# P2962: Erdős's Problem on Sums of a Prime and Powers of 2

- ID: `P2962`
- Reference: `erdos-problem-10`
- Page: https://theoremdb.org/statements/P2962
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$?

### Context

This problem was posed by Paul Erdős. It asks whether every integer greater than 1 can be expressed as a prime plus a bounded number of powers of 2. Related results include: Gallagher proved that for any $\epsilon > 0$ there exists $k(\epsilon)$ such that $\mathcal{S}_{k(\epsilon)}$ has lower density at least $1 - \epsilon$; Granville and Soundararajan conjectured that 3 powers of 2 suffice for all odd integers greater than 1 and 4 powers suffice for all even integers greater than 0; Grechuk found that $1117175146$ is not the sum of a prime and at most 3 powers of 2; and there are infinitely many even integers not representable as a prime plus 2 powers of 2.

### Problem setup

- **Definition (For a non-negative integer $k$, the set $\mathcal{S}_k$ consists of all natural numbers of the form $p + \sum_{i=1}^{m} 2^{a_i}$ where $p$).** For a non-negative integer $k$, the set $\mathcal{S}_k$ consists of all natural numbers of the form $p + \sum_{i=1}^{m} 2^{a_i}$ where $p$ is prime, $0 \leq m \leq k$, and $a_1, \ldots, a_m$ are non-negative integers.
- **Remark.** This problem was posed by Paul Erdős. It asks whether every integer greater than 1 can be expressed as a prime plus a bounded number of powers of 2. Related results include: Gallagher proved that for any $\epsilon > 0$ there exists $k(\epsilon)$ such that $\mathcal{S}_{k(\epsilon)}$ has lower density at least $1 - \epsilon$; Granville and Soundararajan conjectured that 3 powers of 2 suffice for all odd integers greater than 1 and 4 powers suffice for all even integers greater than 0; Grechuk found that $1117175146$ is not the sum of a prime and at most 3 powers of 2; and there are infinitely many even integers not representable as a prime plus 2 powers of 2.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$? [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Current status and unresolved remainder).** OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 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 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 10 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 10 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.

### Open directions

- **Route 1** (reported): Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 10: For a non-negative integer $k$, let $\mathcal{S}_k$ denote the set of natural numbers that can be written as $p + 2^{a_1} + 2^{a_2} + \cdots + 2^{a_m}$ where $p$ is a prime number, $m \leq k$, and $a_1, a_2, \ldots, a_m$ are non-negative integers (with the empty sum of powers of $2$ equal to $0$). Does there exist some $k$ such that $\mathcal{S}_k = \mathbb{N} \setminus \{0, 1\}$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-10`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 10, maintained status record. Erdős Problems record 10, checked 2026-08-01. Problem 10; status field and linked bibliography https://www.erdosproblems.com/10
   - Also cited at Problem 10; 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - For Erdős's Problem on Sums of a Prime and Powers of 2: Records the current open status and links the literature attached to this exact numbered problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 10 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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's Problem on Sums of a Prime and Powers of 2: Pins the maintained database snapshot used for this release's dated status decision.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 10. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/10.lean:L40; theorem erdos_10; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/10.lean#L40
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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's Problem on Sums of a Prime and Powers of 2: Supplies the pinned formal declaration whose human-readable editorial statement is published here.