The \(\mathbb{Q}\)-Korselt set of \(pq\) (Q2221015)

If \(\alpha = \alpha_1/\alpha_2\in \mathbb{Q},\,\, \alpha \ne 0\), and \(N\ne\alpha\)\, a positive integer. \(\alpha\)\, is said to be a \(N\)\,-Korselt base (or alternatively \(N\)\, an \(\alpha\)\,-Korselt number) if \(\alpha_2p-\alpha_1\vert \alpha_2N-\alpha_1\)\, for all prime \(p\vert N\). This definition was introduced (for \(\alpha \in \mathbb{Z}\)) by \textit{K. Bouallégue} et al. [Int. J. Number Theory 6, No. 2, 257--269 (2010; Zbl 1214.11013)] as a generalization of Carmichael numbers.\par Let \(A\)\, be a subset of \(\mathbb{Q}\)\, and \(A-\mathcal{K}\mathcal{S}(N)\)\, the \(A\)\,-Korselt set of \(N\): the set of all \(\alpha \in A-\{0,N\}\)\, such that \(N\)\, is an \(\alpha\)\,-Korselt number. The present paper characterizes \(\mathbb{Q}-\mathcal{K}\mathcal{S}(pq)\),\, the \(\mathbb{Q}\)-Korselt of \(N=pq,\,\, p<q,\,\, p,q\)\, two primes. This also completes the study of \(\mathbb{Z}-\mathcal{K}\mathcal{S}(pq)\)\, in a previous paper of \textit{O. Echi} and \textit{N. Ghanmi} [Int. J. Number Theory 8, No. 2, 299--309 (2012; Zbl 1288.11007)] \par Section 2 studies conditions relating \(\alpha_1\)\, and \(\alpha_2\)\, with \(p\)\, and \(q\), so that \(\alpha\)\, can be a \(pq\)\,-Korselt base and Section 3 provides the looked for structure of \(\mathbb{Q}-\mathcal{K}\mathcal{S}(pq)\). First Theorem 3.2 solves the case \(A=\mathbb{Z}\)\, and \(q <2p\), an open problem in the mentioned paper of Echi and Ghanmi. Then the main Theorem 3.3 completely characterizes the \(\mathbb{Q}\)\,-Korselt set of \(N=pq\).