[#P3000] Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages

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Problem. Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?

A finite mathematical diagram showing totient values connected to their least preimages. — Totient values connected to their least preimages.

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Definitions and notation

1Context

This problem originates from Paul Erdős and concerns the distribution of values of Euler's totient function and the size of their minimal preimages. The totient function is surjective onto the even integers greater than 2, but the structure of its fibers $\varphi^{-1}(\{a\})$ and the growth of minimal elements within them remains subtle.

2Problem setup

Definition 1 (Euler's totient function $\varphi(n)$). Euler's totient function $\varphi(n)$ is the number of positive integers up to $n$ that are relatively prime to $n$.

Definition 2 (For a set $S \subseteq \mathbb{N}$, an element $m \in S$). For a set $S \subseteq \mathbb{N}$, an element $m \in S$ is the least element of $S$, denoted $\min S$, if $m \leq s$ for all $s \in S$.

Definition 3 (For a function $f : X \to Y$ and a subset $B \subseteq Y$, the preimage $f^{-1}(B)$). For a function $f : X \to Y$ and a subset $B \subseteq Y$, the preimage $f^{-1}(B)$ is the set $\{x \in X : f(x) \in B\}$.

Remark 1. A sequence (or net) of real numbers $x_\alpha$ tends to infinity, written $\lim x_\alpha = \infty$, if for every real number $M$ there exists an index beyond which all terms satisfy $x_\alpha > M$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?

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 51 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?^[1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 51 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?	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 51 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?^[1] Relevance to this problem For erdos problem 51, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 51 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let. 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 51 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 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?	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 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?^[1] Relevance to this problem For Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages, record erdos-problem-51-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 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. 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 51: Let $\varphi$ denote Euler's totient function, which counts the number of integers in $\{1, 2, \ldots, n\}$ that are coprime to $n$. For a positive integer $a$, let $\varphi^{-1}(\{a\})$ denote the set of all positive integers $n$ such that $\varphi(n) = a$, and when this set is nonempty, let $n_a$ denote its least element. Does there exist an infinite set $A$ of positive integers such that $\varphi^{-1}(\{a\})$ is nonempty for every $a \in A$, and the ratio $n_a / a$ tends to infinity as $a$ grows without bound through elements of $A$? That is, does there exist an infinite set $A \subseteq \mathbb{N}$ and a function $n : A \to \mathbb{N}$ such that for every $a \in A$, the integer $n(a)$ is the least element of $\varphi^{-1}(\{a\})$, and $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$? 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 51 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 51 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.

ProblemExistence of an Infinite Set of Totient Values with Superlinear Minimal Preimages

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

“Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages.” TheoremDB. P3000. Problem statement; statement text SHA-256 149f20596cbaf881d6f07fc1d070e774c7f80de0c664abcfc83e64af187e4c35. https://theoremdb.org/statement/?ref=P3000

BibTeX

@misc{theoremdb-problem-149f20596cbaf881d6f07fc1d070e774c7f80de0c664abcfc83e64af187e4c35,
  title = {{Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 149f20596cbaf881d6f07fc1d070e774c7f80de0c664abcfc83e64af187e4c35},
  url = {https://theoremdb.org/statement/?ref=P3000}
}

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.

1References

Packet source. Erdős Problems database, Problem 51, maintained status record. Erdős Problems record 51, checked 2026-08-01. Problem 51; 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 51; 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 Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages: 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 51 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 Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 51. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/51.lean:L36; theorem erdos_51; 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 Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages: 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.

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[#P3000] Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages

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3What counts as a solution

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1Context

2Problem setup

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Recent contributions

2See also

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Background

Definitions

What counts as a solution

What's known

Formal statement

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