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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)

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    $\begingroup$ $25-16$ is a square. $\endgroup$
    – Bernard
    Apr 11, 2020 at 11:19
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    $\begingroup$ $5^2-4^2=3^2$ or $13^2-12^2=5^2$, ecc. $\endgroup$
    – gpassante
    Apr 11, 2020 at 11:21
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    $\begingroup$ 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! :) $\endgroup$
    – Mr Pie
    Apr 11, 2020 at 11:28
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    $\begingroup$ I have no idea what Q1 means. What does it mean to "get a rectangle" here??? $\endgroup$
    – David C. Ullrich
    Apr 11, 2020 at 11:52
  • $\begingroup$ @DavidC.Ullrich you are right, I updated my question. $\endgroup$
    – Muflix
    Apr 11, 2020 at 12:26

1 Answer 1

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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.

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  • $\begingroup$ 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. – $\endgroup$
    – Muflix
    Apr 11, 2020 at 11:29
  • $\begingroup$ @Muflix Square minus square can be a square. A rectangle with a square area makes it, well, a square! :) $\endgroup$
    – Mr Pie
    Apr 11, 2020 at 11:31
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    $\begingroup$ @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 ? $\endgroup$
    – Muflix
    Apr 11, 2020 at 11:36
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    $\begingroup$ @Muflix I am dumdum facepalm $\endgroup$
    – Mr Pie
    Apr 11, 2020 at 11:38

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