Note that $$(1^2+1)(2^2+1)=10=3^2+1\ \ \mbox{and}\ \ (1^4+2^4)(5^4+6^4)=8^4+13^4.$$ Today I tried to find positive integers $x,y,z$ satisfying $(x^4+1)(y^4+1)=z^4+1$ but failed. In view of this situation, I pose the following two questions.
Question 1. Let $n>2$ be an integer. Does the equation $$(x^n+1)(y^n+1)=z^2+1,\ \ \mbox{i.e., }\ \ x^n+y^n+x^ny^n=z^2\tag{1}$$ have positive integer solutions?
Remark 1. I have checked $(1)$ for $n=3,4,5$ and $x,y=1,\ldots,3000$, and I have not found any positive integer solution of $(1)$. Fermat ever proved that $x^4+y^4=z^2$ has no positive integer solution.
It is easy to see that $$(x^2-1)((x+1)^2-1)=(x(x+1)-1)^2-1.$$
Question 2. Let $n>1$ be an integer. Does the equation $$(x^n-1)(y^n-1)=z^2+1,\ \ \mbox{i.e., }\ \ x^ny^n-x^n-y^n=z^2\tag{2}$$ have positive integer solutions?
Remark 2. I have checked $(2)$ for $n=2,3,4,5$ and $x,y=1,\ldots,3000$, and I have not found any positive integer solution of $(2)$.
I conjecture that the equation $(1)$ has no positive integer solution for each integer $n>2$, and that the equation $(2)$ has no positive integer solution for each integer $n>1$. It seems not difficult to solve the equation $(2)$ for $n=2$.
Your comments are welcome!
