Skip to main content

Math: Factoring Numbers

I was thinking this morning about factoring numbers. I wonder if it might sometimes be helpful to use real numbers to gain an interesting perspective in order to solve certain problems involving integer numbers (i.e. number theory problems). For instance, I was thinking about factoring large numbers.

For every natural number, C, (that isn't equal to 0 or 1), there are an infinite number of pairs of positive, real numbers A and B for which A * B = C. For instance, 6 = 1.0 * 6.0 = 2.4 * 2.5 = 2.0 * 3.0 = ... I wonder if playing around with pairs of real numbers like (2.4, 2.5) can lead you to pairs of integer numbers like (2, 3).

Imagine all the pairs (A, B) for which A * B = C. Let's create a way to graph all such pairs in a funny sort of way. Let's pick a bunch of A's going from 0 to C. For each different A, we can calculate B via C / A. Let's consider the parts of A and B that are to the right of the decimal point to see if one pair, (A, B) can lead us to another pair (A', B') which are integers (i.e. have only zeros to the right of the decimal point). In fact, let's see if we can come up with a numerical analysis approach where we use estimations to hunt down (A', B').

To do this, let's create a funny sort of three dimensional graph. Here's the pseudo code (assuming we're trying to factor some number, c):
for step in range (1, LARGE_NUM_OF_STEPS + 1):
a = (step / LARGE_NUM_OF_STEPS) * c
b = c / a
x = a - int(a) # x and y will always be in the range [0, 1).
y = b - int(b)
z = floor(a)
draw_point(x, y, z)
If you use a very large number for LARGE_NUM_OF_STEPS, you'll create a funny looking 3D graph. Any place where x = 0 and y = 0, you'll have a pair of integers (a, b) that multiply to equal c.

Naturally, this is an extremely expensive way to factor numbers. However, I'll bet you'd learn a lot about factoring numbers by looking at this graph. In fact, I'm guessing that looking at this graph will lead you to a numerical-analysis-style approximation algorithm for honing in on valid integer pairs (a, b) where a * b = c.

Humans are fairly good at visualizing things in 3D, but it'd be really cool to extend this graph to four dimensions (perhaps using time as the fourth dimension). The fourth dimension would be used for various values of C, from 0 to infinity. I really think that looking at such a graph would help with fast number factoring.

Updated: Fixed a couple of errors pointed out by BMeph.

Comments

Eric Bloch said…
re: initial comment about real numbers.

A lot of the stuff in Knuth's book on Concrete math is based on the principals of using ideas from continous, infinitesimal calculus and applying them to discrete math problems (hence title con-crete). It's a great book.
jjinux said…
"Using ideas from continous, infinitesimal calculus and applying them to discrete math problems" is exactly what I had in mind, and the fact that Knuth beat me to the punch is probably a good sign. Thanks, Eric! :)
BMeph said…
As a member of the Pedantic Police, I feel "Compelled" to hand out some tickets for clarity's sake:

"a = (i / LARGE_NUM_OF_STEPS) * c" should be "a = (step / LARGE_NUM_OF_STEPS) * c", or you can change the for line above it. Being a pedant, I of course prefer the wordier version.

"z = floor(x)" should be "z = floor(a)", since otherwise it'll always be 0.
jjinux said…
You are quite correct. Thank you :)

Popular posts from this blog

Ubuntu 20.04 on a 2015 15" MacBook Pro

I decided to give Ubuntu 20.04 a try on my 2015 15" MacBook Pro. I didn't actually install it; I just live booted from a USB thumb drive which was enough to try out everything I wanted. In summary, it's not perfect, and issues with my camera would prevent me from switching, but given the right hardware, I think it's a really viable option. The first thing I wanted to try was what would happen if I plugged in a non-HiDPI screen given that my laptop has a HiDPI screen. Without sub-pixel scaling, whatever scale rate I picked for one screen would apply to the other. However, once I turned on sub-pixel scaling, I was able to pick different scale rates for the internal and external displays. That looked ok. I tried plugging in and unplugging multiple times, and it didn't crash. I doubt it'd work with my Thunderbolt display at work, but it worked fine for my HDMI displays at home. I even plugged it into my TV, and it stuck to the 100% scaling I picked for the othe

ERNOS: Erlang Networked Operating System

I've been reading Dreaming in Code lately, and I really like it. If you're not a dreamer, you may safely skip the rest of this post ;) In Chapter 10, "Engineers and Artists", Alan Kay, John Backus, and Jaron Lanier really got me thinking. I've also been thinking a lot about Minix 3 , Erlang , and the original Lisp machine . The ideas are beginning to synthesize into something cohesive--more than just the sum of their parts. Now, I'm sure that many of these ideas have already been envisioned within Tunes.org , LLVM , Microsoft's Singularity project, or in some other place that I haven't managed to discover or fully read, but I'm going to blog them anyway. Rather than wax philosophical, let me just dump out some ideas: Start with Minix 3. It's a new microkernel, and it's meant for real use, unlike the original Minix. "This new OS is extremely small, with the part that runs in kernel mode under 4000 lines of executable code.&quo

Haskell or Erlang?

I've coded in both Erlang and Haskell. Erlang is practical, efficient, and useful. It's got a wonderful niche in the distributed world, and it has some real success stories such as CouchDB and jabber.org. Haskell is elegant and beautiful. It's been successful in various programming language competitions. I have some experience in both, but I'm thinking it's time to really commit to learning one of them on a professional level. They both have good books out now, and it's probably time I read one of those books cover to cover. My question is which? Back in 2000, Perl had established a real niche for systems administration, CGI, and text processing. The syntax wasn't exactly beautiful (unless you're into that sort of thing), but it was popular and mature. Python hadn't really become popular, nor did it really have a strong niche (at least as far as I could see). I went with Python because of its elegance, but since then, I've coded both p