Integers simply!

Number Theory Level 4

Find all the integers $x,y,z$ satisfying

$\large x^2 + y^2 + z^2 = 2xyz$ .

Evaluate the $\textbf{sum}$ of all integers of each pair.

$\textbf{Details}$

For e.g. If you get your integers solution as

$(1,2,3)$ and $(2,-1,5)$

Then enter your answer as $1+2+3+2-1+5=\boxed{12}$

1 solution

広務山下
Apr 4, 2018

right side is even.so one of them is only even or every is even.

If one of them is even,right side is multiple of four , left side is not multiple of four.＊[(2q-1)^2=4q^2+4q+1]

so every is even…①

If every is even,x=2p y=2q z=2r 4p^2+4q^2+4r^2=16pqr

p^2+q^2+r^2=4pqr

so p,q,r are even(①)………

that will never end.

so answer is 0.