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On a problem on a generalization of Euler's totient function

This paper proves the conjecture by Büyükaşık et al. that the set of integers $k \geq 1$ for which $\varphi_1(n)$ divides $\varphi_k(n)$ for all $n$ is exactly $\{1, 3, 15\}$ , utilizing a proof strategy developed through extensive interactions with GPT-5.5 Pro.

Original authors: John M. Campbell

Published 2026-06-02

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: John M. Campbell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite box of numbered tiles, from 1 to $n$ . Some of these tiles are "special" because they don't share any common factors with the number $n$ (other than 1). Mathematicians call these "coprime" numbers.

For centuries, we've had a famous rule called Euler's Totient Function ( $\phi$ ). It simply counts how many special tiles are in the box. If you have 10 tiles, the special ones are 1, 3, 7, and 9, so the answer is 4.

The New Game: Summing Powers

In this paper, the author (John Campbell) and some colleagues are playing a more complex game with these same special tiles. Instead of just counting them, they ask: "What happens if we raise each special tile to a power $k$ and add them all up?"

If $k=0$ , we are just counting (the original rule).
If $k=1$ , we add the numbers: $1 + 3 + 7 + 9$ .
If $k=2$ , we square them: $1^2 + 3^2 + 7^2 + 9^2$ .

This sum is called $\phi_k(n)$ .

The Big Question: The "Divisibility Club"

The researchers asked a very specific question about these sums. They wanted to find a special "club" of numbers (let's call the club $D_1$ ).

The Rule for the Club:
A number $k$ gets into the club if, for every single possible box size $n$ , the sum of the special tiles raised to the power of 1 ( $\phi_1$ ) always divides evenly into the sum of the special tiles raised to the power of $k$ ( $\phi_k$ ).

Think of it like this: If you have a pile of cookies ( $\phi_1$ ), can you always split a larger pile of cookies ( $\phi_k$ ) into exact, whole-number groups of the first pile, no matter how many cookies are in the original pile?

The Mystery

Previous mathematicians (Büyükaşık et al.) did some heavy lifting. They proved that the club is finite (it doesn't go on forever) and they did some computer calculations that suggested the club only has three members: 1, 3, and 15.

They said, "We're 99% sure these are the only ones, but we can't prove it."

The Solution: A Human-AI Team-Up

John Campbell stepped in to solve the mystery. He used a mix of deep mathematical tools (like Bernoulli numbers, which are like secret codes hidden in number patterns) and a very unique partner: GPT-5.5 Pro, an advanced AI.

How they solved it:

The Detective Work: Campbell didn't just guess. He used a logical "trap." He showed that if a number $k$ is in the club, it must follow very strict rules.
The "Odd" Requirement: First, he proved that any number in the club must be an odd number. Even numbers were kicked out immediately.
The "Prime" Filter: He then used a series of logical tests involving prime numbers (numbers divisible only by 1 and themselves). He showed that if a number is too big or has the wrong "shape," it fails the divisibility test for some specific box size $n$ .
The Elimination:
- He proved that if $k$ is in the club, $k+1$ must be a power of 2 (like 2, 4, 8, 16, 32...).
- This narrowed the list down to numbers like 3, 7, 15, 31, 63, etc.
- Then, he used more complex patterns to show that 7, 31, 63, and all the bigger numbers fail the test.
- Only 1, 3, and 15 survived the gauntlet.

The Result

The paper concludes that the "Divisibility Club" ( $D_1$ ) contains exactly the numbers {1, 3, 15}. No more, no less.

A Note on the "AI"

The author is very transparent about how he did this. He admits that he had extensive conversations with an AI (GPT-5.5 Pro) to help brainstorm the proof steps. However, he emphasizes that the AI was just a tool. The author did all the heavy lifting of checking, correcting, and verifying every single step. He takes full responsibility for the math, ensuring the final proof is solid and human-verified.

In short: The paper solves a long-standing puzzle about a specific type of number pattern, proving that only three specific numbers have a unique "divisibility superpower," using a blend of traditional math and modern AI assistance.

Technical Summary of "On a problem on a generalization of Euler's totient function"

Problem Statement
This paper addresses a conjecture regarding a family of generalizations of Euler's totient function, $\phi(n)$ , introduced by B¨uy¨uka¸sik et al. (2024). The generalized function is defined as:
$\phi_k(n) = \sum_{\substack{1 \le a \le n \\ (a,n)=1}} a^k$
where $\phi_0(n) = \phi(n)$ . The authors define the set $D_s$ for a fixed non-negative integer $s$ as:
$D_s = \{k \ge s : \forall n \ge 1, \phi_s(n) \mid \phi_k(n)\}$
B¨uy¨uka¸sik et al. proved that $D_s$ is finite for any $s \ge 0$ and conjectured that $D_1 = \{1, 3, 15\}$ . While they provided computational evidence, the conjecture remained open. The primary objective of this work is to provide a rigorous proof that $D_1 = \{1, 3, 15\}$ .

Methodology
The proof proceeds by establishing the reverse inclusion $D_1 \subseteq \{1, 3, 15\}$ , as the inclusion $\{1, 3, 15\} \subseteq D_1$ was already established by B¨uy¨uka¸sik et al. The methodology relies on a synthesis of classical number theory tools and a novel structural argument involving Bernoulli numbers and polynomials.

Reduction to Odd Integers: The authors first demonstrate that if $k \in D_1$ and $k > 1$ , then $k$ must be an odd integer. This is derived from the condition $\phi_1(3) \mid \phi_k(3)$ , which implies $3 \mid (1 + 2^k)$ , forcing $k$ to be odd.
Bernoulli Sums and Integrality: The core of the argument involves the sequence $C_k(v)$ , defined via Bernoulli numbers $B_{2j}$ :
$C_k(v) = \frac{2}{k+1} \sum_{j=0}^{(k-1)/2} \binom{k+1}{2j} B_{2j} (1-2j)^v$
The authors prove a key lemma: if $C_k(v)$ is an integer for all $v \ge 1$ , then for every even integer $h \in [2, k-1]$ , the quantity $\frac{2k!}{h!}B_h$ must be an integer.
Construction of Squarefree Integers: To establish that $C_k(v) \in \mathbb{Z}$ for all $v$ , the authors utilize Dirichlet's theorem on primes in arithmetic progressions. They construct squarefree integers $n$ composed of primes congruent to $1 \pmod L$ (where $L$ is the least common multiple of denominators of specific Bernoulli-related terms). By analyzing the divisibility of $\phi_k(n)/\phi_1(n)$ for these specific $n$ , they deduce that $C_k(\omega(n))$ must be an integer, where $\omega(n)$ is the number of distinct prime factors of $n$ . Since $\omega(n)$ can be any integer $v \ge 1$ , $C_k(v)$ is an integer for all $v$ .
Application of the von Staudt–Clausen Theorem: Using the integrality condition derived in the previous step, the authors apply the von Staudt–Clausen theorem, which characterizes the denominators of Bernoulli numbers. This leads to a critical property: for any even $h \in [2, k-1]$ and any odd prime $p$ such that $(p-1) \mid h$ , there must exist a multiple of $p$ in the set $\{h+1, \dots, k\}$ .
Modular Contradictions: The authors translate this property into a strict inequality involving modular residues: for every odd prime $p \le k$ $p \leq k$ ,
$k \pmod p < k \pmod{p-1}$
By testing specific primes and analyzing the structure of $k+1$ $k + 1$ , they systematically eliminate possibilities for $k$ $k$ :
- They prove $k+1$ cannot have any odd prime divisors, implying $k+1 = 2^a$ .
- They show $a$ must be even, leading to $k = 2^{2\lambda} - 1$ .
- Through modular arithmetic with primes $p=7, 11, 29$ , they demonstrate that $\lambda \ge 3$ leads to contradictions with the strict inequality derived above.

Key Contributions and Results

Proof of Conjecture: The paper successfully proves that $D_1 = \{1, 3, 15\}$ .
Lemma on Bernoulli Sums: The authors establish a new connection between the integrality of specific Bernoulli sums $C_k(v)$ and the divisibility properties of factorials and Bernoulli numbers.
Structural Characterization: The work provides a detailed characterization of the integers $k$ for which $\phi_1(n)$ divides $\phi_k(n)$ , reducing the problem to a finite set of modular constraints.

Significance and Claims
The paper claims to resolve an open conjecture from recent literature (B¨uy¨uka¸sik et al., 2024). The authors explicitly state that the conjecture had remained open prior to their work. A distinctive feature of the paper's methodology is its reliance on "extensive interactions with GPT-5.5 Pro" during the exploratory and proof-development stages. The author notes that while AI suggestions were used, all mathematical content was substantially revised, corrected, and independently verified by the human author, who assumes full responsibility.

The paper concludes modestly, suggesting that extending these techniques to evaluate $D_s$ for $s > 1$ represents a natural direction for future research, without proposing specific experimental applications or broader implications beyond the scope of this arithmetic problem.