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Friday, November 07, 2008
The Sunya Machina - Part III
The central theme in this series of notes (previously:Part I & Part II) is that the beginning of the number line (0,1,2,3) is a machine. It is shown as a generator that spits out the rest of the numbers until infinity. I have used the metaphor of cellular automatons and finite state machines, from my limited knowledge. As I learn more everyday, I will try and describe the developments here.

The Fundamental Theorem of Arithmetic states very simply that:
Every integer n ≥ 2 either is a prime or can be expressed as a product of primes. The factorization into primes is unique except for the order of the factors.
- from Elementary Number Theory with Applications 2e, Thomas Koshy

Such a factorization of any number is called the 'canonical decomposition' of the number, for instance 2520 = 2^3 x 3^2 x 5 x 7 where (2,3,5,7) are primes that make up 2520's internal structure. So perhaps large integers are not unlike organic molecules, made up of simpler components arranged in unique compositions, which gives them their particular properties. Such as, 2520 has a trailing zero because two of it's prime factors are 2 and 5 (2x5=10). The very existence of composite numbers appears to be just a classification tool, since prime numbers are the bricks that make up ALL numbers, big and small. How are molecules held together? By nuclear forces.

So shouldn't there be a bridge that links number theory to particle physics?

In this regard, eminent Japanese scientist Akio Sugamoto (he has co-written several papers with Nobel Laureate Makoto Kobayashi) submitted a paper on 24 Oct, 2008 titled "Factorization of number into prime numbers viewed as decay of particle into elementary particles conserving energy":
In number theory, how a number factorizes into prime numbers is a key issue, while in particle physics how a particle decays into elementary particles is also a key issue. These two key issues are intimately related, if we identify the energy E(n) of a particle labeled by a positive integer n = 1, 2, 3, . . . is proportional to ln(n). Then, factorization of a number into prime numbers can be viewed as the energy conservation law.

While I'm slowly becoming conversant in number theory, the mathematics Sugamoto-san uses in the paper is beyond me at this stage, and I'll leave it to better mathematics geeks. However, it does re-assert my idea that the generation of the number line is connected to physical systems and the mathematical models we have created to understand them.

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2:40 AM