Residue Classes Having Tardy Totients

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $ϕ(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty$ as $k\to\infty$. Here, $ϕ(n)$ is the Euler function. This answers a question raised in \cite{FS}. We also show that every congruence class containing an even integer contains infinitely many values of the Carmichael function $λ(n)$ and the least such $n$ satisfies $n\ll m^{13}$.

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