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Claim. The case $k=2$ of Problem 979: $\limsup_n f_2(n)=\infty$, where $f_2(n)$ counts the solutions of $n=p_1^2+p_2^2$ in primes. P. Erdős, On the sum and difference of squares of primes, J. London Math. Soc. 12 (1937), 133-136, cited as [Er37b] on the problem page, proves that for infinitely many $n$ the equation $n=p^2+q^2$ has more than $n^{c/(\log\log n)^2}$ solutions in primes, for an absolute $c>0$. The sequel, On the sum and difference of squares of primes, II, J. London Math. Soc. 12 (1937), 168-171 (library home erdos_1937_sum_difference_squares_primes), sharpens the count to more than $n^{c/\log\log n}$ solutions for infinitely many $n$, by Brun's method, used through the Brun--Titchmarsh bound for primes in arithmetic progressions.

Covers. The case $k=2$, with a lower bound on the size of $f_2(n)$ along a sequence of $n$. The cases $k\ge3$ are not touched.

Depends on. No page of this wiki.

Acceptance. Refereed: both parts appeared in the Journal of the London Mathematical Society, volume 12 (1937), Part I in issue 46 (April 1937) and Part II in issue 47 (July 1937). Not reviewed under the corpus's rule: the site credits [Er37b] with the case $k=2$, but it labels the problem OPEN, so that remark is commentary on an open problem and not an acceptance that settles it.