Recall that a perfect power has the form $x^m$ with $m,x\in\{2,3,\ldots\}$. Motivated by Fermat's result that the equation $x^4+y^4=z^2$ has no positive integer solution, here I ask the following question.

**Question 1.** Can $x^4+y^4+1$ with $x,y\in\mathbb N=\{0,1,2,\ldots\}$ be a perfect power?

Based on my computation, I conjecture that $x^4+y^4+1$ with $x,y\in\mathbb N$ can never be a perfect power.

**Question 2.** Can we find $x,y\in\mathbb N$ such that $x^4+y^4+1=\prod_{i=1}^kp_i^{a_i}$ for some $a_1,\ldots,a_k\in\{2,3,\ldots\}$ and distinct primes $p_1,\ldots,p_k$?

Via a computer I find no $x^4+y^4+1$ with $x,y\in\{0,1,\ldots,8000\}$ of the form $\prod_{i=1}^kp_i^{a_i}$ with $p_1,\ldots,p_k$ distinct primes and $a_1,\ldots,a_k\in\{2,3,\ldots\}$. Perhaps, Question 2 has a negative answer. Of course, a negative answer to Question 2 implies a negative answer to Question 1.

Your comments are welcome!

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