Talk:Centered square number

I don't inderstand what is meant by represents a square with a dot in the center and all other dots surrounding the center dot equidistantly.

I picture like what is shown in figurate numbers would be useful.

--- User:Karl Palmen 16 June 2004

Those are more like ASCII art than pictures. I'm fooling around with Adobe Illustrator to try to create some nice, professional-looking diagrams. In the meantime, here is some ASCII art to illustrate the concept. And if you come up with a better word description please edit the article accordingly. PrimeFan 21:03, 17 Jun 2004 (UTC)

*   *     *       *
   ***   ***     ***
    *   *****   *****
         ***   *******
          *     *****
                 ***
                  *

I've added the diagram to the main page. One can replace it with a better diagaram later if you want.

I don't regard represents a square with a dot in the center and all other dots surrounding the center dot equidistantly as a correct desription. Surely this would describe

*    *     * *      *      * *
    ***     *             *   *
     *     * *    * * *     *  
                          *   *
                    *      * *

where the applicable distances are the square roots of 0,1,2,4,5, etc.

--- User:Karl Palmen 18 June 2004

You're right about the word description. As for the diagrams, I can't figure out how to make them in Adobe Illustrator. I do have to admit that in the case of centered square numbers, ASCII art is slightly more satisfactory than it is in the case of other figurate numbers. PrimeFan 22:07, 20 Jun 2004 (UTC)

I've corrected the description and made use of the concept of taxicab geometry in so doing

distance = abs(x+y)

rather than

distance = sqrt(x^2+y^2)

Now I've thought of a diagram that shows the (x-1)^2 + x^2 formula. I'll put these in and correct the formulae.

--- User:Karl Palmen 21 June 2004

Context

The introduction to this article provides insufficient context for those unfamiliar with the subject matter.

Well, there's some references added to the page now, but there's still a lack of context to the page. There is no explanation of what this term is used for, who uses it, or what it means. IOW, even assuming they left this article knowing what it is, I think the casual reader would still be left wondering what to do with it. I certainly don't. Nor since it was mentioned, do I consider the articles for Figurate number or Centered polygonal number to be especially enlightening. I'm not seeing any clear explanation of what this is used for, and I'm not sure that either of those two give a clear enough explanation. (I might even say that figurate number's sections on square roots are a bit too textbook like for that matter). Glad somebody added some references though. FrozenPurpleCube 18:24, 29 July 2007 (UTC)

First, i am here a visitor. I am not an active Writer in the english Wikipedia, because i am german, and i don't think, that my english is so good, to write Articles in english.

Second, what are Triangular numbers for, or Carmichael numbers or Lucas sequences or Figurate numbers or Prime numbers?

I think Prime numbers, Carmichael numbers and Lucas sequences are intersting (at least for me). The most mathematical "Muggel" would ask, what are Carmichael numbers are good for. They wouldn`t see the Beauty in this. And the Centered square numbers are part of this Beauty.

There are links to other mathematical topics:

A. Square number: A centered square number is the sum of two consecutive square numbers

A square number is the sum of two consecutive triangular numbers

B. Triangular number: 4*n.triangular_number+1 is a centered square number, 2*n.triangular_number + (n-1).triangular_number + (n+1).triangular_number is a centered square number.

C. Pythagorean tripel. Pythagorean tripel with consecutive square numbers a and b are represent by the centered square numbers. So 3^2 + 4^2 = 5^2 and 20^2 + 21^2 = 29^2.

D. Prime numbers (and Pseudoprimes): Many of the centered Square numbers are prime numbers. And the other Centered square numbers are, except of the one, pseudoprime numbers (to a any base a). Some of them are euler pseudoprimes: 25 (bases: 7, 18), 145 (bases:12, 17, 28, 41, 46, 57, 59, 86, 88, 99, 104, 117, 128, 133), 265 (bases: 23, 52, 54, 76, 83, 107, 129, 136, 158, 182, 189, 211, 213, 242), 481 (bases: 6, 8, 10, 11, 14, 23, 27, 29, 31, 36, 38, 43, 45, 47, 48, 51, 60, 63, 64, 66, 68, 73, 75, 80, 82, 84, 85, 88, 97, 100, 101, 103, 105, 110, 112, 119, 121, 122, 125, 134, 137, 138, 140, 142, 147, 149, 154, 158, 159, 162, 171, 174, 175, 177, 179, 184, 186, 191, 193, 196, 199, 211, 212, 214, 216, 223, 228, 230, 232, 233, 236, 245, 248, 249, 251, 253, 258, 265, 267, 269, 270, 282, 285, 288, 290, 295, 297, 302, 304, 306, 307, 310, 319, 322, 323, 327, 332, 334, 339, 341, 343, 344, 347, 356, 359, 360, 362, 369, 371, 376, 378, 380, 381, 384, 393, 396, 397, 399, 401, 406, 408, 413, 415, 417, 418, 421, 430, 433, 434, 436, 438, 443, 445, 450, 452, 454, 458, 467, 470, 471, 473, 475), ...

I think, you (we) should show the Beauty od the mathematic. I would be better to search for a context, than to say "i don't see there a context, so there is no context". --Arbol01 23:01, 29 July 2007 (UTC)

I did look for a context. I don't see it. Saying things are "beauty" in math, well, that *may* be so, and if you can find a source that says that's why mathematicians find it interesting, that might mean something. Nor does just finding it interesting yourself mean anything. I don't find it interesting. Sorry, but a page should explain what it means in better terms. If you're a foreign-language speaker and don't feel up to explaining it yourself, there's a simple solution. Leave it to somebody else. Myself, I don't feel up to explaining it either. FrozenPurpleCube 16:29, 30 July 2007 (UTC)

As for the examples of other pages, if you are saying that those pages also don't have context explaining what they are, that doesn't mean this page is excused from doing things properly. It means those pages need to be improved. FrozenPurpleCube 16:29, 30 July 2007 (UTC)

I think I understand Arbol01's point. Centered square numbers are interesting in a similar way to how prime numbers were interested before we knew they had any practical application (e.g., cryptography).

As far as I know, centered square numbers have no practical application whatsoever (contrast them to centered hexagonal numbers). So, the answer to "who uses it" is math aficionados who are unconcerned with practical applications.

For me, the graphics are enough to explain "what it means," and the equations are just icing on the cake. PrimeFan 22:25, 30 July 2007 (UTC)

I've rewritten the lead paragraph and made some general tweaks to the article in an attempt to make it more understandable to people unfamiliar with the field. I would very much welcome any feedback on the changes, as well as any improvements you can think of. In the end, I'm not sure how much more context one can really provide: basically, they're just a bunch of numbers that someone has seen fit to define and name. There really isn't anything more to it than that. —Ilmari Karonen (talk) 00:37, 31 July 2007 (UTC)

Well, if that's all there is to it, if you can find a source for that claim (perhaps the existing ones say that, I don't know), then that may be the best you can do. I think it's a little too buried now, and should be moved up in the lead to show it more clearly, but I'm not sure of the best way to do that. FrozenPurpleCube 17:32, 31 July 2007 (UTC)

I think Ilmari Karonen's edits are good enough for providing "context." Whoever doesn't care for the graphics ought to be satisfied that the topic is in elementary number theory. Whoever cares to read further would either be delighted to learn the properties of these numbers or think they already knew them subconsciously. PrimeFan 18:34, 31 July 2007 (UTC)

Yes, they may be acceptable for providing that context, however, that doesn't mean the location is the best. I think placing them under the graphics is distracting. If nobody objects to moving them, I'll do so. FrozenPurpleCube 19:25, 31 July 2007 (UTC)

Phrasing

@Steelpillow (talk · contribs): In Centered square number:

• Is the first figure's caption:

"Centered square numbers, highlighted in red, are in found in the center of odd rows of Floyd's triangle -- taking 25 as an example, it is the sum of a 16 (yellow rhombus formed by shearing a square) and the next smaller square, 9 (sum of blue triangles)."

correctly phrased, & can it be simplified, please? :-P

• Is the following sentence:

"Another way the centered square numbers can be expressed are [...]."

correctly phrased, please?

In advance, thank you very much for your answers! --JavBol (talk) 18:01, 2 December 2021 (UTC)

I have corrected the obvious grammatical howler you highlight. But I know nothing about these numbers, all I can say is that the caption is appalling:

1. It is too long. Whatever it is trying to say should be said in the main text, and the caption should merely identify the aspect of the text that the figure is illustrating.
2. It is utterly unintelligible. For example it mentions 16 and 9 but these numbers are not present in the figure. If it needs to highlight what is in the figure, then it needs to highlight what is in the figure and not something else entirely.

I wish you all the best in sorting it out. — Cheers, Steelpillow (Talk) 18:21, 2 December 2021 (UTC)