Let's see...
So, before I begin something fucking awesome:
Yes, he did play his own baseline.
Ok, now that that is out of the way time for my newest obsession: number factorization.
I just watched a video about Andrew Wiles and his proof of the Taniyama-Shimura conjecture. (There by affirming Fermat's last theorem.) I got a bit of a rush watching it and I'm envious to understand how that feels solving a problem that one is so ‘intimate’ with. At that moment I realized that there was a problem that I was obsessed with: factoring numbers. I do it all the time and don’t even realize, I see the time on a clock and I imagine it to be a decimal number and factor it. At RGIS I always factored my counts and I do the same at Borders/SBC when I see the price for anything. Now, I kinda have a place to start looking but in all reality I have no idea what I’m doing.
So, what I know:
Even numbers are easy to factor.
Odd ones are hard.
Profound, no?
The reason for even number is fairly simplistic, they share a type of symmetric we call modular congruence, namely n = n’ (mod 2). Pretend that equal sign is a congruence symbol and I’ll explain the rest. All the previous statement said is that when you divide any two even numbers (n, n’) by 2 you’ll get the same remainder, namely 0. I know it’s fairly obvious, but it’s a start, because it’s this symmetry that makes even numbers so easy to manipulate.
All that is governed has symmetry. That’s what I’m starting to believe. So my question can be stated as thus: Is there an analogous operation for odd numbers or are they destined to remain computationally intractable?
I know that this is intimately connected with the nature of the primes as any and every number factors uniquely into primes and the prime 2 divides a set of number with a cardinality equal to that of the natural numbers and that set is denser than any other set of numbers that satisfy the congruence n = n' (mod p). That’s what anyone whose even studied a small amount of number theory knows. Where I go next is the weird part.
I’m starting to believe that the positional number system is intimately connected with the problem of number factorization.
Huh? If you think about this it sounds like a very stupid idea. Compare it to trying to determine some laws of electrodynamics by examining the numerical properties of the speed of light. It’s obvious that its speed is independent of any unit we assign to it and thus pointless to study the units in favor of other properties.
The radix of a number is not a unit; it’s a series of questions. They both take a measure, but the former can relay critical information about the object under scrutiny.
In base ten you can ask up to 11 questions about a number. The first question is, can this number be divided by 10n? If not, go on to the next position. However, if the answer's yes, then ask: can 10n * 9 divide this number? How about 10n * 8? How about… you keep asking until you get to 10n * 1 then once you get a yes you then ask, does this number have a remainder? If yes move on to the next position, if no then quit.
Let’s have a concrete example, 19110? Is 19110 divisible by 102? Yes? Is it divisible by…. 102 * 1? Yes. Does it have a remainder, yes. So, thus far we have: 10000000011boolean schema or b.s-10 or 111b.s-10. Ok, 9110. Is it divisible by 101? Yes. Is it divisible by 101 * 9? Yes. Does it have a remainder, yes. So, now we have: 111 11000000001b.s-10. 110 is the same as 10010, except for 100 and lack of remainder, so lets just tack it on at the end. 111 11000000001 110b.s-10 = 19110 in base ten and it can be reduced to 11 1100000000 1compressed b.s-10 if we just look for a remainder instead of asking. It would be much simpler to do it in base 2 over its bull shit representation as 101111112 = 19110. Each 1 or 0 is a question asking, is this number divisible by 2n? In general you only need bull shit representation for radix > 2. As can be seen, regardless of base the questions relay the same information about the object under scrutiny. Information about that number's divisibility and thus about its factors.
At the moment this is all I've got. But I know that I can start looking at modular functions as a start. If I can show that there is such a symmetry for odd numbers then I can use that to create an algorithm for the fast factorization of any odd number, including ones constructed from grotesquely large primes. Ciphers whose security is dependent on the difficulty of either discrete logarithm or the factorization of large numbers would no longer be secure. The converse is that if I find that such a symmetry cannot exist those ciphers are secure against all classical (non-quantum) computers.
Division and symmetry. Yay.
Click to viewYes, he did play his own baseline.
Ok, now that that is out of the way time for my newest obsession: number factorization.
I just watched a video about Andrew Wiles and his proof of the Taniyama-Shimura conjecture. (There by affirming Fermat's last theorem.) I got a bit of a rush watching it and I'm envious to understand how that feels solving a problem that one is so ‘intimate’ with. At that moment I realized that there was a problem that I was obsessed with: factoring numbers. I do it all the time and don’t even realize, I see the time on a clock and I imagine it to be a decimal number and factor it. At RGIS I always factored my counts and I do the same at Borders/SBC when I see the price for anything. Now, I kinda have a place to start looking but in all reality I have no idea what I’m doing.
So, what I know:
Even numbers are easy to factor.
Odd ones are hard.
Profound, no?
The reason for even number is fairly simplistic, they share a type of symmetric we call modular congruence, namely n = n’ (mod 2). Pretend that equal sign is a congruence symbol and I’ll explain the rest. All the previous statement said is that when you divide any two even numbers (n, n’) by 2 you’ll get the same remainder, namely 0. I know it’s fairly obvious, but it’s a start, because it’s this symmetry that makes even numbers so easy to manipulate.
All that is governed has symmetry. That’s what I’m starting to believe. So my question can be stated as thus: Is there an analogous operation for odd numbers or are they destined to remain computationally intractable?
I know that this is intimately connected with the nature of the primes as any and every number factors uniquely into primes and the prime 2 divides a set of number with a cardinality equal to that of the natural numbers and that set is denser than any other set of numbers that satisfy the congruence n = n' (mod p). That’s what anyone whose even studied a small amount of number theory knows. Where I go next is the weird part.
I’m starting to believe that the positional number system is intimately connected with the problem of number factorization.
Huh? If you think about this it sounds like a very stupid idea. Compare it to trying to determine some laws of electrodynamics by examining the numerical properties of the speed of light. It’s obvious that its speed is independent of any unit we assign to it and thus pointless to study the units in favor of other properties.
The radix of a number is not a unit; it’s a series of questions. They both take a measure, but the former can relay critical information about the object under scrutiny.
In base ten you can ask up to 11 questions about a number. The first question is, can this number be divided by 10n? If not, go on to the next position. However, if the answer's yes, then ask: can 10n * 9 divide this number? How about 10n * 8? How about… you keep asking until you get to 10n * 1 then once you get a yes you then ask, does this number have a remainder? If yes move on to the next position, if no then quit.
Let’s have a concrete example, 19110? Is 19110 divisible by 102? Yes? Is it divisible by…. 102 * 1? Yes. Does it have a remainder, yes. So, thus far we have: 10000000011boolean schema or b.s-10 or 111b.s-10. Ok, 9110. Is it divisible by 101? Yes. Is it divisible by 101 * 9? Yes. Does it have a remainder, yes. So, now we have: 111 11000000001b.s-10. 110 is the same as 10010, except for 100 and lack of remainder, so lets just tack it on at the end. 111 11000000001 110b.s-10 = 19110 in base ten and it can be reduced to 11 1100000000 1compressed b.s-10 if we just look for a remainder instead of asking. It would be much simpler to do it in base 2 over its bull shit representation as 101111112 = 19110. Each 1 or 0 is a question asking, is this number divisible by 2n? In general you only need bull shit representation for radix > 2. As can be seen, regardless of base the questions relay the same information about the object under scrutiny. Information about that number's divisibility and thus about its factors.
At the moment this is all I've got. But I know that I can start looking at modular functions as a start. If I can show that there is such a symmetry for odd numbers then I can use that to create an algorithm for the fast factorization of any odd number, including ones constructed from grotesquely large primes. Ciphers whose security is dependent on the difficulty of either discrete logarithm or the factorization of large numbers would no longer be secure. The converse is that if I find that such a symmetry cannot exist those ciphers are secure against all classical (non-quantum) computers.
Division and symmetry. Yay.