Lower bounds for linear forms in two p-adic logarithms

We prove explicit lower bounds for linear forms in two p- adic logarithms. More specifically, we establish explicit lower bounds for the p-adic distance between two integral powers of algebraic numbers, that is, | Lambda | p = | alpha b 1 1 - alpha b 2 2 | p (and corresponding explicit upper bounds for v p ( Lambda )), where alpha 1 , alpha 2 are numbers that are algebraic over Q and b 1 , b 2 are positive rational integers. This work is a p-adic analogue of Gouillon's explicit lower bounds in the complex case. Our upper bound for v p ( Lambda ) has an explicit constant of reasonable size and the dependence of the bound on B (a quantity depending on b 1 and b 2 ) is log B , instead of (log B ) 2 as in the work of Bugeaud and Laurent in 1996. (c) 2024 The Author. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons .org /licenses /by -nc -nd /4 .0/).