Talk:Factorization/Integer/Deduction

The introduction

 * Factorization of uneven integers. [Factorization in Periods] - The problem of factorizing uneven integers known to be composed of two prime factors has essentially been solved algebraically, subject to the two relevant equations not being identities and having satisfactory solutions. The period or periods of the prime factors are not needed. The solution is found for the lesser prime factor, q[1] by solving a very limited number of equations, [N - 1] at the very most, for the unknown, q[1].

I moved this page to its present title, and avoided using "uneven" in the title, because if this is a general method for factorizing uneven integers, it is also a general method for factorizing all integers, by simply finding the largest uneven factor through repeatedly dividing by two until the result is odd.

The mention of what is not needed in the introduction could be confusing. With some topics, if I don't thoroughly understand the first material, I will nevertheless read the rest, because that understanding might arise from how the topic is later treated. However, with mathematics, it is crucial to build understanding on a solid and clear foundation, and I've seen errors creep in when this was not done, due to unwarranted, but perhaps natural, assumptions creeping in.

So, first of all, N is not defined. Now, it is certainly the number to be factored, but that is not stated. Eliminating the specification of the number to be factored as even, as I expect would be done in a more mature version of this essay, I would replace the first substantive sentence of the essay with


 * A solution to the problem of factorizing an integer, N, known to be composed of two prime factors, is here solved algebraically.

Then the reference to "(N - 1)" becomes crystal clear.

However, I'm not then encouraged to find that "at the very most," the method requires "solving ... (N - 1) equations." (To my understanding an experience, parentheses would be used here, or just "N - 1." But I might be wrong about that, what I know is that the brackets are grating for me.)

Given that ordinary methods of factoring require testing at most the square root of N, rounded down, i.e., if the number is less than 10,000, the largest prime factor must be less than 100, having to solve up to 998 equations is less than appealing.

Is there something I'm not understanding here (as to the first paragraph. I definitely don't understand the rest of the essay, because looking at it, it's a wall of text, and when the first paragraph isn't clear, I give up on the rest. So will most other readers, I suspect. --Abd (discuss • contribs) 21:24, 1 February 2014 (UTC)

NOTATION USED
Then the author introduces notation. The notation used is nonstandard. That creates a barrier to understanding, for anyone who has experience with standard notation. If we want to use ASCII text, instead of the markup that allows full standard notation, using a period for multiplication is disconcerting. An asterisk is standard. (An elevated period is more standard, but is not so readily available from an ASCII keyboard.)

The asterisk is routinely used in spreadsheet programs.

Further, indices are generally placed in parenthesis, not brackets. Thus if P is a prime number, in a set, P(N) would indicate the Nth prime in the set. If there is a function F, operating on a value M, then the result of the operation is represented as F(M).

Exponentiation is typically indicated by using a carat, ^, as indicated. However, again, it is standard to enclose the exponent in parentheses, if it is necessary to set off a multi-term exponent. I.e, 2^2 = 4, 2^(1 + 1) equals 4 as well. There is no need for a special notation using curly braces. Unlike the situation with brackets and parentheses in ordinary text, parentheses are simply nested. Again, this is standard spreadsheet practice. If it is desired to set off nested parentheses, then brackets, curly brackets, and angle brackets may be used, the latter at some risk of confusion with "less than" and "greater than."

Essentially, if the author is going to use ASCII notation, for ease of entry and editing, then I suggest standard notation be used.

In particular, I suggest a rigorous development of the exposition. That has not been done in a clear and simple manner. As an example, "F.D.F." is used. It stands out because it is capitalized. Reviewing the paragraph where it is first used, I see a term, "finite discrete functions." It wasn't so easy to find, because it was not capitalized and it was split, in my display, across a line break. Three problems:  ... Finite Discrete Functions (FDFs) ...
 * If an acronym or abbreviation is used, the first time it is used, the full term should be spelled out, and followed by the abbreviation in parentheses. If the abbreviation is the initial letters, to be used in capitals, it is traditional to capitalize the original spelled out version, to make the connection clear and easier to see. So the mention would be:
 * The periods then are not used in the acronym, nor is an apostrophe and s needed for the plural.
 * "Finite discrete function" is not defined, nor wikilinked. I find the term as being used in a few places in Wikiversity. Checking it out, all I saw had been created by the same author, and were not defined where used. Bad Idea. Yes, the meaning might be obvious to some, but ... "finite discrete function" found no direct hits on a Wikipedia search. "Discrete function" is redirected to Sequence and that may contain an explanation of what the author means. Or not.

Personally, I apologize for not watching and handling the contributions of this author; I'd started to look at this in 2011, but was then blocked from editing this site for almost two years. The result could be quite a bit of cleanup necessary, though with author cooperation, as I expect, it won't be so bad.

My goal is to allow the author great freedom of expression, while, at the same time, channeling effort into creating educational resources of use to others as well as the author. My comments here are on a Talk page. As I have placed this resource, the author has, my view, full editorial control of the page, for better or worse. I am not yet addressing Disclosures. --Abd (discuss • contribs) 21:24, 1 February 2014 (UTC)

edits to resource
Generally, authors of attributed resources, neutrally placed as essays or studies, are not signed. In addition, only clarifying copy edits, such as formatting or spelling corrections, may be made to such resources, absent author approval or some broader consensus. I have edited the resource, now, beyond moving it into place as a subpage. The author may revert me, though I don't recommend it. In the first such edit, I attributed the essay explicitly at the top, and added a note that this was Original research. I also removed two signatures at the bottom. We do not ordinarily sign mainspace resources, though we may sometimes sign individual comments found in mainspace resources. When there are multiple edits to some page, we do not sign the page multiple except with an explanation. Multiple signatures at the bottom do not clarify, they confuse, unless there are explanatory notes. While essays are handled differently than other mainspace resources, all edits are already effectively signed in page history, and, there, the details of each edit are shown. --Abd (discuss • contribs) 14:05, 9 February 2014 (UTC)

Undefined terms
The nonstandard notation, given that there is widely-used notation that is different, makes it far more difficult to follow what is being written. However, then, terms are used that are not defined ab initio. I still don't know what Q[N] refers to. Is it a function of N, N being an uneven integer? Or is it the Nth uneven integer? In which case Q[N] = 2.N-1.

Ah, it grates, using that notation for what would otherwise be Q(N) = 2*N-1, or, often, for improved readability, addition and subtraction operators, and sometimes multiplication or division operators, are separated by spaces.

Q(N) = 2 * N - 1 ... which will be interpreted by any spreadsheet correctly. however, more explicitly, this might be expressed as:

Q(N) = (2 * N) - 1, to distinguish it from Q(N) = 2 * (N - 1).

Exponentiation is not. The best known notation is that used in spreadsheet programs.

SHAW, if your purpose in writing is only for your own study and reflection, it might be better in your user space. If it is for others, you may need to develop skill in communication, in writing so that it is readily understood. Mathematics writing is not like writing essays in English. Often understanding is communicated in English through repetition and a series of examples; in mathematics, each piece, read in sequence, should generate explicit and clear understanding of what has been covered so far. It is possible that I could understand your work by reading all of it, over and over, but I'd be doing the ultimate readers a disservice by jumping the gun like that. --Abd (discuss • contribs) 14:38, 9 February 2014 (UTC)

example
As I've written, I find the notation used unintelligible. There is an example in the resource of factoring, Factorization/Integer/Deduction (notice that creating section headers on the page allows linking to a specific section. It also allows editing one section at a time, which is far more convenient.)

The example: Factor 742529.

With the blizzard of unfamiliar notation, I find it impossible to follow the example. SHAW, on the subpage, /742529/, please show this with simple calculations or deductions, starting only from the number itself, and showing how each step is done. You have in your explanation, first the actual factors, then you have some numbers of unexplained origin: 873, 140, 35. --Abd (discuss • contribs) 18:40, 17 March 2014 (UTC)

general comments
SHAW, You talk about factorization of uneven numbers, and treat 2 as a special case. In fact, 2 is not a special case, it is like all the other possible factors; the only difference is that in expressing the number, we have divided it repeatedly by ten, subtracting powers of 10, until the remainder is less than 10, and from this we know if the number is factorable by 2, 5, or 10. In a different number base, 2 would be just as difficult as three. In base 3, 3 would be easy!

You note that


 * Taking an approximation, 861.7 to the square root of 742529, classical factorization requires precisely one hundred and forty-eight long and shorter prime [excepting division by prime two [2], that is in a disjoint class of its own] decimal divisions at the most to find the prime factors.

The classical factorization divides by every prime number up to the square root. There are 149 primes below 861.7, including 2. By excluding 2, you make your work apply only to numbers already pre-tested for division by 2. That's all. --Abd (discuss • contribs) 18:40, 17 March 2014 (UTC)