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

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

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 $\displaystyle\lim_{\substack{a \to \infty \\ a \in A}} \frac{n(a)}{a} = \infty$?

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 $\{x \in X : f(x) \in B\}$.

Definition 4 (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 $x_\alpha > M$). 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 $x_\alpha > M$.

Remark 1. 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.

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

Current 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

Record	Kind	Assessment
	Result	Supported
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 →
	Trace	Reported
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 materialContext, examples, and computations

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 connectTyped relations and evidence flow

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

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

All nonnegative limits of normalized consecutive-prime gapsnumber theory
Decidability of positivity for linear recurrence sequencesnumber theory
Erdős Problem 152: isolated sums in Sidon setsnumber theory

How to cite

TheoremDB contributors, “Existence of an Infinite Set of Totient Values with Superlinear Minimal Preimages,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-51

This page as plain text: erdos-problem-51.md

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.

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

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance