Problem Statement
Let $k\geq 2$, and let $f_k(n)$ count the number of solutions to\[n=p_1^k+\cdots+p_k^k,\]where the $p_i$ are prime numbers. Is it true that $\limsup f_k(n)=\infty$?
Categories:
Number Theory
Progress
Erdős [Er37b] proved this is true when $k=2$, and also when $k=3$ (but this proof appears to be unpublished).Source: erdosproblems.com/979 | Last verified: January 19, 2026