A finiteness theorem for a class of exponential congruences

Vâjâitu, Marian; Zaharescu, Alexandru

doi:10.1090/S0002-9939-99-04822-4

A finiteness theorem for a class of exponential congruences
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by Marian Vâjâitu and Alexandru Zaharescu PDF

Proc. Amer. Math. Soc. 127 (1999), 2225-2232 Request permission

Abstract:

For given elements $\alpha _1,\ldots ,\alpha _k$ and $\beta$ belonging to the ring of integers $\mathcal {A}$ of a number field we consider the set of all $k-$tuples $(a_1,\ldots ,a_k)$ in $\mathbb {N}^k$ for which $\sum _{i=1}^{k}\alpha _i\beta ^{a_i}$ divides $\sum _{i=1}^{k}\alpha _i z^{a_i}$ for any $z\in \mathcal {A},$ and prove under some reasonable assumptions that the set of solutions is finite.

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