# P2984: Additive Complements to the Primes with Sub-Log-Squared Density

- ID: `P2984`
- Reference: `erdos-problem-32`
- Page: https://theoremdb.org/statements/P2984
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$?

### Context

This problem belongs to additive number theory and concerns the minimal growth rate of additive complements to the set of prime numbers.

### Problem setup

- **Definition (A set $A \subseteq \mathbb{N}$).** A set $A \subseteq \mathbb{N}$ is an additive complement to the primes if for all sufficiently large natural numbers $n$, there exist a prime $p$ and an element $a \in A$ such that $n = p + a$.
- **Definition (For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = o(g(N))$ as $N \to \infty$).** For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = o(g(N))$ as $N \to \infty$ means that $\lim_{N \to \infty} \frac{f(N)}{g(N)} = 0$.
- **Definition (For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = O(g(N))$ as $N \to \infty$).** For functions $f, g : \mathbb{N} \to \mathbb{R}$, the notation $f(N) = O(g(N))$ as $N \to \infty$ means that there exists a constant $C > 0$ such that $|f(N)| \leq C|g(N)|$ for all sufficiently large $N$.
- **Remark.** This problem belongs to additive number theory and concerns the minimal growth rate of additive complements to the set of prime numbers.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 32 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$? [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 32 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 32 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 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 32 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 32 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 32: A set $A \subseteq \mathbb{N}$ is called an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and some element $a \in A$. Erdős proved that there exists an additive complement to the primes whose counting function satisfies $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$. Does there exist an additive complement $A$ to the primes such that $|A \cap \{1, \ldots, N\}| = o((\log N)^2)$ as $N \to \infty$? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-32`, 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 32, maintained status record. Erdős Problems record 32, checked 2026-08-01. Problem 32; status field and linked bibliography https://www.erdosproblems.com/32
   - Also cited at Problem 32; 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 Additive Complements to the Primes with Sub-Log-Squared Density: 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 32 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 Additive Complements to the Primes with Sub-Log-Squared Density: 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 32. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/32.lean:L69; theorem erdos_32; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/32.lean#L69
   - 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 Additive Complements to the Primes with Sub-Log-Squared Density: Supplies the pinned formal declaration whose human-readable editorial statement is published here.