Lehmer's Totient Problem

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.

For an integer $n\geq1$, Euler's totient is $\varphi(n)=\#\{a\in\{1,\ldots,n\}:\gcd(a,n)=1\}$. An integer is composite if it is greater than $1$ and not prime. In the standard Lehmer problem the domain is $n>1$; with the convention $\varphi(1)=1$, omitting this domain would introduce the unrelated trivial exception $n=1$.

Statement recorded in the supplied source.

For every integer $n>1$, does \[ \varphi(n)\mid n-1\quad\Longrightarrow\quad n\text{ is prime} \] hold? Equivalently, is there no composite integer $n$ satisfying $\varphi(n)\mid n-1$? [S2]

Short English statement

Can a composite integer have its totient divide one less than itself?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1242 (NT-035). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Manuel Norman, A group-theoretical approach to Lehmer's totient problem, arXiv:2106.11781v1 (2021), abstract and Section 1. https://arxiv.org/pdf/2106.11781.