具有迟缓项的剩余类
Residue Classes Having Tardy Totients
摘要：
我们以一种有效的方式证明了存在一系列同余类$a_k\pmod {m_k}$，使得同余类$ϕ(n)\equiv a_k\pmod {m_k}$的最小解$n=n_k$存在并满足$\log n_k/\log m_k\to\infty $作为$k\to\infty$。这里，$ϕ(n)$是欧拉函数。这回答了\cite{FS}中提出的一个问题。我们还表明，每一个同余类包含一个偶数包含无限多个值的卡迈克尔函数$λ(n)$和最少这样的$n$满足$n\ll m^{13}$。
英文摘要：
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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