Do any odd noncototients exist

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 $\varphi(n)=\#\{a\in\mathbb N:1\le a\le n,\ \gcd(a,n)=1\}$. A positive integer $m$ is a noncototient if $n-\varphi(n)\ne m$ for every $n\in\mathbb N$.

Statement recorded in the supplied source.

Does there exist an odd positive integer $m$ such that $n-\varphi(n)\ne m$ for every $n\in\mathbb N$? [S2]

Short English statement

Can a positive odd integer fail to occur as the difference between a number and its totient?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9400026 (AMR-093-0026). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Wikipedia contributors, List of unsolved problems in mathematics, revision 1366636767 (2026) Number theory, Do any odd noncototients exist. https://en.wikipedia.org/w/index.php?title=List_of_unsolved_problems_in_mathematics&oldid=1366636767#Number_theory.
• S3 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. 1. https://arxiv.org/pdf/1907.09847.
• S4 OEIS Foundation and contributors, The On-Line Encyclopedia of Integer Sequences, A005278: Noncototients: numbers k such that x - phi(x) = k has no solution., official data snapshot 693b5499c9d2 (2026) Name, definition and comments. https://raw.githubusercontent.com/oeis/oeisdata/693b5499c9d248f570dd2d979e8bd1d7eecf689b/seq/A005/A005278.seq.