Crazy Math

The number 1. I've been thinking about it alot recently, due to the fact that it "acts" different than the other numbers. for instance, in the case of the primes they all have two factors -- the number one and themselves. These aren't useful (well, that's the prevailing opinion anyway...) factors, but they do point out the main difference between the number one and the other real numbers. Namely, one divided by itself is itself. While we can get any other integer to behave like the number one with respect to BigOmega (by performing what was described a few posts below as "unit addition"), none will act exactly the same. Any other integer will create a pattern where the least number of factors any member can have is two. In this sense, the only way to produce prime numbers is by performing unit addition with the number one. All the other "primes" using other units will have at least two factors. This is not counting, of course, our "universal" factors -- one and the number itself. The really interesting part of all this is that the concept of any number divided by itself equals one applies to all sorts of suspects outside the set of positive integers; rational numbers, irrational numbers, imaginary numbers, complex numbers, etc. PI divided by PI is one, even though no end has been found to the digits in PI. My gut tells me that part of the answer to knowing a number's factor base is found in a solid understanding of what the concept of one really means, and how to disassemble its behaviour and duplicate it elsewhere. That's an odd concept, transmuting the logical behaviour of 1 to another number, say, 22. But who says it's impossible? Perhaps it is, but 1 is still an interesting number (even if it is the loneliest).