notes

In here are the list of things I did for a particular hour or day. Also included here are the screenshots of games I played, or videos I watched or listened to, or just random things I stumbled upon. I'll occasionally write down what I'm thinking, or things I'm planning to do.

[ 2024-03-22 ]

15:11 [link] [context] #math

Despite what non-binding nonsense I said yesterday, I'm still stuck on my notebook, slowly chipping away more dirt and mud. Maybe the fun hasn't worn off yet, or maybe I'm just too lazy to do anything else. Mucking around seems quite easy in comparison.

In any case, I should write down something. Not sure where to start, I should list down the seemingly useful formulas and equations that I found or made:

Firstly, I can decrement/increment any factors with any number

 For any number n:
  xy = (x+n)y - ny
  xy = (x-n)y + ny

for example:

  3(5) = (3+7)5 - 7(5)

I haven't found any use for it, but I thought that was useless and interesting.

Another is this one:

 x + y + ... = a(xa~ + ya~ + ...)
             = a~(xa + ya + ...)

This one is actually handy, it allows me (1) to "flip" the ~ inside the terms, as (2) well as "borrow" factors even if the terms can't be really factored. In this case, I also started wondering what does factoring really means. I'll get back to this later.

Another one is this:

 xy~ = 1 + (x-y)y~

This is also useful and helped me simplify some tricky equations.

Also this:

   ax + by + cz + ...
 = (a + b + c + ...)(x + y + z + ...) - q

where q = ay + az + bx + bz + cx + cy + ... Basically, the sum of all possible combination of terms not in [ax, by, cz] What does this mean and why is this useful? I don't know, I haven't really found any use for this, but it's saying I could factor out any sum of terms as a sum of their factors, minus some other shit.

Related to this is the following observation (but not yet proven): For (my favorite) example, I have

 469 = 7(67)
     = 4(10^2) + 6(10) + 9
     = 2(2*5)^2 + 2*3(2*5) + 3*3
       ^--a       ^--b       ^--c

I noticed that the factors can be expressed as the sum of some factors. Meaning a factor of 469 can be expressed as any sum of factors from a, b, c. For instance, 7 = 2 + 2 + 3. This seems to hold for the examples I tried, then I thought, maybe because it's trivially true and doesn't really mean anything. Then I found that 425 doesn't hold for this conjecture. But it is true for all the product of prime numbers I tested.

Is it useful? Initially, it thought it could be, but then, the search space is still quite large(r) when going for the brute force method of trying the all the different combination of factors.

So I'm trying the different angle, by looking for alternative or (even more) general ways of factoring, such as

 ax + by + cz = (a + b + c)(x + y + z) - q

What's even my end goal here? Nothing grand, I just want to make my own long division algorithm. I want, at a glance, say hey this number is factorable by these numbers. I don't think I'm smart enough to wreak havoc and render any cryptogaphic algoritms useless or less effective. It's a nice fantasy, but really, I'm just having some fun learning exploring some numeric properties. Again, nothing new here that any number theorist haven't thought of.

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