An algebra problem by U Z
x , y , and z are integers such that
x 2 + y 2 + z 2 = 2 x y z .
Find the value of
1 + x 3 0 0 1 + y + 2 8 z ( x 5 + y 8 + z 1 0 ) .
The answer is 0.
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2 solutions
x=0 ,y=0 ,z=0 are the simplest solution of given equation on substituting ans is 0
but what about these cases
1) if 2 of them are odd and one is even
2) if 2 of them are negative odd integers and one is even
3) if all of them are even
suppose we write the equation as
y z x + x z y + y x z =2
then would you simply keep x=0 ,y=0 ,z=0 ?
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then what's the intended solution?
then please give the genuine solution
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First of all notice that all x,y,z must be even else we will have residue issues on both sides. Then by Fermat's method of inifinte descent, we can prove that the only solutions are (x,y,z) = (0,0,0).