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==Question on Perfect Squares==
For which n does the following to hold:

1<sup>2</sup> ± 2<sup>2</sup> ± 3<sup>2</sup> ± ... ± n<sup>2</sup> = 0?

where it is possible to choose either + or - in any of the cases.  

My guess is that this relation will not hold for any natural n, and I would guess its proved by contradiction, but formulating this proof seems quite difficult. Any suggestions welcome, or even related material to look at. 

--[[Special:Contributions/84.70.242.151|84.70.242.151]] ([[User talk:84.70.242.151|talk]]) 15:20, 9 March 2009 (UTC) William Kitchen

:The purpose of this [[WP:TALKPAGE|talk page]] is to discuss improvements to the article [[Square number]]. You can ask questions about mathematics at [[Wikipedia:Reference desk/Mathematics]]. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 16:27, 9 March 2009 (UTC)
:There are plenty of solutions. The first is 1<sup>2</sup> + 2<sup>2</sup> - 3<sup>2</sup> + 4<sup>2</sup> - 5<sup>2</sup> - 6<sup>2</sup> + 7<sup>2</sup> = 0. Here are more according to a computer search:

:''n'': Terms for which to add the square
:7: 1,2,4,7
:8: 1,4,6,7
:11: 1,3,4,5,9,11
:12: 1 to 8, and 11
:15: 1 to 6, and 8,10,13,14
:16: 1 to 4, and 6,7,8,12,13,16
:19: 1 to 9, and 11,12,18,19
:20: 1 to 10, and 12,16,17,19
:23: 1 to 12, and 14,15,17,19,21
:24: 1 to 14, and 16,17,19,23
:27: 1 to 15, and 18,20,21,22,24
:28: 1 to 16, and 22,24,25,26
:31: 1 to 19, and 22,25,27,30
:32: 1 to 20, and 22,25,29,30
:35: 1 to 25, and 29,33
:36: 1 to 22, and 25,27,29,32,33
:39: 1 to 26, and 28,30,33,36
:40: 1 to 26, and 30,34,37,38
:43: 1 to 28, and 32,33,35,36,37
:44: 1 to 31, and 35,38,40
:47: 1 to 32, and 37,39,41,43
:48: 1 to 32, and 35,36,39,41,43
:51: 1 to 35, and 42,43,44,48
:52: 1 to 36, and 38,43,46,50
:55: 1 to 40, and 45,46,47
:56: 1 to 40, and 49,51,54
:59: 1 to 40, and 42,48,50,54,59
:60: 1 to 42, and 46,52,56,58
:63: 1 to 46, and 48,56,61
:64: 1 to 47, and 50,56,58
:67: 1 to 52, and 55
:68: 1 to 49, and 51,52,59,66
:71: 1 to 50, and 52,58,60,63,66
:72: 1 to 51, and 53,54,57,63,71
:75: 1 to 54, and 60,65,69,72
:76: 1 to 55, and 60,65,68,72
:79: 1 to 57, and 61,62,64,65,67
:80: 1 to 61, and 65,72
:83: 1 to 60, and 62,78,80,83
:84: 1 to 60, and 62,63,74,80,84
:87: 1 to 63, and 65,70,74,76,77
:88: 1 to 66, and 74,75,80
:91: 1 to 66, and 69,71,80,81,83
:92: 1 to 69, and 71,85,88
:95: 1 to 70, and 74,76,92,93
:96: 1 to 69, and 71,73,75,81,86,89
:99: 1 to 74, and 76,77,82,89
:100: 1 to 75, and 83,94,100
:Note that the sum of all squares must be even to have a chance. [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 17:14, 9 March 2009 (UTC)

== Merge with [[Square (algebra)]] ==
Both these articles represent the same function, so I see no reason why they shouldn't be merged. I also wish for [[Cube (algebra)]] to moved to Cube number so we have something that can fit in [http://en.wikipedia.org/wiki/Category:Figurate_numbers Category:Figurate numbers] that can coincide with other polyhedral numbers such as [[Tetrahedral number]]s and [[Centered cube number]]s. The idea of having two articles for square numbers and yet one for cubes seems odd to me. [[User:Robo37|Robo37]] ([[User talk:Robo37|talk]]) 10:05, 23 November 2009 (UTC)

I agree with this because it is in the same "grade"  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Froogle1099|Froogle1099]] ([[User talk:Froogle1099|talk]] • [[Special:Contributions/Froogle1099|contribs]]) 00:44, 4 February 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

I disagree because perfect squares and squares in general have incredibly different properties from a set theory point of view. Proving things with perfect squares is completely different because they have a much more restricted set of properties.  <span style="font-size: smaller;" class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/128.252.78.87|128.252.78.87]] ([[User talk:128.252.78.87|talk]]) 02:16, 27 September 2010 (UTC)</span><!-- Template:UnsignedIP --> <!--Autosigned by SineBot-->

:I agree with a merge: as it stands [[Square (algebra)]] is mostly about perfect squares anyway. --[[User:Physics is all gnomes|Physics is all gnomes]] ([[User talk:Physics is all gnomes|talk]]) 22:30, 2 January 2011 (UTC)

== Sum of digits ==
Shouldn't it be mentioned that the sum of the digits of a square number must be 1, 4, 7 or 9? I don't know the exact term in English, but this is the same as saying that the rest if one divides by nine must be 0, 1, 4, 7. I don't have any proof of this property though.
[[User:Vittorio Mariani|Vittorio Mariani]] ([[User talk:Vittorio Mariani|talk]]) 13:20, 11 December 2009 (UTC)

:It's mentioned at [[Digital root#Some properties of digital roots]]. I'm not sure it is worth mentioning in [[Square number]]. It follows from [[modular arithmetic]] that the property only has to be checked for 0^2 to 8^2. Suppose you write an arbitrary integer as (9n+k) where 0≤k<9. (9n+k)^2 = (9n)^2+18nk+k^2. Divided by 9 it must give the same [[remainder]] as k^2 divided by 9, because 9 divides (9n)^2+18nk. Similar rules for possible values of the remainder when a square is divided by any other integer d can be given. Just list the remainder when dividing 0^2 to (d-1)^2 by d (actually you can limit it to 0^2 to floor(d/2)^2 for symmetry reasons). [[User:PrimeHunter|PrimeHunter]] ([[User talk:PrimeHunter|talk]]) 13:58, 11 December 2009 (UTC)

:Maybe just mentioning wouldn't do bad, I won't insist anyway :) --[[User:Vittorio Mariani|Vittorio Mariani]] ([[User talk:Vittorio Mariani|talk]]) 12:48, 22 February 2010 (UTC)

== Pattern In squares ==
when you are using squares here is an easy patters.

1x1=1
(+3)
2x2=4
(+5)
3x3=9
(+7)
4x4=16

As you should see by now each time the factor goes up the solution goes up by an odd number.  <small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:Froogle1099|Froogle1099]] ([[User talk:Froogle1099|talk]] • [[Special:Contributions/Froogle1099|contribs]]) 00:51, 4 February 2010 (UTC)</span></small><!-- Template:Unsigned --> <!--Autosigned by SineBot-->

== Huh??? ==
This article informs us that
: 
:: An easy way to find square numbers is to find two numbers which have a mean of it, 21<sup>2</sup>:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22 × 20 = 440 and  440&nbsp;+&nbsp;1<sup>2</sup> = 441.
: 
That could bear translation into English. [[User:Michael Hardy|Michael Hardy]] ([[User talk:Michael Hardy|talk]]) 18:56, 16 March 2010 (UTC)

:I think what it means is that to manually calculate <math>n^2</math>, it is sometimes easier to calculate <math>(n+b)(n-b) + b^2</math> for some ''b''.  For example, if one would like to square 39, it is easier to calculate 38·40+1 = 1521 than to multiply 39·39 directly.  —[[User:Dominus|Dominus]] ([[User talk:Dominus|talk]]) 19:07, 16 March 2010 (UTC)

:I deleted it from the article, but I will not be offended if you think it is worth putting back.  —[[User:Dominus|Dominus]] ([[User talk:Dominus|talk]]) 19:11, 16 March 2010 (UTC)

== Hexadecimal section ==
Added the Hexadecimal section to Properties chapter. Contains factorization without any division! May be interesting for young math geeks. Please correct me if some place is unclear.  [[User:Neeme Vaino|Neeme Vaino]] ([[User talk:Neeme Vaino|talk]]) 09:15, 26 March 2010 (UTC)

== "Uses" section ==
[{{fullurl:Square number|diff=510626176}}] 24 hours to demonstrate the relevance of <span style="font-size:300%">integer</span> square numbers to "the real number system" and statistics. Otherwise, I will henceforth reduce such edits of {{user|Anita5192}}, possibly with my <span style="text-decoration:underline; color:blue">[rollback]</span> link. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 19:14, 3 September 2012 (UTC)

* Do you actually have [[WP:rollback|rollback]]? Your link seems to be a ''fake'', though even if you did, you have '''[[WP:rollback#How it works|no right]]''' to use it so your edits dominate others. 
* There is no need for such a confrontational approach as you have done so here ("<code><nowiki><span style="font-size:300%">integer</span></nowiki></code>"). 
* Furthermore are you able to [[Talk:Square number#Merge with Square (algebra)|'''''read''''' what others have written, when they '''supported the merge''']]? You have [[Wikipedia talk:WikiProject Mathematics#Square (algebra)|deceptivley re-labelled]] the situation to give the impression that they ([[user:Robo37|Robo37]], [[user:Froogle1099|Froogle1099]], [[user:Physics is all gnomes|Physics is all gnomes]]) "disagreed" (ok yes ''one'' IP disagreed but he/she was outnumbered anyway according to JBL), yet you have '''failed to point to the link to where they explicitly state so'''. ''Where'' is the link??

This is appalling. You are simply fighting and complaining for your own way '''against''' the rules - given your burning desire to overwhelm [[user:Anita5192|Anita5192]]. [[User:Maschen|Maschen]] ([[User talk:Maschen|talk]]) 20:45, 3 September 2012 (UTC)
: Of course, it's a fake link ''here''. I'm not a moron to put my own privilege to a publicly readable page ☺<br/>The accusation that I "fight '''against''' the rules" requires evidences. There was no deception, only a part of statement was missing before the [[semicolon]] character. Note that I would not have such a grievance about [[user:Joel B. Lewis‎]] were the merger ''procedurally'' accurate. Probably, I fight against opinions of 4 users (although Joel B. Lewis‎ virtually renounced his position, and Physics is all gnomes commented only what [contemporarily] "stands Square (algebra) is mostly about"), but opinions of 4 users are not the rules of Wikipedia, anyway. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 21:36, 3 September 2012 (UTC)
----
The content of the [{{canonicalurl:Square number|oldid=510637141}}#Uses present "Uses" section] is off-topical, because explicitly mentions "the system of real numbers". It belongs to the topic "[[square (algebra)]]", not to the topic "square number", which is defined as an integer or, in a very general sense, [[rational number]]. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 07:55, 4 September 2012 (UTC)

== Formatting ==
Well, yet another problem. Let us compare the article's formatting:
{| class=wikitable
 ! colspan=2 |Before&nbsp;Incnis&nbsp;Mrsi and after&nbsp;Anita5192
 ! colspan=2 |After Incnis Mrsi
 ! rowspan=2 |Comments
 |-
 !code
 !rendered
 !code
 !rendered
 |-
 |<code><nowiki>√9&amp;nbsp;=&amp;nbsp;3</nowiki></code>||√9&nbsp;=&nbsp;3
 |<code><nowiki>{{sqrt|9}}&amp;nbsp;=&amp;nbsp;3</nowiki></code>||{{sqrt|9}}&nbsp;=&nbsp;3
 |No [[vinculum (symbol)|vinculum]] before Incnis Mrsi
 |-
 |<code><nowiki>m = 1<sup>2</sup> = 1</nowiki></code>||m = 1<sup>2</sup> = 1
 |<code><nowiki>{{mvar|m}} = 1<sup>2</sup> = 1</nowiki></code>||{{mvar|m}} = 1<sup>2</sup> = 1
 |{{mvar|m}} become italicized, like in a text run. Reverted by Anita5192
 |-
 |<code><nowiki>(''n''&amp;nbsp;&amp;minus;</nowiki><br/>&amp;nbsp;1)-th</code>||(''n''&nbsp;&minus;&nbsp;1)-th
 |<code><nowiki>{{math|(''n'' − 1)}}-th</nowiki></code>||{{math|(''n'' − 1)}}-th
 |
# NBSPs look exaggeratedly wide, [[thin space]]s look fine
# Anita5192's variant is [[obfuscated]]
 |-
 |<code><nowiki>in base 10</nowiki></code>||in base 10
 |<code><nowiki>in [[base 10]]</nowiki></code>||in [[base 10]]
 |Relevant internal link removed by Anita5192
 |-
 |<code><nowiki>if ''k² &amp;minus; m''</nowiki></code>||if ''k² &minus; m''
 |<code><nowiki>if {{math|''k''<sup>2</sup> − ''m''}}</nowiki></code>||if {{math|''k''<sup>2</sup> − ''m''}}
 |
# [[Wikipedia:MOSMATH#Superscripts and subscripts]] explicitly forbids the use of "²" except in limited circumstances
# Anita5192's variant can [[word wrap]]
 |-
 |<code><nowiki>''k'' ≥ √''m''</nowiki></code>||''k'' ≥ √''m''
 |<code><nowiki>{{math|''k'' ≥ }}{{sqrt|{{mvar|m}}}}</nowiki></code>||{{math|''k'' ≥ }}{{sqrt|{{mvar|m}}}}
 |
# No vinculum
# Can word wrap
 |-
 |<code><nowiki>pp. 30-32</nowiki></code>||pp. 30-32
 |<code><nowiki>pp.&amp;nbsp;30–32</nowiki></code>||pp.&nbsp;30–32
 |
# Improper use of [[hyphen-minus]]; see [[WP:–]]
# Can word wrap
 |}
How exactly are aforementioned Anita5192's codes better than mine? [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 20:27, 3 September 2012 (UTC)

:As said, the ordinary markup is cleaner than {{tl|mvar}}. It is not possible to copy and paste the text into another edit window (if needed) using the templates, using the ordinary markup makes that easy. [[User:Maschen|Maschen]] ([[User talk:Maschen|talk]]) 20:45, 3 September 2012 (UTC)
:: As said by whom? I do not understand completely what do you speaking about. If you want to copy and paste between edit forms, then there is no difference between "the ordinary markup" and template formatting, and you certainly have to realize this. Assuming you speak about copying from a ''rendered'' page (i.e., a web page in the browser) to an edit form, which is another possibility, please, demonstrate how is it possible to easily copy and paste the ordinary markup such as <span style="font-size:200%">m = 1<sup>2</sup> = 1</span> or <span style="font-size:200%">(''n''&nbsp;&minus;&nbsp;1)-th</span> to an edit form. Do you assert that by copying from this zoomed text on a web page will you obtain a workable wiki code, with superscripts and italics? Maybe, you could even demonstrate this? In which browser? In any way, this consideration have little to do with concrete, pronounced shortcomings which I listed. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 21:36, 3 September 2012 (UTC)
----
I do not insist on {{tl|math}} and {{tl|mvar}} and do not have a strong preference towards {{tl|sqrt}} at the expense of &lt;math>. Just [[WP:Manual of Style/Text formatting|fix not less formatting errors]] than [{{canonicalurl:Square number|diff=510620927}} I fixed], and I'll give up my version. If nobody will do it&nbsp;– sorry, but my version is the best in the article's history as of now. Do improve the article, if you do. But if you do not&nbsp;– please, do not hinder me with this job. [[User:Incnis Mrsi|Incnis Mrsi]] ([[User talk:Incnis Mrsi|talk]]) 07:55, 4 September 2012 (UTC)

Note: White gaps between squares serve only to improve visual perception.
There must be no gaps between actual squares._who is the idiot that needed to be told this?[[Special:Contributions/121.127.198.87|121.127.198.87]] ([[User talk:121.127.198.87|talk]]) 04:00, 20 July 2014 (UTC)

== Way to represent the nth perfect square ==

Example: 1 + 2 + 3 + 4 + 3 + 2 + 1 = 4 ^ 2 = 16

4 is in the center of sum, and 16 is the 4th perfect square.

Is it reasonable to add this to the article?

[[User:Azurean72|Azurean72]] ([[User talk:Azurean72|talk]]) 04:17, 19 March 2023 (UTC)
:This article is not for your own original research. Each addition must be supported by published sources that establish the notability of the result. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 10:40, 19 March 2023 (UTC)
:It's quite a trivial result, as splitting the sum gives (in this case) four columns each totalling 4{{pb}}1 + 2 + 3 + 4 +{{pb}}3 + 2 + 1{{pb}}so it doesn't seem worth including, even if any reliable source has bothered to provide such a simple and obvious proof. [[User:Certes|Certes]] ([[User talk:Certes|talk]]) 13:25, 19 March 2023 (UTC)

== '''perfect''' square ==

I came here looking for (obviously historical) info on why they are called ''perfect''. [[User:Jae|jae]] ([[User talk:Jae|talk]]) 18:56, 13 August 2023 (UTC)

:Perfect squares are called that because they are integer square numbers. The recent fad of calling perfect squares ''square numbers'' is silly—all positive real numbers can be regarded as square numbers because they have a square root, i.e., they are ''squares'' of their roots. π² is obviously the square of π. My guess is that the trend of using the term ''square numbers'' to describe perfect squares arose from the brainless impulse to find things to improve, which typically leads to changes that are not improvements. [[User:Aboctok|Aboctok]] ([[User talk:Aboctok|talk]]) 22:09, 22 August 2024 (UTC)
::I guess (I have no source) that "square number" is a much older phrase than "perfect square". In fact, it dates from the time where "number" meant only [[natural number]]. I guess also that the qualificative "perfect" is used because perfect squares are those (natural) numbers such that the square root is perfectly represented by a (natural) number. However, this terminology dates from the time where most mathematics were written in Latin, and the use of the term "perfect" can be a side effect of the translation in English. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 07:41, 23 August 2024 (UTC)
::{{Re|Aboctok}} Yes, all positive (actually non-negative) real numbers are square numbers in the context of real numbers. But here, the context is the set of integers. If you consider that you should not have an implicit context, then the term ''perfect square'' is also ambiguous, because it is also used for non-integers, like [https://www.learnalberta.ca/content/memg/Division03/Perfect%20Square/index.html here], where 102.01 is regarded as a perfect square because (10.1)² = 102.01 exactly. See also the footnote saying: "Some authors also call squares of rational numbers perfect squares." So I don't see anything silly by just saying "square number". Note that there's the same terminology issue in French. — [[User:Vincent Lefèvre|Vincent Lefèvre]] ([[User talk:Vincent Lefèvre|talk]]) 19:48, 26 August 2024 (UTC)