Carmichael's totient function conjecture

In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function ${\displaystyle \varphi (n)}$, which counts the number of integers less than and coprime to ${\displaystyle n}$. It states that, for every ${\displaystyle n}$ there is at least one other integer ${\displaystyle m\neq n}$ such that ${\displaystyle \varphi (m)=\varphi (n)}$. Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.