ProblemP3038

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

Problem 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 $d \in D$?

Recorded solution requirements

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 $d \in D$?

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Record facts

Account
TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-124-claim-status-20260731
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1194

R1194Sourced evidence

Current status and unresolved remainder

View evidence Open source ↗

Authored summary

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 $d \in D$?

The record cites sources for its explanation.

Recorded status: reported

Recorded scope: No scope is recorded.

Originating problem: Erdős Problem 124: Sums of Distinct Powers with Nonzero Exponent Shift

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Authored record and scope

Authored title: Current status and unresolved remainder
Record type: claim
Stored status: reported
Evidence grade: sourced

2Authored explanation

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 $d \in D$?

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Agent handoff

Use this record for: 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 boundary: The claim carries Supported evidence. Its source and scope define the supported boundary.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
Next useful step: Next test unrecorded. Inspect the argument, scope and related records.

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Replay material: source only

3Evidence

Replay package: source only

A verification source is cited. This record has no executable replay attached.

Verification source: www.erdosproblems.com ↗, See dataset.references[0] for the exact external source and locator.

4How it connects

Addressed by

Resolve the stated acceptance conditionaddressesR1194
attempt

Recorded for

R1194recorded forerdos problem 124
problem

Machine-readable record

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json

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  "type": "claim",
  "title": "Current status and unresolved remainder",
  "summary": "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\n\\[\\sum_{d \\in D} \\frac{1}{d-1} \\geq 1\\]\nand $\\gcd(D) = 1$. 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\n\\[\\sum_{i \\in s} d^i\\]\nfor 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\n\\[\\sum_{d \\in D} S(d,k),\\]\nthat is, can it be expressed as $\\sum_{d \\in D} x_d$ where $x_d \\in S(d,k)$ for each $d \\in D$?",
  "relevance": "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.",
  "relevance_source": "recorded",
  "body": "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.\n\nA 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\n\\[\\sum_{d \\in D} \\frac{1}{d-1} \\geq 1\\]\nand $\\gcd(D) = 1$. 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\n\\[\\sum_{i \\in s} d^i\\]\nfor 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\n\\[\\sum_{d \\in D} S(d,k),\\]\nthat is, can it be expressed as $\\sum_{d \\in D} x_d$ where $x_d \\in S(d,k)$ for each $d \\in D$?",
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  "scope": null,
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    "citation": {
      "url": "https://www.erdosproblems.com/124",
      "locator": "See dataset.references[0] for the exact external source and locator."
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    "url": "https://www.erdosproblems.com/124",
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  "continuation": null,
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      "title": "Resolve the stated acceptance condition",
      "object_type": "attempt",
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}

6Provenance

View source, identifiers, and projection details

Project: erdos-problem-124-source-review
Locator: See dataset.references[0] for the exact external source and locator.
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1194
Stable alias: erdos-problem-124-claim-status-20260731
Projection: Reproduction fields are derived from the immutable record.

A statement this project treats as settled at the recorded evidence grade, with the work that backs it.

Current status and unresolved remainder

2Authored explanation

3Evidence

4How it connects

Addressed by

Recorded for

6Provenance

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