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Feb 11, 2015 at 2:57 PM, <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.
Rich
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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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