http://news.illinois.edu/news/15/0209innercore_XiaodongSong.html http://www.nature.com/ngeo/journal/vaop/ncurrent/full/ngeo2354.html Allegedly the earth has an inner core inside its core. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
Allegedly the earth has an inner core inside its core.
I wonder how big you have to be for you get an inner-inner-inner core of neutronium. Suppose the whole earth were an iron ball or the sun? Is there a size that collapses to have neutron matter at the center without becoming neutron matter all the way through? My best guess is that some of it will become neutron matter and the rest will blow away under the energy of falling toward the neutron matter. Whit
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
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
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
--------
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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e = 10.111111111111111111... in factorial base. The integer portion left of the decimal(!) point has weights ... 2!, 1!, and the fraction part has weights 1/2!, 1/3!, ... We exclude the useless 0! weight from both sides, and the redundant 1/1! from the fraction. Dan's example with 1/e is .020406080a0c0.... Rich ----- Quoting Neil Sloane <njasloane@gmail.com>:
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
--------
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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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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On Feb 11, 2015, at 11:20 AM, Daniel Asimov <asimov@msri.org> wrote:
(**) f = c_1 / 2! + c_2 / 3! + . . . + c_k / k! + . . . with 0 <= c_j <= j for all j
This was in the American Mathematical Monthly ca. 1964, and just happened to be something I was working on at the time: PUZZLE: Suppose that in (**) the coefficients c_j are periodic: For some n and for all j, c_(j+n) = c_n . Evaluate the fraction f in terms of c_1, . . ., c_n. --Dan
P.S. The puzzle I just posted suggested another one (new to me), of which I have no idea of the answer, or more accurately have too many things to do right now, so can't think about it. (Posting to math-fun is a very successful technique for procrastinating.) Suppose the integer coefficients c_j of this infinite series: (***) x = c_1/1 + c_2/2 + c+3/3 + . . . + c+k/k + . . . are periodic (for some n in Z+ and all j in Z+,, c_j = c_(j+n)), and that the series converges. The c_j are otherwise arbitrary. Can x be evaluated? --Dan P.S. As may be obvious, I had too much coffee this morning.
On Feb 11, 2015, at 12:57 PM, Daniel Asimov <asimov@msri.org> wrote:
On Feb 11, 2015, at 11:20 AM, Daniel Asimov <asimov@msri.org> wrote:
(**) f = c_1 / 2! + c_2 / 3! + . . . + c_k / k! + . . . with 0 <= c_j <= j for all j
This was in the American Mathematical Monthly ca. 1964, and just happened to be something I was working on at the time:
PUZZLE:
Suppose that in (**) the coefficients c_j are periodic: For some n and for all j,
c_(j+n) = c_n
.
Evaluate the fraction f in terms of c_1, . . ., c_n.
participants (6)
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Dan Asimov -
Daniel Asimov -
Neil Sloane -
rcs＠xmission.com -
Warren D Smith -
Whitfield Diffie