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I wondered if a square minus a square could be a square ? When I put question into equation, I have

aa - bb = cc (when condition is a > b) *I don't know what is the correct notation of a mathematical condition

Q1: Is that right, that we always got a rectangle ?

Q2: Is there some easy way how to prove this ?

Q3: If answer to Q1 is true, is it also true when we remove the condition ?


Update

enter image description here

Q1 does not make sense, because we always get two rectangles, not one. But we can join these two rectangles to composed one. (which can have same square area as some other square)

Muflix
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    $25-16$ is a square. – Bernard Apr 11 '20 at 11:19
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    $5^2-4^2=3^2$ or $13^2-12^2=5^2$, ecc. – gpassante Apr 11 '20 at 11:21
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    There are many general cases. For example: for all $a$ and $b$, we have $$(a^2+b^2)^2-(a^2-b^2)^2=(2ab)^2$$ or for all $c$, we have $$(2c^2\pm 2c+1)^2-(2c^2\pm 2c)^2=(2c\pm 1)^2$$ (where the $\pm$ signs are not independent of each other). And it doesn't just stop there with the difference of two squares equalling a single square. Such general equations can involve abundances of square numbers! :) – Mr Pie Apr 11 '20 at 11:28
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    I have no idea what Q1 means. What does it mean to "get a rectangle" here??? – David C. Ullrich Apr 11 '20 at 11:52
  • @DavidC.Ullrich you are right, I updated my question. – Muflix Apr 11 '20 at 12:26
  • It now makes less sense than previously! Because $5^2-4^2$ is a square, but that picture applies perfectly well with $a=5$ and $b=4$, giving two rectangles. – David C. Ullrich Apr 11 '20 at 12:52
  • @DavidC.Ullrich I think about it like this: if square one were your land and you sold part of the land (square two) to someone else, on square area you have now, you wouldn't be able to build a square-footed house, even though (or because) the house had the same square area which left you (square one minus square two). – Muflix Apr 11 '20 at 13:21
  • So 55−44 is a square (but can be two rectangles as well), but not if first and second square is a land, you cannot move to fit a new square. You can sell the land and buy new square-footed land with the same square area as you sold. – Muflix Apr 11 '20 at 13:30
  • I understand that that's how you're thinking about it. But the question was whether $a^2-b^2$ can be a perfect square, and that simply has nothing at to do with whether you can build that square house! (For example, for the second time: $5^2-4^2$ is a square, even though you still see two rectangles in that picture.) – David C. Ullrich Apr 11 '20 at 13:30
  • @DavidC.Ullrich I didn't argue about that, because my Q1 is misleading as you correctly pointed out. – Muflix Apr 11 '20 at 13:32

1 Answers1

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Restrict to the set of natural numbers.

If $a^2-b^2 = c^2$, then $a^2=b^2+c^2$. The triples $(a,b,c)$ with this property are called Pythagorean triples. There are infinitely many of them.

Wuestenfux
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  • Thank you, I can see it now. I visualized it wrong. Becase square minus square cannot be a square but the resulting rectangle will have same square area. – – Muflix Apr 11 '20 at 11:29
  • @Muflix Square minus square can be a square. A rectangle with a square area makes it, well, a square! :) – Mr Pie Apr 11 '20 at 11:31
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    @Mr Pie But, the rectangle with square area 16 (where a = 2 and b = 8) has same square area as the square where a = 4. So the square area is the same, but one is rectangle and one is square, right ? – Muflix Apr 11 '20 at 11:36
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    @Muflix I am dumdum facepalm – Mr Pie Apr 11 '20 at 11:38