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Claim. The case $k=3$ of Problem 979. Yukai Wang and Xu Zhang, Many Representations as Sums of Three Prime Cubes, arXiv:2608.19262v1 (18 August 2026), prove in their Theorem 1.1 that there are integers with arbitrarily many representations as sums of three cubes of primes, that is, $\limsup_n F_3(n)=\infty$, where $F_k(n)$ counts the unordered representations $n=p_1^k+\cdots+p_k^k$ by primes with repetitions allowed. The abstract names the classical Hecke equidistribution theorem for the CM Fermat cubic as the principal input, with standard estimates for primes in arithmetic progressions and elementary counting for the rest. Their Theorem 1.2 shows that infinitely many positive integers have two genuinely distinct unordered representations as sums of four fourth powers of distinct primes, so $\limsup_n F_4(n)\ge2$; the authors say the cubic argument does not extend to $k=4$.

Covers. The case $k=3$ (Theorem 1.1). Theorem 1.2 settles no instance of the problem, since it bounds $\limsup F_4$ only below by $2$; the cases $k\ge4$ remain open.

Depends on. No page of this wiki.

Standing. The preprint was linked in the problem's thread on 22 August 2026 and was not filed on the site's proof-claims tab. The site's label is OPEN, no reviewer is named, and the preprint is not refereed, so the claim stays claimed. The other claims of the case $k=3$ are Erdős's unpublished claim and Kitamura's Lean proof.