Monthly Archives: January 2006

Thoughts on Number Theory

The conversation about life frequency has lead me on a short intellectual stroll. I’ve wound up reflecting on number theory, a subject with which I am only tangentally familiar, but one which holds me in a certain fascination. As we will see, this fascination is in turn of interest.

The notes on the January 12th meeting indicate my perceived connection between life frequency and cellular automata. My history of playing (off and on) with cellular automata goes back a very long way. In fact, it was the Game of Life distributed with the original Macintosh systems that first sparked my interest in cellular automata (no, I never owned a Mac, but the University of Rochester bought their first personal computer lab from Apple – I found those machines very painful and confusing to use, but that’s another story). In college, I spent a bit of time collecting information about these systems, and then, after college, I spent plenty of Perkin-Elmer’s time (unbeknowst to them), continuing this research. It was in the course of pursuing this interest that I first read Steven Wolfram’s papers on the topic, which is where he got his start in his theory of “a new kind of science” as he calls it. Subsequently (according to Wolfram), he decided he needed a more powerful system with which to explore his ideas, and he set out to build it. The result was Mathematica, which is by far my favorite software system. Wolfram then went on to publish his “New Kind of Science”, which he views as the completion of his quest for a unifying principle for complex systems of all kinds, including the Universe as a whole. In my opinion (this would be a separate topic), he falls far short of a theory of anything in this work, though he does make a slew (a big slew) of interesting observations.

Which leads me back to number theory (admittedly after Kim asked me one night what a Mersenne number is). Number theory also appears to consist of a huge pile of facts, which I’ll call observations, a somewhat smaller pile of recognized patterns within those observations (hypotheses), and a relatively small pile of proven relationships (theorems). The topic of number theory is (rather obviously) numbers, specifically positive integers; however, in advanced subfields of number theory this is generalized to include any “number system”. For our present purpose it is sufficient to consider just plain “elementary” number theory (which is far from elementary).

To get flavor for what elementary number theory contains, here is a very small taste of definitions, theorems, and assertions:

A Mersenne prime is a prime number of the form 2^n-1. The first few are 3,7,31,127,8191,131071,524287,2147483647. There are only 43 known Mersenne numbers. The largest one known has over 9 million digits. The 35th-43rd were “discovered” using a distributed processing computer system (GIMPS) that is downloaded and used similarly to the SETI screensaver.

Unproven hypothesis: There are an infinite number of Mersenne primes

A perfect number is equal to the sum of its divisors. The first couple are 6=1+2+3, 28=1+2+4+7+14.

Theorem: For every perfect number, there is a Mersenne prime, and vice-versa.

Unproven hypothesis: There are no odd perfect numbers. What is proven is that there are none less than 10^300 (1 followed by 300 zeros).

Theorem: Every even perfect number > 6, is of the form 1+9Tn, where Tn is a triangular number, which by definition is of the form 0.5n(n+1), with n=8j+2, j a positive integer.

Theorem: Every even perfect number is the sum of consecutive positive integers starting with 1 and ending with a Mersenne prime.

Theorem: The only perfect number of the form 1+x^3 is 28.

Had enough? To give you some idea of the breadth of number theory, there are definitions of each of the following types of numbers, each with its own theorems and inter-relationships to various other types:

Abundant Number, Amicable Numbers, Deficient Number,e-Perfect Number, Harmonic Number, Hyperperfect Number, Infinitary Perfect Number, Multiperfect Number, Multiplicative Perfect Number, Pluperfect Number, Pseudoperfect Number, Quasiperfect Number, Semiperfect Number, Smith Number, Sociable Numbers, Sublime Number, Super Unitary Perfect Number, Superperfect Number, Unitary Perfect Number, and, last but not least: Weird Numbers.

One more example: There are two Sublime Numbers known to man: 12, and
608655567023837898967037173424316962265783077335188597
0528324860512791691264.

We do not know if any odd sublime numbers exist.

If you are laughing by now (I was), ask yourself why. What is it about this field of math that seems ludicrous?

I’ll have more to say on this.

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Archive.org

I listened yesterday to the Audible program “Library of Congress Series on Digital Future: Lecture 2”, given by Brewster Kahle. This was a talk promoting the idea of making all of the contents of the Library of Congress available to the world through digitization. As a “starter project” of sorts, the author referred to www.archive.org, which is an amazing treasure trove of material, including text, audio and video – all freely available.

My favorite section at the moment is the presidential recordings, which include a moderate set of audio files, mostly speeches, but also secret tapes of the Oval Office (not Nixon’s, however). Last night I listened to a conversation between JFK and MacArthur – quite interesting, somewhat surprising that MacArthur appeared to be generally Democratic, at least in that conversation. MacArthur was clearly indicating that the US needed to regain the initiative in the Cold War; Kennedy was giving excuses for why some of what the General proposed had been rejected.

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Commenting on a Page Solved

As you can see, there is more than one “template” through which the blog can be displayed (about 30 to choose from). I had a hunch that the “features” available might vary by template, and I was right. This is the least unattractive (to my eye) of the formats that allow commenting on pages.

If you go to any of the pages, you should now see an area at the bottom of the page in which to write your comment. All comments are moderated, meaning that I have to read and approve them before they appear on the site, so don’t be concerned when your comments do not appear immediately.

I’m going to test this with Bob’s emailed comments on the December 30th meeting later tonight or tomorrow.

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Meeting notes: January 5, 2006

I’ve added notes on our January 5th meeting. Still can’t figure out how to create a link to allow comments on a “page” as opposed to a “post”, though in editing a “page”, I can set a parameter that supposedly allows comments.

At any rate, I suggest that each time I add a page, or edit a page, I’ll write a post (like this one), to which you can add comments.

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Meeting Notes

Notes on the weekly meetings of the Connecticut Objectivism Discussion Group shall be posted in a section under “Pages”, which can be seen toward the lower left of the main screen. These shall be grouped by month. Notes from last week’s meeting will be posted later this evening.

Please be aware that these “notes” are not meeting minutes. They are meant to both record what topics were discussed with the major opinions that were expressed, and to expand upon those topics. Particularly when more than a day or two has elapsed since the meeting, I will only recall the portions that I found interesting. This means that these “notes” are at best heavily biased toward my own opinions, and at worst may completely misrepresent the opinions of others.

Group members are strongly encouraged to make comments when I have misrepresented the content of the meetings!

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Organization Details

Having begun to play a bit more with the system, it appears that the posting area is not going to be particularly useful for allowing a gradual refinement of written material. Instead, a hierarchy of pages can be constructed and edited as necessary. This looks much more promising, and I’ve begun with a sketch of my thoughts on Sovereignty. The plan will be to organize these pages into a set of categories, with essays contained within each category. In addition, there is a “category” construct in the system allowing each post and page to be attached to one or more categories, which can then be used as search terms.

In the posting section, I’ll provide details such as the current post, as well as any “diary” material that I believe may be of interest either to myself or to others of a like mind.

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