Talk:Square number

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One of the 500 most frequently viewed mathematics articles.

Fredrik, couldn't you also leave the other method there? It might be useful for less experienced people who cannot understand the method that you have written. - 18:13 Feb 15, 2005 (GMT)

Okay -- Sorry about that, I didn't realize. Jacquerie27

note on layout: don't use FONT tags, especially not to specify "windings" as the font - not all users will have it. -- Tarquin 22:11 Apr 21, 2003 (UTC)

I have seen now how it was supposed to look; earlier my browser was set "ignore fonts". - Patrick 23:27 Apr 21, 2003 (UTC)

Can we add another formula? This one is for finding the next square number in the sequence from any square number. I have written it and await approval/disapproval. It can be found here: User:Whiteheadj/Square Number. Enjoy! --Whiteheadj 20:46, 16 November 2005 (UTC)

Showing numbers are not perfect squares

I want to see if I can come up with some criteria that the digits of a number must follow every piece of for it to be a perfect square. Look at Talk:241 (number) and it will show you what piece of criteria can be used to rule it out as a perfect square. In turn, however, there still are some numbers that meet all of the criteria that are not perfect squares, the smallest of these is 721. Anyone know how to rule out 721?? Georgia guy 22:49, 24 February 2006 (UTC)

The simplest reason why 721 cannot be square is that it is divisible by 7 but not 49. As you point out, divisibility by seven can be determined by the digits; a similar test can be worked out for forty-nine. But a general test, in this manner, would probably be infinitely long, since it would requite a test for every prime. Septentrionalis 19:35, 13 September 2006 (UTC)
I couldn't follow the discussion at 241, but perhaps what he is getting at is that the decimal representation of a square must end in 0, 1, 4, 5, 6, or 9. This test doesn't rule out 241 or 721. But one can take it further: the decimal representation of a square must end in 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89, or 96; this rules out 78% of all integers as possible squares---although still not 241 or 721. But taking it one step further, one finds that the decimal representation must end in one of the 159 patterns 000, 001, ..., 489, ..., 976, 984, or 996, thus ruling out 84.1% of all integers as possible squares, and including 241 and 721. -- Dominus 03:21, 14 September 2006 (UTC)
The discussion at 241 is a demonstration that 241 = 7n + 3, for some n. Squares must be 49n, 7n+1, 7n +2,, or 7n+4 (proved by enumeration in base 7) Septentrionalis 01:23, 15 September 2006 (UTC)
When it comes to 721, I used the rule about adding 5 times the last digit to the rest of the number (72 + 5 = 77; 7 + 35 = 42) and this proves it is not a perfect square. However, even with all this criteria, there still are some numbers that meet the criteria but that are not perfect squares, the smallest of which is 1009. Georgia guy 00:55, 19 September 2006 (UTC)
And, in turn, I expanded the criteria to include divisibility by 11 (which the remainder must be 0, 1, 3, 4, 5, or 9) and it ruled out 1009, but didn't rule out 1969. Georgia guy 16:30, 19 September 2006 (UTC)
Now, I expanded to include divisibility by 13 and the remainder must be 0, 1, 3, 4, 9, 10, or 12. This does rule out 1969, but even then, there is a larger number, 5769, that still meets all the criteria that is not a perfect square. Georgia guy 16:28, 11 October 2006 (UTC)

Negative squared isn't positive

The fact that any negative number squared becomes positive is actually false. If you ever see the expression -3^2 evaluated as 9, that's incorrect. The exponentiation is always done before the negation unless there are parentheses there to indicate otherwise.

However, there are some contexts in which it _looks_ like texts are saying that -3^2 = 9, but a closer inspection will either reveal a subtle interpretation or a misunderstanding. For instance, what is the difference between the following statements:

  "If I take negative three and square it, I get nine."
  "If I square negative three, I get nine."
  "If I evaluate negative three squared, I get negative nine."
  "If I take the opposite of three squared, I get negative nine."

All of the above statements are correct. The reason some of them end up with 9 as the answer and some end up with -9 is that some of the statements have groupings implied in their phrasing. The first two statements translate into algebraic notation as (-3)^2 = 9, the third statement translates to -3^2 = -9, and the fourth statement translates to -(3^2) = -9. This has been taken from the webite

Chen's theorem

Chen Jingrun showed in 1975 that there always exists a number P that between n2and (n+1)2. See also Legendre's conjecture and Big Omega function.

This is dubiously relevant to the article. We cannot include every theorem that ever dealt with squares, so why this one?

However, my real objection is to the wording. User:WATARU may be proud that he knows what the Big Omega function is, but that is no reason to require our readers to do so. To say that there is either a prime or a product of two primes between two consecutive squares is both shorter and more comprehensible. Septentrionalis 19:31, 13 September 2006 (UTC)

Something else

a^2 - b^2 = a + b if b+1=a

Figuring out squares

Look at this! 0^2 is 0, 1^2 is 1, 2^2 is 4, 3^2 is 9, 4^2 is 16, etc. Well, i figured out that if you take the answer to a square and add an odd number that is next in the sequence starting at 1, you get the next square. Example: 0(first square) + 1(first odd number) = 1(second square), 1 + 3(second odd number) = 4(third square)! This continues on. my dad did the mathimatical proof. y=(x+1)^2 -x^2, where x = the number before the number you want to square and the y = the odd number to add. So if x = 10 (this means we are trying to get 11), then y=(10+1)^2 -100, or y=(11)^2 -100. The answer is 21. 100 + 21 = 121 = 11^2! If you factor this out, it becomes y = x^2 + 2x + 1 - x^2, or y = 2x +1. Plug in 10 for x, and you get y = 20+1, or y = 21. This is an easy way to find hard square numbers. Isn't that amazing? Dogmanice 03:56, 7 December 2006 (UTC)

Unfortunately, that's a well known property, it already appears in the article, and it doesn't really assist in finding squares except for rather specific cases. Calculating something like 7253482 takes less than a minute with long multiplication but is virtually impossible to calculate with this method. -- Meni Rosenfeld (talk) 10:32, 8 December 2006 (UTC)

0 zeroth square or first square??

0 is the square of 0, but is the first, not the zeroth, number in the list of square numbers in this article. Georgia guy (talk) 23:57, 1 March 2008 (UTC)

In [1] I have defined 0 as the zeroth square number. I think this use of zeroth is accepted and makes the article consistent. PrimeHunter (talk) 02:34, 2 March 2008 (UTC)

Square theory

I made this equation back in 7th grade, and I couldn't think of any thing that it applies to in the real world. Could you guys help me.


This equation will transfer x^2 into y^2 if x is a positive whole number and y is a positive whole number larger than x.

By the way, I'm still in 7th grade. —Preceding unsigned comment added by SuperCockroach (talkcontribs) 19:13, 13 September 2008 (UTC)

Sum of consecutive odds

Perfect squares are also the sum of a number of consecutive odd integers starting at 1. The number of odd integers to be summed is the value of their root. For example, 1=1, 1+3=4, 1+3+5=9, 1+3+5+7=16, 1+3+5+7+9=25, etc. Shouldn't this be in the article somewhere? -- (talk) 13:48, 12 January 2009 (UTC)

It's already mentioned clearly in Square number#Properties. PrimeHunter (talk) 15:03, 12 January 2009 (UTC)

Question on Perfect Squares

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.

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

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)
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)

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 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)

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

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 (talk) 02:16, 27 September 2010 (UTC)

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)

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. 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). PrimeHunter (talk) 13:58, 11 December 2009 (UTC)
Maybe just mentioning wouldn't do bad, I won't insist anyway :) --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. —Preceding unsigned comment added by Froogle1099 (talkcontribs) 00:51, 4 February 2010 (UTC)


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)

I think what it means is that to manually calculate , it is sometimes easier to calculate 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)
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)

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

"Uses" section

[2] 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)

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)

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)

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)


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

Before Incnis Mrsi and after Anita5192 After Incnis Mrsi Comments
code rendered code rendered
√9&nbsp;=&nbsp;3 √9 = 3 {{sqrt|9}}&nbsp;=&nbsp;3 9 = 3 No vinculum before Incnis Mrsi
m = 1<sup>2</sup> = 1 m = 12 = 1 {{mvar|m}} = 1<sup>2</sup> = 1 m = 12 = 1 m become italicized, like in a text run. Reverted by Anita5192
(n − 1)-th {{math|(''n'' − 1)}}-th (n − 1)-th
  1. NBSPs look exaggeratedly wide, thin spaces look fine
  2. Anita5192's variant is obfuscated
in base 10 in base 10 in [[base 10]] in base 10 Relevant internal link removed by Anita5192
if ''k² &minus; m'' if k² − m if {{math|''k''<sup>2</sup> − ''m''}} if k2m
  1. Wikipedia:MOSMATH#Superscripts and subscripts explicitly forbids the use of "²" except in limited circumstances
  2. Anita5192's variant can word wrap
''k'' ≥ √''m'' k ≥ √m {{math|''k'' ≥ }}{{sqrt|{{mvar|m}}}} km
  1. No vinculum
  2. Can word wrap
pp. 30-32 pp. 30-32 pp.&nbsp;30–32 pp. 30–32
  1. Improper use of hyphen-minus; see WP:–
  2. Can word wrap

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

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)
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)

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)

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

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