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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