Problem 979

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 $\limsup f_k(n)=\infty$ ?

Source: erdosproblems.com/979

Number theory

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Open

No claim settles this problem.

Open. The site labels the problem OPEN (the page was last edited on 19 September 2025) and credits Erdős with the case $k=2$ [Er37b] and with a proof of the case $k=3$ that appears to be unpublished.

Problem 979

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