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Claim. The case $k=3$ of Problem 979: $\limsup_n f_3(n)=\infty$, where $f_3(n)$ counts the solutions of $n=p_1^3+p_2^3+p_3^3$ in primes. In his 1965 survey, Erdős, Paul, Some recent advances and current problems in number theory, Lectures on Modern Mathematics, Vol. III, Wiley (1965), 196--244 (cited as [Er65b] on the problem page; library home erdos_1965_recent_advances_current_problems_number_theory), Erdős states on p. 224, after the case $k=2$, that he can also prove the case $k=3$, that the proof is unpublished, and that it seems to need special properties of the primes. The site's curator quoted the sentence in the problem's thread (second link above). Wang and Zhang report that Erdős and Graham, Old and New Problems and Results in Combinatorial Number Theory (1980), p. 47, repeat the assertion without a proof.

Covers. The case $k=3$. The cases $k\ge4$ are not touched.

Depends on. No page of this wiki.

Standing. Claimed, not accepted: no proof by Erdős has appeared, so there is nothing to review or referee, and the site describes the proof as unpublished. The page is dated to the first day of 1965, the survey's year. Later proofs of the case $k=3$ are Kitamura's Lean proof and Wang and Zhang's preprint; neither reconstructs Erdős's argument.