# Diophantine equations and the number 4

Suppose $$x^2 + y^2 + z^2 + 1 = 4xyz,$$ where $$x,y,z$$ are integers satisfying the inequalities $$1 < x \le y \le z$$. Find the minimum possible value of $$z$$.

(Extra credit: prove that the equation $$x^2 + y^2 + z^2 + 1 = mxyz$$ has no integer solutions if $$m$$ is a positive integer not equal to $$4$$.)

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