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On the generalized Pillai equation

✍ Scribed by Reese Scott; Robert Styer

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
234 KB
Volume
118
Category
Article
ISSN
0022-314X

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✦ Synopsis

We show that the equation Β±a x Β± b y = c (where the Β± 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 1b y 1 | = |a x 2b y 2 | for b = 2 or 3 and prime a not a base-b Wieferich prime.

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