Journal of the
Korean Mathematical Society JKMS

A problem raised by Selfridge and solved by Pomerance asks to find the pairs $(a, b)$ of natural numbers for which $2^a-2^b$ divides $n^a-n^b$ for all integers $n$. Vajaitu and one of the authors have obtained a generalization which concerns elements $\alpha_1, \ldots, \alpha_k$ and $\beta$ in the ring of integers $\mathbf A$ of a number field for which \begin{equation*} \sum_{i=1}^k \alpha_i \beta^{a_i} \quad \text{divides} \quad \sum_{i=1}^k \alpha_i z^{a_i} \ \text{for any} \ z \in \mathbf A. \end{equation*} Here we obtain a further generalization, proving the corresponding finiteness results in a multidimensional setting.

Keywords: exponential congruences, algebraic integers, polynomials of several variables

MSC numbers: 11A07