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Can somebody fill me in on a visual explanation for the following:

If there exist integers $x, y$ such that $x^2 + y^2 = c$, then there also exist integers $w, z$ such that $w^2 + z^2 = 2c$

I know why it is true (ex. take $w = x-y, z = x+y$), but I would think there is a visual explanation hiding somewhere because of squared terms (we can make squares!!)

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6 Answers 6

up vote 25 down vote accepted

It will take me forever to post the diagram so here is a description.

Draw the circle with centre $(0,0)$ and radius $\sqrt c\,$. Locate the point $(x,y)$ on this circle: by assumption, $x$ and $y$ are integers. Draw the tangent to this circle starting at $(x,y)$ and having length $\sqrt c\,$. This will give a point distant $\sqrt{2c}$ from the origin (because we have a right angled isosceles triangle), and the point will have integer coordinates because it is obtained from $(x,y)$ by a displacement of $(y,-x)$ or $(-y,x)$, depending which way we drew the tangent.

Update: see another answer for the picture. Thanks Oleg!

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Doubt: How did you calculate the displacement? –  Anant Mar 20 at 17:17
    
The line perpendicular to a point $(x, y)$ has a slope of $\frac{-x}{y}$. So, a displacement of $(-y, x)$ or $(y, -x)$ is contained by that perpendicular and is the same length as the original displacement from the origin of $(x, y)$ –  Michael T Mar 20 at 19:19
    
Or more simply: the tangent is perpendicular to the radius, so the $x$ coordinate becomes the $y$ coordinate and vice versa, with a little care to get the signs right. –  David Mar 20 at 21:01
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In terms of the areas of squares:

enter image description here

Edit: Thanks to Michael T for pointing out a simpler rearrangement.

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Hey, squares! This was the original idea I had in mind. Looks very nice, thank you. –  Michael T Mar 21 at 19:59
    
I have found what I believe is a simpler construction (and have thus answered my own question) –  Michael T Mar 21 at 21:52
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@user92774 I've edited the diagram to incorporate your construction. Thanks! –  Anant Mar 21 at 22:51
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Hint $\ $ The transformation is simply multiplication by $\, 1+i\,$ in the complex plane

$$(1+i)(x+yi)\, =\, x\!-\!y + (x\!+\!y)i $$

Thus if $\ |x+yi| = x^2+y^2 = c \ $ then

$$ (x\!-\!y)^2+(x\!+\!y)^2 =\, |x\!-\!y+(x\!+\!y)i| = |(1+i)(x+yi)|\, =\, 2\,|x+yi|\,=\,2c$$

Geometrically, multiplication by $\ 1+i = \sqrt{2}\, e^{\large i\pi/4}\,$ may be visualized as an expansion by the factor $\,\sqrt{2}\,$ composed with a rotation by $\,\pi/4.\,$ Let's visualize this in the diagram from Oleg567's answer, excerpted below. Let $\,\rm\color{#c00}{v = x+y\, i}\,$ be the lower red vector. Multiplying it by $\,1+i\,$ yields $\, (1+i){\rm v = v} + i\rm v,\,$ where $\,\rm\color{#c00}{i\,v = -y + x\, i}\,$ is the rotation of $\,\rm v\,$ by $\,\pi/4.\,$ Adding these red vectors yields the result $\,\rm\color{blue}{v + i\, v = x-y + (x+y)\,i}.$

$\qquad\quad$ enter image description here

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This is very nice. Always appreciate when complex numbers are used in clever ways like this. –  Michael T Mar 21 at 19:10
    
Reminded me of this explanation, which opened my eyes to the beauty of complex numbers :) –  Anant Mar 22 at 8:25
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I have devised what I believe to be a simpler construction of Anant's squares argument and have thus shared it here:

enter

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Very nice! Silly that I didn't spot it :) –  Anant Mar 21 at 22:26
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2 Circles

$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$

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$\sqrt{c}$ is the radius of smaller circle; $\sqrt{2c}$ is the radius of bigger circle; $(\sqrt{c})^2+(\sqrt{c})^2=(\sqrt{2c})^2$ (triangle is $45^\circ$ right triangle). –  Oleg567 Mar 20 at 12:05
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source ( code)? –  user13107 Mar 20 at 16:05
    
@Oleg567 What if $c$ isn't an integer? –  Jack M Mar 20 at 18:07
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@JackM That's impossible, since x and y are integers. –  Red Alert Mar 20 at 18:34
    
@user13107: earlier [rough] image was made via MS Word (to draw circle, line, show Grid :); last [accurate] image was made via C/C++ (Windows API, GDI-functions: MoveToEx(), LineTo, Ellipse(), ...). –  Oleg567 Mar 21 at 10:59
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If you imagine an integer-sized coordinate grid (as in Oleg567's answer) overlaid with another coordinate grid that has been shrunk by a factor of $\sqrt2$ and rotated by $45^{\circ}$ about the origin, the original unit vectors still have integer coordinates.

So the point $(x,y)$ also has integer coordinates in the new system, being a linear combination of unit vectors. Its distance to the origin is now $\sqrt2$ times the old distance, because we did shrink the unit vectors.

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