Friday, January 07, 2005

Happy New Year's Day. 2004 was the first year in many that felt somewhat like a full year - an eventful one.

'tis good tidings and a busy schedule that bids me to write my updates so few and far in between - didn't a buddhist once mention that the spaces between the notes make the music ?

More Random Thoughts For You All:
Infinite sets are fun, and even weirder are infinite sets derived from non-repeating irrational numbers such as pi ... in fact, for those of you who know of Edmund Landau's notation for the Prime Counting Function (pi(x)), here's an interesting rationalization (read: rationalization, NOT a proof) for his use of that particular symbol:

Take an irrational non-repeating number such as pi, and treat that number as an infinite sequence - one that can be written as a partitioned set of strings, where all strings in the set hold a trait in common. In particular, assume each string to represent a prime number, created by appending digits from pi (as found in the sequence) into the current result until the result is a prime not yet counted, whereupon it is added to the count, and we move on to the next string of digits. Since this set can be represented as the set of all prime numbers, we may order this set, starting with the smallest prime - thus giving us a mapping of the set of prime numbers to the set of positive integers.
We may call this function (when it is found at last) "The xth Holy Grail of Mathematics" where x is a variable (after all, there are far more than just one of these in mathematics) .
But less dramatically, since we have shown that the sequence of digits from pi can be associated with the set of all prime numbers, we can all (or maybe just all of one person) sleep more easily knowing that there is some justification in the use of the symbol pi as the Prime Counting Function (pi(x)). (which is simply a function that gives the number of primes less than a given integer.)

Even weirder: Take a second number in that category of irrational non-repeating numbers, such as phi ( for the function phi(x) ), and partition its equivalent sequence in the same manner as shown above ... we now have a 1-to-1 mapping from one set to the other, showing them to be of equal size ... bizarre, eh ? Just what meaning could this hold for phi and pi ? Well, as you will see below, my conjecture is that it doesn't prove anything, because we can map these irrational numbers in any function that we want ... allowing the range of phi(x) to be the set of all even numbers, and the range of pi(x) to be the set of all odd numbers ... then of course, the set of all positive integers must be equal to the union of these sets phi and pi, thus showing them to be subsets where the positive integer set is equal to the partition { range(phi) , range(pi) } !
This is just idle wondering.

Something a little more concrete (and perhaps more productive), but not by much:
Now here's something more concrete (and redundant) I've been wondering about for awhile:
Using an array of logarithms, where the lookup is based off of the index of the array, how does a computer's built-in multiplication compare to the equivalent via logarithm addition and lookup ? And what about treating these numbers as logarithm polynomials ? And what about programming structures created for arbitrarily large numbers? (grouped with modular arithmetic and bit-shifting, too?) How does it affect prime number verification ? Especially coupled with the knowledge that all prime numbers above ten have a least significant digit of 1, 3, 7, or 9 ?
If you haven't yet guessed by my overly specific questions, I'm going to find out. I know it's been done many many times before by other programmers, but I haven't been able to find in-depth results online anywhere - hence, I'm going to experiment and post the results. Just keep in mind, it's another project added to the stack (the stack that should be a queue)
and, if you-know-who is willing, then it may just get done. After all, I had so many projects I wanted to finish before the start of Spring semester, which now looms over the hills less than a week away.

Happy New Year, everyone ! And remember :
A good teacher makes boring stuff exciting, and a bad teacher makes exciting stuff boring
(I didn't create this particular aphorism; I heard it from someone else and had to share.)
;-)