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

- ID: `P3000`
- Reference: `erdos-problem-51`
- Page: https://theoremdb.org/statements/P3000
- Record maturity: Reviewed problem with recorded work

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

### Context

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.

### Problem setup

- **Definition (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 (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 (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 (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.** 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.

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

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

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

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-51`, 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 51, maintained status record. Erdős Problems record 51, checked 2026-08-01. Problem 51; status field and linked bibliography https://www.erdosproblems.com/51
   - 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
   - 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 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 51 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 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 51. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/51.lean:L36; theorem erdos_51; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/51.lean#L36
   - 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 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.