---

Claim. S. A. Burr, P. Erdős, R. L. Graham and W. Wen-Ching Li, Complete sequences of sets of integer powers, Acta Arith. 77 (1996), no. 2, 133--138 (the source card is burr_1996_complete_sequences_sets_integer_powers). For a set $A$ of integers greater than $1$, $\mathrm{Pow}(A;s)$ is the set of powers $a^j$ with $a\in A$ and $j\ge s$. Section 3 states the largest integer that is not a sum of distinct elements of $\mathrm{Pow}(A;1)$ for four sets $A$: $581$ for $\{3,4,7\}$, which the authors derive from the Mignotte--Waldschmidt lower bound on $|3^p-4^q|$; $111$ for $\{3,5,7,13\}$; $16$ for $\{3,6,7,13,21\}$; and $78$ for $\{3,4,5\}$, the last offered as an example of what the authors call their "limited computational experience" with sets whose reciprocal sum exceeds $1$. The paper prints none of the computations. Each of the four sets has $\sum_{a\in A}1/(a-1)\ge1$ (with equality for the first three) and greatest common divisor $1$, so each is an admissible tuple of [[problems/diophantine_problems/E0124/_index|Problem 124]], and a sum of distinct elements of $\mathrm{Pow}(A;1)$ is a sum $\sum_ic_ia_i$ with $a_i\in P(d_i,1)$, one term per base.

Covers. The second question at $k=1$ for the four tuples $\{3,4,7\}$, $\{3,5,7,13\}$, $\{3,6,7,13,21\}$ and $\{3,4,5\}$: yes, every integer above the stated value is represented. Not covered: exponents $k\ge2$ and every other tuple, and the first question. The site's commentary says the authors proved the conjecture for $\{3,4,7\}$; the result covers only $k=1$.

Depends on. No page of this wiki.

Acceptance. Refereed: Acta Arithmetica 77 (1996), no. 2, 133--138. The site's commentary (page last edited 1 December 2025) credits the $\{3,4,7\}$ case to this paper but labels the problem OPEN, so no reviewed evidence is listed. The formal-conjectures statement file for the problem states the $\{3,4,7\}$ case at $k=1$ as erdos124.ne_zero_three_four_seven, marked research solved and credited to this paper, with no formal proof. The computations were not reconstructed in this corpus.