Problem 124

For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$ . Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that

\sum_{1\leq i\leq r}\frac{1}{d_r-1}\geq 1.

Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i\in P(d_i,0)$ ?

If we further have $\mathrm{gcd}(d_1,\ldots,d_r)=1$ then, for any $k\geq 1$ , can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i\in P(d_i,k)$ ?

Second statement

For any $d\geq 1$ and $k\geq 0$ let $P(d,k)$ be the set of integers which are the sum of distinct powers $d^i$ with $i\geq k$ . Let $3\leq d_1<d_2<\cdots <d_r$ be integers such that

\sum_{1\leq i\leq r}\frac{1}{d_i-1}\geq 1.

Can all sufficiently large integers be written as a sum of the shape $\sum_i c_ia_i$ where $c_i\in \{0,1\}$ and $a_i\in P(d_i,0)$ ?

Source: erdosproblems.com/124

Number theory
Base representations
Complete sequences

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label is OPEN (page last edited 1 December 2025); the claim pages are named under Current assessment.

Notes

As printed, the condition is $\sum_{1\le i\le r}1/(d_r-1)\ge1$ , that is $r/(d_r-1)\ge1$ . Since $d_r\ge d_1+r-1\ge r+2$ , no admissible tuple satisfies it, and, read as the site words it, both questions hold vacuously. The misprint entered with the site's rewrite of 1 December 2025. The earlier text asked a single question under $\sum_{1\le i\le k}1/(d_i-1)\ge1$ . The rewrite renamed $k$ to $r$ and added the gcd-conditioned second question, after Aristotle's proof of the earlier statement (thread posts of 29 and 30 November 2025). The site's commentary, [BEGL96] and the formal-conjectures statement all read the condition as $\sum_i1/(d_i-1)\ge1$ , and a thread post of 13 May 2026 points out the misprint. The change replaces $d_r$ by $d_i$ in the summand. The first question has a claimed affirmative answer; the second is open apart from the instances the claim pages settle.

BEGL96: Burr, S. A. and Erdős, P. and Graham, R. L. and Li, W. Wen-Ching, Complete sequences of sets of integer powers. Acta Arith. (1996), 133-138.
Er97: Erdős, Paul, Problems in number theory. New Zealand J. Math. (1997), 155-160.
Er97e: Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
Me04: Melfi, Giuseppe, On certain positive integer sequences. Riv. Mat. Univ. Parma (7) (2004), 253-260.

Problem 124

Second statement

Status

Notes

Proof claims

Discussion