Silk Road Mathematics Competition

Firstly, note that the equation $x^2-p{y}^2=1$ has infinitely many solutions in positive integers. (Pell's equation) Then for every prime number $p$ there exist infinitely many positive integers $s$ and $t$ such that $s^2-1=p{t}^2$. Putting $x=s^2-1$, $y=s+1$, $z=s-1$ we have $$\begin{gathered} (x^2+s^2-1)(y^2+s^2-1)(z^2+s^2-1)= \\ =(s^2-1)s^2(s+1)(2s)(s-1)(2s)=((s^2-1)2s^2{)}^2. \end{gathered}$$ It is remained to check that $x,y,z\ne t$. Note that $x=s^2-1=p{t}^2\ne t$. If $y=s+1=t$ then $$(s-1)(s+1)=p{t}^2=p(s+1{)}^2\Rightarrow s-1=p(s+1)>s+1>s-1.$$ If $z=s-1=t$, then $$(s-1)(s+1)=p{t}^2=p(s-1{)}^2\Rightarrow s+1=p(s-1)\ge 2(s-1)>s+1,$$ for $s>3$.