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

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 $N \to \infty$?

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Definitions and notation

1Context

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

2Definitions

Definition 1 (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 2 (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 3 (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 $N$.

3What 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 $N \to \infty$?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (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 $N \to \infty$?^[1]

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Record	Kind	Assessment
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 $N \to \infty$?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For erdos problem 32, pins the dated research frontier: 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. 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 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 $N \to \infty$? Open full record →
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 $N \to \infty$?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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] Relevance to this problem For Additive Complements to the Primes with Sub-Log-Squared Density, record erdos-problem-32-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 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 $a \in A$. 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 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 $N \to \infty$? 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 →
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Notes and companion material

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.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemAdditive Complements to the Primes with Sub-Log-Squared Density

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Merging of orbits under adding the product of nonzero digitsnumber theory
Sharp multipliers for balanced binary productsnumber theory
Prime exceptions to connectivity of the Markoff graphnumber theory

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Plain text

“Additive Complements to the Primes with Sub-Log-Squared Density.” TheoremDB. P2984. Problem statement; statement text SHA-256 f7de131559fce225da3f5e4ff197a3b84c22fecf2b834cd4f116363e7e631df9. https://theoremdb.org/statement/?ref=P2984

BibTeX

@misc{theoremdb-problem-f7de131559fce225da3f5e4ff197a3b84c22fecf2b834cd4f116363e7e631df9,
  title = {{Additive Complements to the Primes with Sub-Log-Squared Density}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f7de131559fce225da3f5e4ff197a3b84c22fecf2b834cd4f116363e7e631df9},
  url = {https://theoremdb.org/statement/?ref=P2984}
}

Plain text: Built Markdown snapshot

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

1Lean verification

1References

Packet source. Erdős Problems database, Problem 32, maintained status record. Erdős Problems record 32, checked 2026-08-01. Problem 32; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 32 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 Additive Complements to the Primes with Sub-Log-Squared Density: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 32. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/32.lean:L69; theorem erdos_32; 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 Additive Complements to the Primes with Sub-Log-Squared Density: 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.

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

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