Ohh, squares

Two positive integers ( x , y x, y ) satisfy that x y + 1 xy+1 is a perfect square.

True or false?

There always exist a positive integer ( z z ), where all of the numbers below are perfect squares. y z + 1 x z + 1 x y + x z + y z + 1 \begin{aligned} yz & + 1 \\ xz & + 1 \\ xy+xz+yz & + 1 \end{aligned}

False True

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