How many sets of integers $ \{x,y,z\} $ satisfy bo
Algebraic simultaneous equations problem
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number-theory
algebraic-geometry
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0What did you try? – 2017-01-31
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0one dumb approach -- substitute $z=54-x-y$ into the quadratic equation, simplify and change variables, to eliminate the cross term $xy$ and this should reduce into 2 complete squares – 2017-01-31
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0Bruteforce answer: [(1,26,27),(2,21,31),(5,15,34),(9,10,35)] – 2017-01-31
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0What is "bo" and how does one satisfy it? – 2017-02-01
1 Answers
1
Hint
$$\underbrace{(x+y+z)^2}_{2916}=\underbrace{x^2+y^2+z^2}_{1406}+2z\underbrace{(x+y)}_{54-z}+2xy$$ thus $$z^2-54z+(755-xy)=0$$ as a result $$z=27+\sqrt{xy-26}$$ or $$z=27-\sqrt{xy-26}$$