Friday, June 18, 2004
Errors, errors, aren't they fun?
So, my somewhat scatterbrained post from yesterday (I was noticing several logical nonsequitirs, among other delights) also contained an interesting error, and one which kinda throws a wrench in the works. It's like this: All is fine and good with the symbolic representation of the composition of some set, up until n = 5. However, we run into a problem in n = 5. Namely, we can't say that Cd3 (i.e. the third degree composite) is odd. n = 5 is this set -- {32..65}. Now, the smallest n-degree odd composite is 3^x. The first few values are 3, 9, 27, 81... If you've noticed, there is no member of {32..63} with a factorization of 3^x. Since 3^x is the smallest possible odd composite of degree x, the only possible members of {Cd3} in {32..63} have to be even, since the range of the function BigOmega is smooth. No biggie, since that's the case we can just figure out whether or not 3^x lies in a set and work things out from there. The problem, though, lies in the symbolic representation I laid out in the last post. In it, I defined Cdn as representing an odd composite of degree n. It turns out this is not always the case -- Cdn could also represent an even composite. This means the set of all {Cdn} can include both even or odd numbers, that it overlaps with the even members and thus cannot be rigidly defined as meaning "an odd composite of degree n". Instead, the definition should be "a composite of degree n". This gums up our ability to count symbolically represented even and odd numbers in these sets. I.e., if we look at {E, P} (which has a symbol count of 2), there are two members represented symbolically. However, if we look now at {{{E, P} * 2} & P} * 2 | {Cd2} & {P} (which has a symbol count of 5), we can't say with a certainty whether Cd2 contains even numbers, odd numbers, or both. By definition all even numbers have been defined symbolically and we cannot assume that, say, {Cd2} doesn't include {P * 2}, which means we cannot say with certainty that the symbol count equals 5. Since we can't accurately count the symbolic representation of the set, we cannot say with accuracy how many symbolic representations there are in the set _without_ figuring out first if that particular set includes some number 3^x.
I kind of knew this was true in the back of my mind, but it wasn't until I worked out the symbolic representation for {32..63} that I realized the obvious error. That doesn't mean, though, that all is lost. Sometime in the next few days I'll post about an interesting consequence of the disjointed overlap of n = y and 3^x. We already know that, with the exception of the set {2, 3} we never have a set where the largest odd composite is 3^n. This would mean that 2^n and 3^n were in the same set, which they cannot be since the _smallest_ number with n possible factors is 2^n, and therefore 3^n has to occur in at least the next set, i.e. n + 1. Until then, you might want to consider the set of numbers output by performing BigOmega for each member of the set {2..256}. It will figure prominently in the next post...
I kind of knew this was true in the back of my mind, but it wasn't until I worked out the symbolic representation for {32..63} that I realized the obvious error. That doesn't mean, though, that all is lost. Sometime in the next few days I'll post about an interesting consequence of the disjointed overlap of n = y and 3^x. We already know that, with the exception of the set {2, 3} we never have a set where the largest odd composite is 3^n. This would mean that 2^n and 3^n were in the same set, which they cannot be since the _smallest_ number with n possible factors is 2^n, and therefore 3^n has to occur in at least the next set, i.e. n + 1. Until then, you might want to consider the set of numbers output by performing BigOmega for each member of the set {2..256}. It will figure prominently in the next post...
