Is the number of coprime solutions of "n-Pythagorean" equation $x_0^n = x_1^n + \dots + x_n^n$ finite or infinite for $n>3$ ?
Equivalently, does the hypersurface $\{X_0^n = X_1^n + \dots + X_n^n\} \subset \Bbb Q P^{n+1}$ contain infinite number of rational points?
I know the Ramanujan's infinite series for $n=3$
Using elliptic curves, it can be shown there are infinitely many rational points for the case $n=4$ and $n=5$ with the constraint,
$$x_0+x_1+x_2+\dots +x_n = 0\hskip1.3in$$
$$x_0^4 = x_1^4+x_2^4+ \dots +x_4^4,\quad \text{by Jacobi-Madden}$$
$$x_0^5 = x_1^5+x_2^5+ \dots +x_5^5,\quad \text{by Lander}\hskip0.6in$$
Without the constraint, there is also,
$$x_0^7 = x_1^7+x_2^7+ \dots +x_\color{red}8^7,\quad \text{by Choudhry}\hskip0.4in$$
a polynomial identity with coefficients as high as $10^{1180}$, though this has an extra term re your post.
