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Claim. Theorem 2 (p. 2) is printed as "For every $0<\epsilon\le1$, and every integer $k\ge1$, there exists an integer $N=N(\epsilon,k)$ such that for every prime $p\ge2$, and every integer $0\le a\le p-1$, there exist integers $x_1,\ldots,x_N$ such that $1\le x_i\le p^\epsilon$, and $a\equiv\frac1{x_1^k}+\cdots+\frac1{x_N^k}\pmod p$" (Integers 4 (2004), #A20, p. 2, as in arXiv v2; arXiv v1 and the copy on the author's papers page print $x_1,\ldots,x_n$). With $k=1$ this is the question of Problem 1180 for $0<\epsilon\le1$, with $C_\epsilon=N(\epsilon,1)$ and repetition allowed, read with at most $N$ summands. As printed, with exactly $N$ summands, the statement holds only for large $p$: for a prime with $p^\epsilon<2$, and for $p=2$ at every $\epsilon\le1$, the only admissible $x_i$ is $1$, and a sum of exactly $N$ of them is the single residue $N\bmod p$. The proof covers the finitely many small primes by enlarging $N$, which works when $N$ bounds the number of summands, and the one-line completion of Shparlinski's and Glibichuk's pages (a residue $a$ is the sum of $a$ copies of $1=1^{-1}$) gives them; Glibichuk's introduction restates the theorem for sufficiently large $p$. For $\epsilon>1$ the set $\{n^{-1}:n\le p^\epsilon\}$ contains the set for $\epsilon=1$, so $C_\epsilon=C_1$ serves (the monotonicity remark of the problem page). The resulting $N$ is not explicit. The theorem is compiled on the result page theorem_2; the digest is on the card croot_2004_sums_reciprocal_powers_modulo_prime.

Argument, in outline. The Bourgain--Katz--Tao sum-product estimate is applied to the sums of inverse $k$th powers of small primes, and an exponential-sum lemma then writes every residue as $t_1t_2+t_3t_4+\cdots+t_{15}t_{16}$ with the $t_i$ in a set $T$ of sums of inverse $k$th powers, which the paper counts as at most $16h^2$ terms $1/(qq')^k$ (pp. 2--5). The outline records the structure only; no step is checked. The paper's introduction states the Erdős--Graham question in the monograph's form and credits its first affirmative answer to Shparlinski, through Karatsuba's result in the simplified form of Friedlander and Iwaniec; that answer is on Shparlinski's page.

Acceptance. Refereed: Ernie Croot, Sums of the form $1/x_1^k+\cdots+1/x_n^k$ modulo a prime, Integers 4 (2004), Paper A20, per the journal's volume contents and the Zenodo deposit of the paper; the journal's text, deposited at Zenodo under CC-BY-4.0, agrees with arXiv:math/0403360v2 (21 October 2004); arXiv v1 and the copy on the author's papers page differ from it in the introduction and Theorem 2. The page is named by the first arXiv posting, v1 of 22 March 2004, under the title "Reciprocal power sums modulo a prime". Not reviewed: the site's curator names Croot, in the problem's commentary, only for the earlier bound of $(\log p)^{3+o(1)}$ summands from his 1999 paper, not for this theorem, so no reviewed evidence is listed. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki; the theorem is proved in the refereed paper linked above.