Problem 979

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claims/: The 4 claim pages of Problem 979, one per claimant's result; the problem's standing derives from them.

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

Status. 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19 September 2025) and credits Erdős with the case $k=2$ [Er37b] and with a proof of the case $k=3$ that appears to be unpublished.

Source. erdosproblems.com/979, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #979, https://www.erdosproblems.com/979.

References.

• [Er37b] Erdős, Paul, On the Sum and Difference of Squares of Primes. J. London Math. Soc. (1937), 133-136.
• [Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244; the problem on p. 224. Library home: erdos_1965_recent_advances_current_problems_number_theory.

Formalization. Statement in formal-conjectures.

Current assessment

Open. The question asks whether $\limsup_n f_k(n)=\infty$ for every $k\ge2$. The case $k=2$ is Erdős's refereed theorem of 1937, an accepted partial claim on Erdős's 1937 page. The site's [Er37b] is Part I of the paper, J. London Math. Soc. 12 (1937), 133-136, which gives more than $n^{c/(\log\log n)^2}$ representations for infinitely many $n$; the linked library card erdos_1937_sum_difference_squares_primes is Part II, pp. 168-171 of the same volume, which sharpens the count to $n^{c/\log\log n}$.

The case $k=3$ carries three pending partial claims. Erdős wrote in [Er65b] that he could also prove it, in unpublished work (claim page). Kenta Kitamura published a Lean 4 proof on 17 August 2026, presented as independent of Erdős's argument (claim page); since that date the formal-conjectures statement file has named it as the formal proof of its variant erdos_979.variants.k3. This corpus has not built that Lean. Yukai Wang and Xu Zhang posted an unconditional proof on arXiv on 18 August 2026 (claim page); its second theorem, $\limsup F_4(n)\ge2$, settles no instance. None of the three is reviewed, refereed or formalized in this corpus, so all stay claimed.

No result settles any case $k\ge4$; Wang and Zhang's $\limsup F_4(n)\ge2$ is the only recorded result there. The derived standing is open (status open, claim none): the accepted claim is partial and the general question is unsettled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1937_sum_difference_squares_primes
• erdos_1937_sum_difference_squares_primes / theorem_section_1