I'm stuck on a homework problem, which asks whether or not the sum of three squares of integers can equal 999. This is an abstract algebra class, not number theory, and we haven't learned any number theory tools beyond the division algorithm and relative primeness. I'm at a loss how to proceed. The only thing I've noticed is that either all three numbers must be odd, or exactly one of them, but this doesn't seem to lead me anywhere. Any help is appreciated, thank you.
How to know if a sum of squares has integer solutions?
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abstract-algebra
number-theory
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0What do you know about the integers, do they have to be consecutive? – 2017-01-30
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0No restrictions on the three numbers except that they must be integers – 2017-01-30
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0@Jan Oh, you are right. This is even exactly about $n=999$. Thank you! – 2017-01-30
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0@Dietrich Burde How do you make your link look the way it looks (and not line mine). Thank you. – 2017-01-30
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1Ok so Legendre's theorem states that $n=x^2+y^2+z^2$ if and only if $n \neq 4^a (8b+7)$ for integers $a,b$. the fact that this shows that $n$ must be divisible by $2$ is a good starting point. – 2017-01-30
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0@Jan You're right, thank you for directing me to that question. I don't know how I didn't find it – 2017-01-30
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0@Jan Good question. MSE produces this line, if you click on "duplicate". – 2017-01-30
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0@BensonLin Yes, Jan has pointed this out above. – 2017-01-30