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# Repost of unanswered question

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How many ordered triples (x,y,z) of integers are there such that ? Does the question have a geometric interpretation?

MIRB16  Mar 12, 2018
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$$\displaystyle x^{2}+y^{2}+z^{2}=49,$$

is the equation of a sphere radius 7, centered at the origin, so what you are looking for are points on its surface such that all three co-ordinates are integers.

I don't see any way through it other than by trying each integer value of one of the variables in turn.

Choosing z, it's largest value is 7, and that gets us

$$\displaystyle x^{2}+y^{2}=0, \text{ from which we have } x=0\text{ and }y=0.$$

If z = 6,

$$\displaystyle x^{2}+y^{2}=13,\text{ leading to }x=\pm2, y=\pm3,\text{ or }x=\pm3,y=\pm2.$$

If z = 5,

$$\displaystyle x^{2}+y^{2}=24,\text{ and there are no integer solutions for that.}$$

So on through to z = -7.

Guest Mar 12, 2018

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