Coprime mappings on Gaussian lines (Q6546727)

If \(A\) and \(B\) are sets of integers, then a bijection \(f: A\rightarrow B\) is called a coprime mapping if \(a\) and \(f(a)\) are coprime for all \(a \in A\). The existence of coprime mappings between intervals of positive integers has been well studied (see [\textit{C. Pomerance} and \textit{J. L. Selfridge}, Mathematika 27, 69--83 (1980; Zbl 0442.10003); \textit{T. Bohman} and \textit{F. Peng}, Mathematika 68, No. 3, 784--804 (2022; Zbl 1523.11142)]).\N\NThe authors of the paper under review extended this study to coprime mappings between intervals of Gaussian integers on lines in the complex plane. They proved the following results.\N\N\begin{itemize}\N\item[1.] Let \(L\) be a primitive Gaussian line and \(\mathbb{P}(L)\) be its prime set. Then a coprime mapping \begin{center} \(f : [\alpha_k, \alpha_{k+n-1}] \rightarrow [\alpha_{k+n}, \alpha_{k+2n-1}]\) \end{center} exists for all \(k \in \mathbb{Z}\) and all \(n \in \mathbb{N}\) if and only if at least one of the following three conditions holds:\N\begin{itemize}\N\item[(1)] the Gaussian prime \(1+i\) is not in \(\mathbb{P}(L)\);\N\item[(2)] none of \(3\), \(7\), or \(11\) are in \(\mathbb{P}(L)\);\N\item[(3)] none of \(3\), \(7\), or \(19\) are in \(\mathbb{P}(L)\).\N\end{itemize}\N\N\item[2.] Let \(m\) be a positive integer. There are infinitely many Gaussian lines \(L\) with the property that for every integer \(n\) with \(1< n < m\), there are infinitely many integers \(k\) such that no coprime mapping \begin{center} \(f : [\alpha_k, \alpha_{k+n-1}] \rightarrow [\alpha_{k+n}, \alpha_{k+2n-1}]\) \end{center} exists. In particular, no such coprime mapping exists for infinitely many values of \(k\) if \(\mathbb{P}(L)\) contains \(1 + i\) and every prime \(p \equiv 3 \pmod 4\), \(p \leq m\).\N\item[3.] Let \(L\) be a primitive Gaussian line such that every prime \(p \equiv 3 \pmod 4\) in \(\mathbb{P}(L)\) is smaller than \(10\). If \(n \geq 10\), then a coprime mapping exists between any two contiguous intervals on \(L\) of length \(n\).\N\item[4.] Let \(L\) be a primitive Gaussian line such that every prime \(p \equiv 3 \pmod 4\) in \(\mathbb{P}(L)\) is smaller than \(100\). If \(n \geq 102\), then a coprime mapping exists between any two contiguous intervals on \(L\) of length \(n\).\N\end{itemize}\N\NThey also studied coprime mappings on lines in the other imaginary quadratic fields with class number one and prove the following result.\N\NLet \(K\) be an imaginary quadratic field with class number one and \(L\) be a primitive \(\mathcal{O}_K\)-line. For a rational prime \(p\), let \(\pi_p\) be a prime in \(\mathcal{O}_K\) that divides \(p\). Then a coprime mapping exists between any two contiguous intervals on \(L\) of the same length if and only if at least one of the following three conditions is satisfied:\N\begin{itemize}\N\item[1.] the prime \(2\) is not split in \(K\) and \(\pi_2 \not\in \mathbb{P}(L)\);\N\item[2.] the primes \(3\) and \(5\) are not split in \(K\) and \(\pi_3, \pi_5 \not\in \mathbb{P}(L)\);\N\item[3.] for some \(p \in \{11,13,17,19\}\), the primes \(3\), \(7\), and \(p\) are not split in \(K\) and \(\pi_3, \pi_7, \pi_p \not\in \mathbb{P}(L)\).\N\end{itemize}