Tuesday, August 24, 2004
update and indexes
I have not fallen off the face of the earth, although I do now find myself in an interesting place, trying to determine where to go next.
Let's begin with Z, which I define here as the set of all integers greater than 1. Factor all of the members of Z, which is infinite. What do you get? The first answer that might spring to mind is "the set of all primes", which I'll denote as P. O.k., but is that _really_ what you get? Actually, it isn't. What you really get is all the primes, each counted an infinite number of times. For instance, if I completely factor 8, I get 2 * 2 * 2. That's three 2s, not one, and if I factored all the integers I'd wind up with quite a lot of 2s. The same could be said for 3, 5, 7, etc. So, if we throw all our factors into a set we can definte this set as P_2. It's similar to P, in that P is included in P_2, but it is an order of infinity greater than P.
Ah, but we can go further. What if, for instance, we count all the powers of a prime seperately? In this case, a number such as 8 is equal to 2^3. If we now re-order our set we've gained another degree of infinity, for now we have all the prime powers counted infinitely, along with all the primes counted infinitely and the infinity of primes, and I denote this set as P_3.
And this, I think, leads to an interesting and most basic question. Which, if any, of these three sets is "the right one"? I tend towards P_2, but this is only because I don't like the idea of considering each power of a prime as a seperate case. It's an arbitrary distinction to be sure, but since this is my math blog and I'm just thinking out loud here I'll declare, by fiat, that P_2 is my favorite of these three sets.
If that were all we'd have quite a bit to think about, because now there's an interesting issue to address. In the set P_2 we're implying that all the 2s, for instance, are seperate. This is a question that I can't begin to address, but its interesting to consider if that were actually the case. By "actually the case", I mean that making this distinction would have some mathematically significant purpose.
In my mind it does, and this is because by profession I do alot of database programming. In programming databases, one deals often with "relational" data. This means that an object can be related to another object or group of objects. Think of the structure of a company, for instance. The parent company may have divisions, and these divisions can have children. We can draw this relationship, and it'll look like a tree, with the branches representing different "child" divisions. Let's say that we wanted to search through this tree to see if division a is related to some other division, maybe f. We can write a routine that will figure out all of the relations and put them into a table with two columns, one for "parents" and one for "children". Thus f will appear as a "child" and a as a "parent" no matter how many levels may seperate them. If we compare our tree and our two-column table, we see that the table is quite a bit larger than the tree, and both are larger than the number of divisions within the company.
In case you didn't catch the analogy, the "divisions" represent the primes; the "tree" represents all the positive integers. If we took all the "divisions" listed in the two-column table and put them into one set, with each "division" grouped together into its own subset, we'd have a set that's analogous to P_2.
The rules that define the relationships are what cause the generation of the extra information that results in our P_2 set being larger than the set of primes. What are those rules? Is there one algorithm that states them or mimics them? Can such an algorithm be devised? That's the question at the root of the problem of factoring, and I believe that the concept of the set P_2 can help us better approach the problem.
Now, that line of thought has occupied me for the last few weeks. If that were all, it would be fine and good. However, that is not all, and that's why I'm putting up this post.
We have not taken into consideration here the concept of unit addition.
My examination of this subject is incomplete, but here's where my thoughts have been going in the last couple of days. We can construct the set P_2 in a different manner than simply factoring all the integers greater than 1. Instead, let us use unit addition to break apart each integer into its prime component. I say "component" because that's what unit addition is all about -- one number is the "unit", and it is added to itself some n number of times. Take for example the number 6. This number, in unit addition terms, can be broken down two ways -- 2 + 2 + 2 = 6, and 3 + 3 = 6. This is fundamentally different than the factoring case, where we get 3 * 2 = 6, and the resulting set will therefore be different. It won't look different to us (it's still the set of all primes, each prime counted an infinite number of times), but a very interesting thing happens when we look at the individual prime units.
Let us assign an index number to each occurance of a prime in the set P_2 created by "factoring" into primes and P_2 created by reversing the unit addition process. Now, we will recombine the various elements to produce the set Z again, only this time let's add up all the index values for the numbers used. So, in the case of the number 2, the index value is one in both our sets P_2 because in both sets it is the first occurance of the number 2. We have the same story with 3, and now we get to 4. In the case of four, we have a.) the second two multiplied by the third two -- this gives us a combined index value of 5, and b.) the second two plus the third two, which again gives an index sum of 5. We move on to 5, which will have an index sum of 1 in both sets. However, 6 is where we start to run into the fun stuff. In the multiplicative set, we have a.) the fourth two times the second three for an index sum of 6, and b.) either the fourth two plus the fifth two plus the sixth two, which gives us an index sum of 4 + 5 + 6, or 15; or the second three plus the third three, which gives us an index sum of 5.
I think that's enough for tonight; it's late and I need sleep (a crazy day of programming awaits tomorrow). I hope this gives you some food for thought. Who knows if it's actually useful...
Let's begin with Z, which I define here as the set of all integers greater than 1. Factor all of the members of Z, which is infinite. What do you get? The first answer that might spring to mind is "the set of all primes", which I'll denote as P. O.k., but is that _really_ what you get? Actually, it isn't. What you really get is all the primes, each counted an infinite number of times. For instance, if I completely factor 8, I get 2 * 2 * 2. That's three 2s, not one, and if I factored all the integers I'd wind up with quite a lot of 2s. The same could be said for 3, 5, 7, etc. So, if we throw all our factors into a set we can definte this set as P_2. It's similar to P, in that P is included in P_2, but it is an order of infinity greater than P.
Ah, but we can go further. What if, for instance, we count all the powers of a prime seperately? In this case, a number such as 8 is equal to 2^3. If we now re-order our set we've gained another degree of infinity, for now we have all the prime powers counted infinitely, along with all the primes counted infinitely and the infinity of primes, and I denote this set as P_3.
And this, I think, leads to an interesting and most basic question. Which, if any, of these three sets is "the right one"? I tend towards P_2, but this is only because I don't like the idea of considering each power of a prime as a seperate case. It's an arbitrary distinction to be sure, but since this is my math blog and I'm just thinking out loud here I'll declare, by fiat, that P_2 is my favorite of these three sets.
If that were all we'd have quite a bit to think about, because now there's an interesting issue to address. In the set P_2 we're implying that all the 2s, for instance, are seperate. This is a question that I can't begin to address, but its interesting to consider if that were actually the case. By "actually the case", I mean that making this distinction would have some mathematically significant purpose.
In my mind it does, and this is because by profession I do alot of database programming. In programming databases, one deals often with "relational" data. This means that an object can be related to another object or group of objects. Think of the structure of a company, for instance. The parent company may have divisions, and these divisions can have children. We can draw this relationship, and it'll look like a tree, with the branches representing different "child" divisions. Let's say that we wanted to search through this tree to see if division a is related to some other division, maybe f. We can write a routine that will figure out all of the relations and put them into a table with two columns, one for "parents" and one for "children". Thus f will appear as a "child" and a as a "parent" no matter how many levels may seperate them. If we compare our tree and our two-column table, we see that the table is quite a bit larger than the tree, and both are larger than the number of divisions within the company.
In case you didn't catch the analogy, the "divisions" represent the primes; the "tree" represents all the positive integers. If we took all the "divisions" listed in the two-column table and put them into one set, with each "division" grouped together into its own subset, we'd have a set that's analogous to P_2.
The rules that define the relationships are what cause the generation of the extra information that results in our P_2 set being larger than the set of primes. What are those rules? Is there one algorithm that states them or mimics them? Can such an algorithm be devised? That's the question at the root of the problem of factoring, and I believe that the concept of the set P_2 can help us better approach the problem.
Now, that line of thought has occupied me for the last few weeks. If that were all, it would be fine and good. However, that is not all, and that's why I'm putting up this post.
We have not taken into consideration here the concept of unit addition.
My examination of this subject is incomplete, but here's where my thoughts have been going in the last couple of days. We can construct the set P_2 in a different manner than simply factoring all the integers greater than 1. Instead, let us use unit addition to break apart each integer into its prime component. I say "component" because that's what unit addition is all about -- one number is the "unit", and it is added to itself some n number of times. Take for example the number 6. This number, in unit addition terms, can be broken down two ways -- 2 + 2 + 2 = 6, and 3 + 3 = 6. This is fundamentally different than the factoring case, where we get 3 * 2 = 6, and the resulting set will therefore be different. It won't look different to us (it's still the set of all primes, each prime counted an infinite number of times), but a very interesting thing happens when we look at the individual prime units.
Let us assign an index number to each occurance of a prime in the set P_2 created by "factoring" into primes and P_2 created by reversing the unit addition process. Now, we will recombine the various elements to produce the set Z again, only this time let's add up all the index values for the numbers used. So, in the case of the number 2, the index value is one in both our sets P_2 because in both sets it is the first occurance of the number 2. We have the same story with 3, and now we get to 4. In the case of four, we have a.) the second two multiplied by the third two -- this gives us a combined index value of 5, and b.) the second two plus the third two, which again gives an index sum of 5. We move on to 5, which will have an index sum of 1 in both sets. However, 6 is where we start to run into the fun stuff. In the multiplicative set, we have a.) the fourth two times the second three for an index sum of 6, and b.) either the fourth two plus the fifth two plus the sixth two, which gives us an index sum of 4 + 5 + 6, or 15; or the second three plus the third three, which gives us an index sum of 5.
I think that's enough for tonight; it's late and I need sleep (a crazy day of programming awaits tomorrow). I hope this gives you some food for thought. Who knows if it's actually useful...
