Talk:Square number

For which n does the following to hold:

1² ± 2² ± 3² ± ... ± n² = 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.

--84.70.242.151 (talk) 15:20, 9 March 2009 (UTC) William KitchenReply

The purpose of this talk page is to discuss improvements to the article Square number. You can ask questions about mathematics at Wikipedia:Reference desk/Mathematics. PrimeHunter (talk) 16:27, 9 March 2009 (UTC)Reply

There are plenty of solutions. The first is 1² + 2² - 3² + 4² - 5² - 6² + 7² = 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. PrimeHunter (talk) 17:14, 9 March 2009 (UTC)Reply

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 Category:Figurate numbers that can coincide with other polyhedral numbers such as Tetrahedral numbers and Centered cube numbers. The idea of having two articles for square numbers and yet one for cubes seems odd to me. Robo37 (talk) 10:05, 23 November 2009 (UTC)Reply

I agree with this because it is in the same "grade" —Preceding unsigned comment added by Froogle1099 (talk • contribs) 00:44, 4 February 2010 (UTC)Reply

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. —Preceding unsigned comment added by 128.252.78.87 (talk) 02:16, 27 September 2010 (UTC)Reply

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

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. Vittorio Mariani (talk) 13:20, 11 December 2009 (UTC)Reply

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). PrimeHunter (talk) 13:58, 11 December 2009 (UTC)Reply

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

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. —Preceding unsigned comment added by Froogle1099 (talk • contribs) 00:51, 4 February 2010 (UTC)Reply

This article informs us that

An easy way to find square numbers is to find two numbers which have a mean of it, 21²:20 and 22, and then multiply the two numbers together and add the square of the distance from the mean: 22 × 20 = 440 and 440 + 1² = 441.

That could bear translation into English. Michael Hardy (talk) 18:56, 16 March 2010 (UTC)Reply

I think what it means is that to manually calculate ${\displaystyle n^{2}}$ , it is sometimes easier to calculate ${\displaystyle (n+b)(n-b)+b^{2}}$  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. —Dominus (talk) 19:07, 16 March 2010 (UTC)Reply

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

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. Neeme Vaino (talk) 09:15, 26 March 2010 (UTC)Reply

[1] 24 hours to demonstrate the relevance of integer square numbers to "the real number system" and statistics. Otherwise, I will henceforth reduce such edits of Anita5192 (talk · contribs), possibly with my [rollback] link. Incnis Mrsi (talk) 19:14, 3 September 2012 (UTC)Reply

• Do you actually have rollback? Your link seems to be a fake, though even if you did, you have 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 ("<span style="font-size:300%">integer</span>").
• Furthermore are you able to read what others have written, when they supported the merge? You have deceptivley re-labelled the situation to give the impression that they (Robo37, Froogle1099, 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 Anita5192. Maschen (talk) 20:45, 3 September 2012 (UTC)Reply

Of course, it's a fake link here. I'm not a moron to put my own privilege to a publicly readable page ☺
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. Incnis Mrsi (talk) 21:36, 3 September 2012 (UTC)Reply

The content of the 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. Incnis Mrsi (talk) 07:55, 4 September 2012 (UTC)Reply

Well, yet another problem. Let us compare the article's formatting:

How exactly are aforementioned Anita5192's codes better than mine? Incnis Mrsi (talk) 20:27, 3 September 2012 (UTC)Reply

As said, the ordinary markup is cleaner than {{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. Maschen (talk) 20:45, 3 September 2012 (UTC)Reply

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 m = 1² = 1 or (n − 1)-th 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. Incnis Mrsi (talk) 21:36, 3 September 2012 (UTC)Reply

I do not insist on {{math}} and {{mvar}} and do not have a strong preference towards {{sqrt}} at the expense of <math>. Just fix not less formatting errors than I fixed, and I'll give up my version. If nobody will do it – 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 – please, do not hinder me with this job. Incnis Mrsi (talk) 07:55, 4 September 2012 (UTC)Reply

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?121.127.198.87 (talk) 04:00, 20 July 2014 (UTC)Reply

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?

Azurean72 (talk) 04:17, 19 March 2023 (UTC)Reply

This article is not for your own original research. Each addition must be supported by published sources that establish the notability of the result. D.Lazard (talk) 10:40, 19 March 2023 (UTC)Reply

It's quite a trivial result, as splitting the sum gives (in this case) four columns each totalling 4

1 + 2 + 3 + 4 +

3 + 2 + 1

so it doesn't seem worth including, even if any reliable source has bothered to provide such a simple and obvious proof. Certes (talk) 13:25, 19 March 2023 (UTC)Reply

I came here looking for (obviously historical) info on why they are called perfect. jae (talk) 18:56, 13 August 2023 (UTC)Reply

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. Aboctok (talk) 22:09, 22 August 2024 (UTC)Reply

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. D.Lazard (talk) 07:41, 23 August 2024 (UTC)Reply

@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 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. — Vincent Lefèvre (talk) 19:48, 26 August 2024 (UTC)Reply