New flame war idea ??

[skan]

Hello

Some time ago I was trying some new ideas to quickly factorize numbers. Such as rewriting known factorization relations to work directly with the binary coefficients of a number instead of the whole number, and applying statistical heuristics.
I was trying to implement a functional method many years ago but it got too complicated for me and finally I found some other authors trying some of those ideas and the results were not so good.

Today I want to post something different I found, maybe not useful but at least nice graphics. I haven't developed it further.

If you plot Mod[N, k] for all k you get a nice graph like shown at the first picture. (the remainder of the division).
In this example N=272123, but you get similar graphs for other numbers.
It seems that there is a pattern.

[skan]

I you plot the difference of two consecutive remainders you get something much clearer:



I don't know why I can't load several pictures at once.

This is Mathematica notation, and graphics.

[skan]

And if you represent ListPlot[Differences[Mod[N, k], 2] is much clearer, there is a simple pattern.

You need to zoom in to notice that the graph is right, the value alternates between the curves.


We could plot an approximate curve from a few random divisors and then save time by only trying numbers near this curves? or watching the areas where the results diverges...

Is this already implicit in any other method?
Is this just stupid?
I have more strange graphs

The two vertical lines represent the values of the known divisors of 272123. You can notice a gap there, this gap could also be useful.
Regards

[R.D. Silverman]

Originally Posted by skan

Hello

Some time ago I was trying some new ideas to quickly factorize numbers. Such as rewriting known factorization relations to work directly with the binary coefficients of a number instead of the whole number, and applying statistical heuristics.

This last sentence is total nonsense. "factorization relations" is meaningless.
"binary coefficients of a number": Numbers do not have coefficients.
"statistical heuristics": meaningless gibberish. If you want to
discuss mathematics you have to learn to use standard terminology and standard definitions.

I was trying to implement a functional method

You are bandying words with no understanding of what they mean.
What is a "functional method"?

many years ago but it got too complicated for me

Why am I not surprised???

The first thing you need to do is learn some math. You don't know
any. Take a course in number theory. Take one in abstract algebra.
Learn the difference (topologically speaking) between R and Q.
You do not seem to understand what a function is. You do not seem
to understand what a curve is.

If you plot Mod[N, k] for all k you get a nice graph like shown at the first picture. (the remainder of the division).
In this example N=272123, but you get similar graphs for other numbers.
It seems that there is a pattern.

There are no "curves". Curves are continuous. You are working in a discrete
topology. Tell us why you think this graph is relevant to finding k
such that N mod k = 0. Do you imagine that there is some relationship
between N mod k and N mod (k+1)? Tell us what you think it is.

What "patterns" do you think are there? Tell us. Describe them
algebraically. Stop babbling.

[skan]

Originally Posted by R.D. Silverman

This last sentence is total nonsense. "factorization relations" is meaningless.
"binary coefficients of a number": Numbers do not have coefficients.
"statistical heuristics": meaningless gibberish. If you want to
discuss mathematics you have to learn to use standard terminology and standard definitions.



You are bandying words with no understanding of what they mean.
What is a "functional method"?



Why am I not surprised???

The first thing you need to do is learn some math. You don't know
any. Take a course in number theory. Take one in abstract algebra.
Learn the difference (topologically speaking) between R and Q.
You do not seem to understand what a function is. You do not seem
to understand what a curve is.



There are no "curves". Curves are continuous. You are working in a discrete
topology. Tell us why you think this graph is relevant to finding k
such that N mod k = 0. Do you imagine that there is some relationship
between N mod k and N mod (k+1)? Tell us what you think it is.

What "patterns" do you think are there? Tell us. Describe them
algebraically. Stop babbling.

First of all English is not my native language, it's difficult to use the proper words.

When I say "binary coefficients of a number" I mean the coefficients of the number when you express it in base 2.

I know they aren't "curves" in the strict sense of the word, that's why I said that you could "approximate" or "fit" a curve to that points.

Anyway, I was sharing my ideas, maybe they are not good but that doesn't imply you are a rude with me.

I don't what relations would arise from here that's why I share the graphics, maybe somebody else find them interesting.

[Batalov]

Let's move this discussion to Misc.Math, and by definition, it will be immunized* from further "real math" criticism.

By skan's own admission, this is not factoring, just some "graphics which maybe somebody else find interesting".

[cheesehead]

Originally Posted by R.D. Silverman

This last sentence is total nonsense. "factorization relations" is meaningless.
"binary coefficients of a number": Numbers do not have coefficients.
"statistical heuristics": meaningless gibberish. If you want to
discuss mathematics you have to learn to use standard terminology and standard definitions.


You are bandying words with no understanding of what they mean.

Mr. Silverman, you just can't understand why anyone without your mathematical talents would ever stumble, while asking for help, on the way to learning something, can you?

This isn't your classroom. Stop treating this forum as though it were.

Your response was inappropriate, rude and uncouth. You have an enormous "blind spot" about other people and about proper verbal behavior.

The first thing you need to do is learn some math.

No, the first thing is that _you_ need to learn how other people learn something for which they don't have the enormous near-perfect talent that you have.

You do not seem to understand what a curve is.

No, you're the one who can't tell the difference between two different kinds of "curves". I had no trouble at all understanding what the OP referred to ... why do you have difficulty?

There are no "curves". Curves are continuous. You are working in a discrete topology.

Wrong. The OP is working in a learning, not-yet-having-mastered-the-subject context.

But you are blind to that which is obvious to others.

[cheesehead]

Originally Posted by skan

If you plot Mod[N, k] for all k you get a nice graph like shown at the first picture. (the remainder of the division).

In this example N=272123, but you get similar graphs for other numbers.
It seems that there is a pattern.
Attachment 10333

Do you mean that this graph plots k on the x-axis, and Mod[272123,k] on the y-axis? I don't see how that can be.

Since 272123 = 503 * 541, Mod [272123,k] should equal k for k in the range [0,502]. The graph should be a straight line with a slope of 1 for k between 0 and 502.

My conclusion is that your first graph is not simply Mod[N, k], but something else.

I think I could guess what it actually is, given enough time, but not yet.

What is the y-coordinate for k=100, 200, 300, 400, 500, 600, 700, 800, 900 and 1000?

[cheesehead]

Originally Posted by skan

If you plot Mod[N, k] for all k you get a nice graph like shown at the first picture. (the remainder of the division).
In this example N=272123, but you get similar graphs for other numbers.
It seems that there is a pattern.
Attachment 10333

One pattern is that Mod[272123,k] = 0 whenever k= 503 or 541 (or 503*541). But that's no shortcut to factoring, because you already had to do the division to get the Mod[] value. That's no better than trial factoring (sometimes called "sequential division" outside GIMPS).

I think all other visual patterns can be traced back to knowing the factors of N ... but the Mod[] calculations to produce the graph already have revealed those.

In other words: in order to produce these graphs, one has to do enough computation to already have revealed the factors of N before the graph is completed. The patterns shown in the graphs reveal nothing beyond what will already have been known by doing those computations.

Originally Posted by skan

And if you represent ListPlot[Differences[Mod[N, k], 2] is much clearer, there is a simple pattern.

< snip >

Is this already implicit in any other method?

Yes. Trial factoring.

Is this just stupid?

No, it's just a natural consequence of your not yet having already learned enough math to recognize what the patterns mean.

I recall looking, when I was young, at some graphs similar to yours and not yet knowing why the patterns were predictable. (In other words: been there, done that. And I actually do have a mathematics t-shirt, but it's not about graphs.)

Asking about new patterns you notice is just part of learning. But once in a while, after you've learned a lot, you may notice a pattern no one else has actually ever noticed or explained, and it will be your own new discovery when that happens.

The two vertical lines represent the values of the known divisors of 272123. You can notice a gap there, this gap could also be useful.

But the gap is revealed only after one has done enough computation to have already found the factors, thus saving no time in factoring N.

[TheMawn]

+1 Cheesehead. I won't be saying very much on the matter as I recently withdrew from a different fight with someone who similarly rhymes with glasshole and have no interest in starting another one.

To the OP, in particular, don't let this one bother you too much. People here are nice. Mister (Doctor, yes?) Silverman is the exception, not the rule.

[LaurV]

We are now in the misc math subforum. I really appreciate post #8. However I believe that both #7 and #10 posts were TOTALLY uncalled for. Please stop! (it will end with bans, and I don't want to lose a chess player from my team, again! )

On the other hand, I know that Mr. Silverman was ill, I am happy that he is well again, and biting...