Talk:Table of prime factors

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Articles for deletion This article was nominated for deletion on May 25, 2007. The result of the discussion was Keep.

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[edit] Multiplication symbol, 1, sum function

Why the "1·"? That is not needed and a bit odd. - Patrick 11:49 May 6, 2003 (UTC)

Jacquerie27, I've checked the table and it is correct. I agree with Patrick and I propose that we omit the 1s since we know that 1 is a divisor/factor of every integer. I would also change the function FacSum(n) to more 'common' additive function a0(n) (where a0(n) = FacSum(n)-1). Perhaps it is really odd, but in fact it does not really matter if we include 1s or not. I am also glad that you've used · notation for a multiplication instead of ×, although many here use the last one. I prefer the first one, but I do not own wikipedia :-) On the other side at the Table of divisors we should leave 1s as they are, since a 'common' multiplicative arithmetic function τ(n) == d(n) == NumDiv(n) returns them for all n. But nevertheless nice work. I also like colors very much. And I do not know if all browsers show them well. I know there were some problems with the tables for some well-known planets and stars. (See Earth|Sun|Betelgeuse for instance). Best regards. --XJamRastafire 00:46 May 7, 2003 (UTC)
Thanks a lot, and that sounds good. I've got a program to do the tables, so I can create and adapt them easily. Let me know if you'd like to see anything else like this. Your English is v. good, btw, but the approach to no mistakes outside your mother tongue is an asymptotic curve (if I've got that right). Jacquerie27 08:42 May 7, 2003 (UTC)

[edit] Prime factors of One

The prime factors presented in the Table of prime factors are (is) 1. However, I would prefer to see {} or \emptyset as the set of prime factors and {1} as the set of divisors. So, I will include a note that 1 does not have any prime factors and empty the table cells. Gebruiker:Dedalus 08:12, 15 Feb 2005 (UTC)

[edit] WP:FLC

I was wondering if this should be a Wikipedia:Featured list, since it seems to meet most of the criteria: references would be needed, although this is just arithmetic so it is difficult to know what sort of references there would be, and an image would also be good, although, again, what images would be relevant?

One question: is there a reason for the table stopping at 1002, apart from it being the first convenient number greater than 1000 that is divisible by 3 (to fit into 3 columns)? Why not 1000 and 2 or 4 or 5 columns? -- ALoan (Talk) 13:39, 3 August 2005 (UTC)

[edit] too many redlinks

All the redlinks to numbers that fail to meet WP:NUM standards should be removed! Xtifr tälk 09:35, 23 October 2006 (UTC)

[edit] Why sum of prime factors?

Why list the sum of prime factors? It seems to be a rarely used function and I cannot think of practical uses for viewers of this table. Prime factors are meant to be multiplied, not added. Number of prime factors (Big Omega function), and number and sum of divisors (Divisor function) are more used functions. But I would probably prefer to only list the prime factors in the table. Then it really lives up to its name. If any reader is interested in the sum of prime factors then it's easily computed in the head, considering that the second-largest prime factor is at most 31. Another thing: I would prefer the number of rows being a multiple of 100, so there is a simple relation between numbers in the same row. Editing a large computer generated table is hard. Maybe somebody (e.g. me) could make an easily modified public domain program in a common language to generate the table. Or is Wikipedia a bad place for such things? PrimeHunter 15:45, 25 October 2006 (UTC)

[edit] New table format

I made major changes to the table today (November 21 2006). Here is the latest version before these changes.

  • The sum of prime factors was removed because it seemed of no use.
  • The formerly long table was broken into 10 sections, one for each 100 numbers.
  • Multiple factorizations on the same line are now part of separate subtables with interval headings, and spaces between. I think this makes it easier to locate numbers and read the tables. It also means the source becomes in numerical order. I chose 20 lines with 5 factorizations per line.
  • The color in the columns was removed, reducing article size by around 60%. Colors were unnecessary after the subtables were separated by spaces.
  • All exponents now use superscripts and look similar, i.e. special characters for exponents 2 and 3 are not used.
  • Only those number articles which exist or redirect November 19 2006 are linked.

The new tables were generated by a simple PARI/GP program. The tables in this article version were generated by this program version. It's easy to modify the program. I don't recommend manual editing of the tables, but I or my program have no special rights to the article. I watch this talk page and my own pages for suggestions. If you modify the program on your own and insert the output in the article, then I recommend making the modified program available with a link on this page.

If there should later be support for reformatting to one number per line with more details about that number, similar to table of divisors, then I could probably adapt the program. PrimeHunter 02:46, 21 November 2006 (UTC)

[edit] Alternative format, matter analogy

New New table format: I reorganized the first 100 numbers into six columns (for reasons explained in the article). Base 10 is not all that special, other than the prehistoric fact that we evolved with 10 fingers that are convenient to count on. But there are very strong reasons for organizing factored numbers into six columns, as is readily apparent in the new format. The Babylonians knew this. Pythagoreans knew this. Somehow today we have grown blind to such harmonies, with almost the entire globe having been swept over by metric thought.ChrisnHouston 22:17, 15 December 2006 (UTC)

I should note that I stopped reorganizing after 100. You may attribute this to my laziness, but I happen to think that both formats juxtaposed serve as an excellent illustration of number harmony in Base 6 (or Base 12) versus lack of harmony in Base 10.ChrisnHouston 22:26, 15 December 2006 (UTC)

Please use preview to avoid multiple consecutive edits which clutter up the article history.
I will revert your changes soon unless you give very good arguments or other editors support you. My arguments follow.
This has been added to the article:
"Unity is a stronger quality than both prime and composite. In an analogy to matter, prime numbers can be thought of as elements which are used to build composite numbers (which themselves can be thought of as the compounds). The unit is what primes are built with."
It's unclear what is meant by "stronger quality" and by "The unit is what primes are built with". This is mathematics and the article already links to unit (ring theory) which explains the relevant mathematical meaning of unit. Prime factorization is a simple concept. I see no need for a questionable analogy to complicated physics and chemistry. We know today that elements (I assume it refers to atoms) are built of smaller particles. Only some elements can be combined to form compounds but any primes can be multiplied to form composites. There are a finite number of elements but infinitely many primes.
Any prime p is obviously a prime factor in every p'th number. Your format with 6 columns just shows this for the prime factors of 6 (p=2 and p=3): Primes above 6 are on the form 6n +/- 1 (as prime number already says). 10 columns would have shown it for p=2 and p=5: Primes above 10 are on form 10n +/- (1, 3). 30 columns would have shown it for p=2, p=3 and p=5: Primes above 30 are on form 30n +/- (1, 7, 11, 13). And so on. 6 is not mathematically special in this regard and it would be misleading to give readers that impression. 6 is just the product of the 2 smallest primes, and a typical screen has room for 6 columns but not 10 or more (my screen might require a little scrolling to see 6 columns of 3-digit numbers with factorizations).
I definitely think all tables should have the same format. It's confusing otherwise. I think it's more important to have easily accessible information than to use tables of 1000 numbers in total to show one already mentioned example (n=6=2*3 for i=1 and i=5) of something general and trivial: If a*n+i is prime for a>0, then n and i are relatively prime (otherwise their common divisors would divide a*n+i). I find Ulam's spiral much more interesting than seeing primes above 3 in columns 1 and 5 when numbers are organized in rows of 6.
Base 10 is not mathematically special but it's special to humans. Not because we have 10 fingers, but because we have chosen to use base 10 for most things. We are very familiar with base 10 and therefore usually find it much easier to overview information organized around base 10. The primes should obviously remain expressed in base 10.
I think sections should continue to have 100 numbers. 6 does not divide 100, so your section 1-100 ends in column 4. To continue the column pattern , section 101-200 would have to start in column 5 and end in column 2. The earlier system with 5 columns of 20 numbers have each section starting in the first column and ending in the last column. I also think it's easier to find a given number when consecutive numbers are immediately adjacent in columns with headings, instead of being split by factorizations and spaces in rows (and "horizontal" tables would look bad). PrimeHunter 20:43, 16 December 2006 (UTC)
My apologies for cluttering up the article history.
Yes, the analogy isn't perfect. (If it was, then it wouldn't be an analogy, now would it?) The utility of it is that most people don't understand the power of primes. The concept is typically taught in a negative light. "Primes are those numbers that can't be divided." Most students therefore fail to grasp the full scope of what being prime means. But people DO understand the power of elements as building blocks of matter. This is why I expect the analogy to be helpful to many Wikipedians (even some who may be mathematicians).
The analogy also helps to explain why 1 can be excluded from the group of primes. While elements make up matter, people are clear that elements themselves are not the most basic bits of substance. The analogy shows that 1 is more fundamental than a prime. That is what is meant by "stronger quality".
You state the reason for Base 10 being used is not because of our 10 fingers, but because we use it for most things. However, if you examine why that is, it brings you back to the 10 finger counting legacy. You are also discounting Base 6 as being "just the product of the 2 smallest primes." I see much more power in this fact than you are giving credit. Consider the harmonic ratios in music, for instance. The halves and thirds give rise to Base 12 semitones. The product of the two smallest primes drives all of music. Such resonances are also found in orbital mechanics and quantum physics.
Organizing this table into six columns is very special. It has physical meaning beyond the math. Or more properly, our physical world has mathematical structure that governs matter.
And even if you yourself don't see this rationale to be compelling, you may find that others find these changes to be very helpful.
ChrisnHouston 22:10, 16 December 2006 (UTC)
You give no sources to your theory that the analogy makes primes easier to understand and I would be surprised if it's the case. If people understand the mathematics then they don't need a poor analogy, and if they don't understand the mathematics then the many faults in the analogy doesn't appear likely to be helpful. Without a reliable source I also think the analogy violates WP:OR. In any case, articles like prime number, prime factor and prime factorization (all linked from the introduction here) seem better places to explain the terms. I think this article should be about listing prime factorizations in an easily readable format. I understand the mathematical significane of 1 perfectly well, but I still don't see what you mean by "stronger quality" or whether you compare 1 to a proton in your physics analogy (protons are made of quarks). You should cite reliable sources (WP:V and WP:CITE) if you introduce vaguely defined or undefined terms in mathematics articles. This is an exact discipline. I think we do our users a disservice by not showing that, but instead make questionable comparisons to less exact disciplines. Music is a human invention and humans choose (with different opinions and variations over time) which music they like, and what they call music. Physics is an evolving discipline where old theories are sometimes proved wrong. For example, theories about what the smallest "building blocks" are have changed more than once and the question seems far from being settled (maybe it never will).
I stated the reason for base 10 being used in this Wikipedia article: Because it's used almost everywhere else and our users are very familiar with it, unlike base 6. It doesn't seem relevant for this discussion why base 10 is used almost everywhere else (10 fingers is just a guess about that). PrimeHunter 15:46, 17 December 2006 (UTC)
I don't see how any analogy could qualify as research. Analogies are simply aids to learning things that have already been researched. To help you get to that "a-ha" moment. Analogies are inherently lacking fidelity. But that certainly does not preclude their utility.
And I am curious about your attitude toward mathematically rigorous articles in Wikipedia. Wikipedia is by-the-people, for-the-people. A math wiki would be by-the-mathematicians, for-the-mathematicians. Math has a language all in itself. Not many people speak in integrals, let alone subrings, Hermitian adjoints, etc. Any concept, no matter how complex, can be translated into simple terms that are accessible to the masses. There are many who sacrifice accuracy when doing so, but I believe that it is possible to do without dilution to the point of introducing error.
This is what I uphold as the ultimate purpose of Wikipedia: a reference that is accessible to the meat of the bell curve. Whether talking about cross-stitching or quantum mechanics, we can have articles that are meaningful to most everyone. Jargon and symbolic notation is great, but if those aren't explained, then the article is limited to a sliver of the wikipedian population.
So what percentage of wikipedians who aren't sure of the concept of a unit then click through to the ring theory explanation, read through it and see that "vu = uv = 1 sub r", etc and go A-HA! Now I get it! My guess is that the total is zero people. Because if they knew the ring theory thing, they wouldn't have had to click through. Now I suggest to you that a far greater percentage of wikipedians will be able to grasp (or at least glimpse) the concept of 'unit' from the matter analogy. Yes, the analogy doesn't hold up to "atom splitting". It is not meant to. It is meant to help people get to that 'a-ha' moment where they can become aware of the analogy's limitations. Nowhere does it say that "1 is like a proton" (I agree that it would be a can of worms). The analogy is dropped after the element/compound comparison, leaving open the door to the notion that the unit is a mathematical entity that is more fundamental than primes.
Well this is my best persuasive effort for you, so I don't expect that I'll have much more to add here on this. You (and everyone else) can make your decision with what I've offered.
As a parting note, I will offer my contrasting response to your view that math is an exact discipline. Everyone who has studied math to its limits has become aware of its imperfections. To quote Bertrand Russell:
"Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true."
ChrisnHouston 19:17, 19 December 2006 (UTC)
I'm not saying your analogy is "research" in the normal scientific sense, but it claims similarity between two apparently unrelated things and I think that violates WP:OR (and I think the two things are so different that the analogy is poor and misleading, not helpful at all). I agree mathematical writing is not absolutely exact, but it can (and should) come much closer than this. And I don't see how your analogy can help to understand the mathematical meaning of "unit" when you say "unit" isn't even part of the analogy (it's not compared to a proton or anything else). People who want to know more about unit (ring theory) can follow the link. That's what wikilinks are for, and I think more than 0 will follow it and learn something. People can skip the link if they are satisfied by reading that 1 is called a unit and is neither prime nor composite. The analogy makes no attempt, neither exact nor inexact, to describe the mathematical meaning of unit. I have removed it. I have also reverted your changes to the first table section so it again becomes consistent with the other 9, using the same format. No other editor has commented and I don't think you have given good reasons to change an easily readable format. I will however add a more general and informative comment which explains the phenomena you wanted to show for the case n = 6 = 2·3:
"If numbers are arranged in increasing columns of n numbers, then the prime factors of n will occur in the same row each time. This means those rows contain no primes after the prime factors of n. The table columns have 20 = 22·5 numbers, so the prime factors 2 and 5 occur in fixed rows." PrimeHunter 02:27, 20 December 2006 (UTC)

[edit] two suggestions

I have two suggestions: (a) links to OEIS in the "Properties" section, (b) use a template for the formatting of numbers (or do you already do so, with subs: ?) Also, did you have it a try to put the tables horizontally? I think it would well fit also on low-res screens and make the page a bit more compact. Another remark: according to your tables, 219 seems to be a very uninteresting number: it's the only red link up to 760... More seriously, you linked all numbers (producing many red links beyond 760) "as per request" of somebody, but this seems not appreciated by everybody; I'd also frown upon this: finally, your page is about prime decompositions, and "nothing else"; maybe one single link to a page providing similar tables with links to all numbers would do the same job... At the same token, it could be discussed whether WP should duplicate web sites like the "prime curios" etc...

PS: on a second thought, maybe it's not that bad to leave all numbers (red)linked...— MFH:Talk 00:38, 19 April 2008 (UTC)
I have added OEIS links to the sequences in the properties section. No template is used to format numbers. 1000 template calls seems like an unneeded server load. I don't think horizontal tables would look good. How wide to make the page is the same issue for vertical and horizontal tables. To me, the current tables seem compact for their width but still easily readable. There could be 6 or more columns instead of 5 but then many readers would probably need horizontal scrolling. I don't think there is any way to tell how wide the readers window is and format differently depending on it. I have recreated the redirect from 219 (number) to 210 (number) where 219 is mentioned. The redirect was deleted a week ago after some confusion. I like having links on all numbers so it can be seen which of them exist. If links are only made for current articles then updating is needed when new number articles are created. It might be possible to use up to 1000 #ifexist to only make a link if the target exists when the page is rendered, but that would be a big server load and maybe break a limit. The page was kept at Wikipedia:Articles for deletion/Table of prime factors. I'm not sure what you mean by "maybe one single link to a page providing similar tables with links to all numbers would do the same job". Where would you place that single link and how would the linked page be different from the current Table of prime factors? PrimeHunter (talk) 02:41, 19 April 2008 (UTC)