[#P3038] Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift

A finite mathematical diagram showing distinct powers in several bases feeding one sum. — Distinct powers in several bases feeding one sum.

Problem. Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$?

1Context

This problem is a variant of a classical question about representing integers as sums of numbers with restricted digits in multiple bases. The case $k = 0$ was solved affirmatively by Boris Alexeev. The case with $k \neq 0$ and the additional coprimality condition was conjectured by Burr, Erdős, Graham, and Li.

2Problem setup

Definition 1 (For integers $d \geq 2$ and $k \geq 0$, the set $S(d,k)$ of sums of distinct powers (with exponent at least $k$) consists of all natural numbers of the form $\sum_{i \in s} d^i$ where $s$). For integers $d \geq 2$ and $k \geq 0$, the set $S(d,k)$ of sums of distinct powers (with exponent at least $k$) consists of all natural numbers of the form $\sum_{i \in s} d^i$ where $s$ is a finite set of integers satisfying $i \geq k$ for all $i \in s$. Equivalently, these are numbers whose base-$d$ representation contains only digits $0$ or $1$, with all nonzero digits occurring at positions corresponding to exponents at least $k$.

Definition 2 (For a finite set $D$ of natural numbers, $\gcd(D)$). For a finite set $D$ of natural numbers, $\gcd(D)$ denotes the greatest common divisor of all elements of $D$.

Definition 3 (The notation $\sum_{d \in D} S(d,k)$). The notation $\sum_{d \in D} S(d,k)$ denotes the Minkowski sum of the sets $S(d,k)$ over all $d \in D$, consisting of all sums $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$.

Definition 4 (The phrase 'every sufficiently large natural number'). The phrase 'every sufficiently large natural number' means that there exists some natural number $N$ such that the property holds for all natural numbers $n \geq N$.

Remark 1. This problem is a variant of a classical question about representing integers as sums of numbers with restricted digits in multiple bases. The case $k = 0$ was solved affirmatively by Boris Alexeev. The case with $k \neq 0$ and the additional coprimality condition was conjectured by Burr, Erdős, Graham, and Li.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$?

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 124 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$?^[1]

1Packet records

Record	Kind	Assessment
	Result	Supported
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 124 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$?^[1] Relevance to this problem For erdos problem 124, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 124 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 124 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$? Open full record →
	Trace	Reported
attempt · Route 1 Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$?^[1] Relevance to this problem For Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift, record erdos-problem-124-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 124: Let $k$ be a nonzero integer, and let $D$ be a finite set of integers, each at least $3$, such that \[\sum_{d \in D} \frac{1}{d-1} \geq 1\] and $\gcd(D) = 1$. For each $d \in D$ and each integer $k \geq 0$, let $S(d,k)$ denote the set of natural numbers that can be written as \[\sum_{i \in s} d^i\] for some finite set $s$ of integers with $i \geq k$ for all $i \in s$. Does every sufficiently large natural number belong to the set \[\sum_{d \in D} S(d,k),\] that is, can it be expressed as $\sum_{d \in D} x_d$ where $x_d \in S(d,k)$ for each $d \in D$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Related: Claim 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 124 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 124 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connectTyped relations and evidence flow

ProblemErdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-124

This page as plain text: erdos-problem-124.md

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 124, maintained status record. Erdős Problems record 124, checked 2026-08-01. Problem 124; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 124; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 124 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 124. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/124.lean:L58; lemma erdos124.ne_zero; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

[#P3038] Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance