Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$, where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n) = \infty$?

Erdős [Er37b] proved that if $f_2(n)$ counts the number of solutions to $n = p_1^2 + p_2^2$, where $p_1$ and $p_2$ are prime numbers, then $\limsup f_2(n) = \infty$.

[Er37b] Erdős, Paul, On the Sum and Difference of Squares of Primes. J. London Math. Soc. (1937), 133--136.