Carmichael's totient function conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Euler's totient is $\varphi(n)=\#\{a\in\mathbb N:1\le a\le n,\ \gcd(a,n)=1\}$ for $n\in\mathbb N$. For $m\in\mathbb N$, its multiplicity is $A(m)=\#\{n\in\mathbb N:\varphi(n)=m\}$.

Statement recorded in the supplied source.

Is $A(m)\ge2$ for every $m\in\varphi(\mathbb N)$? Equivalently, for every $n\in\mathbb N$, does there exist $n'\in\mathbb N$, $n'\ne n$, with $\varphi(n')=\varphi(n)$? [S2]

Short English statement

Does every totient value come from at least two distinct positive integers?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400002 (AMR-093-0002). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mithun Kumar Das, Pramod Eyyunni and Bhuwanesh Rao Patil, Sparse subsets of the natural numbers and Euler's totient function, arXiv:1907.09847v2 (2020) Introduction, p. 3, statement before Theorem 1.6. https://arxiv.org/pdf/1907.09847.
• S3 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Carmichael's totient function conjecture. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.