[#P3040] Maximal Number of Distinct Prime Factors in Pairwise Sums Products

Problem. Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $|A| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ as $n \to \infty$?

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Definitions and notation

1Context

This problem concerns the growth rate of a function defined through extremal combinatorial number theory. Erdős and Turán established in their first joint paper that $\log n \ll f(n) \ll \frac{n}{\log n}$, and it remains open whether the lower bound can be strengthened to $f(n)/\log n \to \infty$.

2Problem setup

Definition 1 (For a finite set $A \subseteq \mathbb{N}$, the off-diagonal product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$). For a finite set $A \subseteq \mathbb{N}$, the off-diagonal product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ is the product of all sums $a + b$ where $a$ and $b$ are distinct elements of $A$.

Remark 1. For a positive integer $k$, we write $\omega(k)$ for the number of distinct prime factors of $k$.

Remark 2. For functions $g, h: \mathbb{N} \to \mathbb{R}$, we say $g(n) = O(h(n))$ if there exists $C > 0$ and $N \in \mathbb{N}$ such that $|g(n)| \leq C|h(n)|$ for all $n \geq N$.

Remark 3. For functions $g, h: \mathbb{N} \to \mathbb{R}$, we say $g(n) = o(h(n))$ if $\lim_{n \to \infty} g(n)/h(n) = 0$.

Remark 4. We say $g(n) \ll h(n)$ if $g(n) = O(h(n))$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $|A| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ 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 126 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $|A| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ 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 126 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ 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 126 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ as $n \to \infty$?^[1] Relevance to this problem For erdos problem 126, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 126 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126. 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 126 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 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ as $n \to \infty$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ 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 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ as $n \to \infty$?^[1] Relevance to this problem For Maximal Number of Distinct Prime Factors in Pairwise Sums Products, record erdos-problem-126-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 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. 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 126: For each positive integer $n$, let $f(n)$ denote the maximum integer $m$ such that for every subset $A \subseteq \mathbb{N}$ with $\|A\| = n$, the product $\prod_{\substack{a,b \in A \\ a \neq b}} (a + b)$ has at least $m$ distinct prime factors. Is it true that $\displaystyle\frac{f(n)}{\log n} \to \infty$ 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 126 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 126 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.

ProblemMaximal Number of Distinct Prime Factors in Pairwise Sums Products

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

“Maximal Number of Distinct Prime Factors in Pairwise Sums Products.” TheoremDB. P3040. Problem statement; statement text SHA-256 f296d8a555fb74441d1c54cba79b4daa62b900ed0abcdd24207869c0046b11a2. https://theoremdb.org/statement/?ref=P3040

BibTeX

@misc{theoremdb-problem-f296d8a555fb74441d1c54cba79b4daa62b900ed0abcdd24207869c0046b11a2,
  title = {{Maximal Number of Distinct Prime Factors in Pairwise Sums Products}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f296d8a555fb74441d1c54cba79b4daa62b900ed0abcdd24207869c0046b11a2},
  url = {https://theoremdb.org/statement/?ref=P3040}
}

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 126, maintained status record. Erdős Problems record 126, checked 2026-08-01. Problem 126; 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 126; 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 Maximal Number of Distinct Prime Factors in Pairwise Sums Products: 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 126 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 Maximal Number of Distinct Prime Factors in Pairwise Sums Products: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 126. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/126.lean:L41; theorem erdos_126; 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 Maximal Number of Distinct Prime Factors in Pairwise Sums Products: 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.

[#P3040] Maximal Number of Distinct Prime Factors in Pairwise Sums Products

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