Find all positive integers $n$ such that $3^n+5^n = x^3$

Find all positive integers $n$ such that $3^n+5^n = x^3$ for some positive integer $x$.

One solution is $n = 1, x = 2$.

We have $1 < 3^n+5^n \leq 8^n$, so $1 < 3^n+5^n \leq 2^{3n}$. Thus $1 < x \leq 2^n$. How can we continue from here?

Solution 1:

Modulo $9$, if $n>1$, the equation simplifies to $5^n\equiv x^3 \pmod 9$. Because the cubes mod 9 are $0,1,8$, and $5^n \pmod 9$ repeats with period $\phi(9)=6$ with the pattern (starting at $n=0$) $1,5,7,8,4,2,\ldots$, we have that $n$ is a multiple of $3$ (if $n>1$).

However, working modulo $7$, where $3^n+5^n$ also has period $\phi(7)=6$, we see that $3^{3k}+5^{3k}$ is never a cube mod 7. (The cubes are 0,1,6, and $3^n+5^n$ repeats $2,1,6,5,6,1,\ldots,$ but neither $2$ nor $5$ are cubes).

Therefore, if $n>1$, combining both results, we see there are no solutions. However, $n=1, x=2$ is a solution. Hence, it is the only solution.