# Can $x^4+y^4+1$ be a perfect power?

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.

• Numbers as in Question 2 are known as powerful numbers. Related question, which may suggest that the answer is positive but the least example is large. May 15, 2021 at 14:52
• It has solutions in rationals like $$(\frac{95800}{414560})^4+(\frac{217519}{414560})^4+1=(\frac{422481}{414560})^4.$$ May 15, 2021 at 16:31
• May 15, 2021 at 17:59
• Note that $x^4+y^4+1\equiv2{\rm\ or\ }3\bmod5$ (so is not a perfect square) unless $x$ and $y$ are both multiples of five. May 16, 2021 at 1:18
• Please restrict to one question per post. May 16, 2021 at 8:34

To answer question 2: $$346^4+36788^4+1=1831575032204939793=3^3\cdot19^3\cdot179^2\cdot17569^2.$$

• Cool. How did you calculate this? May 15, 2021 at 18:32
• I just ran through possibilities for $x$ and $y$, and then did trial division. Usually with trial division you stop when $p^2>n$, but if you're checking whether a number is powerful then you can stop when $p^5>n$, and then check whether $n$ is a perfect square or perfect cube. May 15, 2021 at 18:45
• One can use PARI/GP with a built-in function ispowerful(). May 15, 2021 at 18:50