A finiteness theorem for a class of exponential congruences

Abstract

For given elements a1,... , ak and 3 belonging to the ring of integers A of a number field we consider the set of all k-tuples (al,... , ak) in Nk for which Ek l caiai divides Ekcaiai for any z E A, and prove under some reasonable assumptions that the set of solutions is finite. The original motivation for this work comes from a problem raised by J. L. Selfridge (see Guy [1], problem B47) who asks for what pairs (a, b) does 2a-2b divide nanb for all integers n. A related (but more difficult) problem proposed by H. Ruderman asks to show that if 2a 2b divides 3a -3b, then 2a 2b divides n nb for all integers n. This was investigated by B. Velez in [6]. While Ruderman's problem is still open, Selfridge's problem was solved by Pomerance [2], who combined results of Schinzel [4] with Velez's work. It turns out that there are exactly 14 solutions. The problem was also solved by Sun Qi and Zhang Ming Zhi [5]. In this paper we show that the above finiteness result is a particular case of a more general phenomenon. Let IC be a number field, A = AKC its ring of integers and U = UKc its group of units. Let c1,... , ak and p be nonzero elements of A. We consider the set of all k-tuples (al,... , ak) in Nk for which