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On the generalized Pillai equation [formula omitted]

Authors
Journal
Journal of Number Theory
0022-314X
Publisher
Elsevier
Publication Date
Volume
118
Issue
2
Identifiers
DOI: 10.1016/j.jnt.2005.09.001
Disciplines
  • Design

Abstract

Abstract We show that the equation ± a x ± b y = c (where the ± signs are independent) has at most two solutions ( x , y ) for given integers a and b both greater than one and c greater than zero, except for listed specific cases. For any prime a > 5 and b = 2 , we show that there are at most two values of c allowing more than one solution to this equation, not counting trivial rearrangements; further restricting a to be a non-Wieferich prime, we improve this result: we show that there are no values of c allowing more than one solution, apart from designated exceptional cases. Finally, we give all solutions to the equation | a x 1 − b y 1 | = | a x 2 − b y 2 | for b = 2 or 3 and prime a not a base- b Wieferich prime.

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