There is no n so that the only integer m with phi(n) = phi(m) is m=n

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 $n\in\mathbb N$, Euler's totient is \[\varphi(n)=\#\{a\in\mathbb N:1\le a\le n,\ \gcd(a,n)=1\},\] so $\varphi(1)=1$.

Statement recorded in the supplied source.

For every $n\in\mathbb N$, does there exist $m\in\mathbb N\setminus\{n\}$ with $\varphi(m)=\varphi(n)$? Equivalently, can any value attained by the totient function have exactly one positive-integer preimage? [S2]

Short English statement

Does every positive integer share its totient value with another positive integer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3100065 (AMR-030-0065). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Florentin Smarandache, On Carmichaël's conjecture (2007 arXiv posting), Introduction and Section4, explicit quantification of Carmichael’s conjecture, pages1 and3. https://arxiv.org/abs/0704.2453.