Problem Statement
Is there an infinite set $A\subset \mathbb{N}$ such that for every $a\in A$ there is an integer $n$ such that $\phi(n)=a$, and yet if $n_a$ is the smallest such integer then $n_a/a\to \infty$ as $a\to\infty$?
Categories:
Number Theory
Progress
Carmichael has asked whether there is an integer $t$ for which $\phi(n)=t$ has exactly one solution. Erdős has proved that if such a $t$ exists then there must be infinitely many such $t$.See also [694].
This is discussed in problems B36 and B39 of Guy's collection [Gu04].
Source: erdosproblems.com/51 | Last verified: January 13, 2026