Powers Which Differ by a Value of One

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The discussion centers on the equation Ax + 1 = By, seeking integer solutions for A and B where x and y are greater than 2. Participants explore the implications of prime compositions in numbers that differ by one, considering the rarity of powerful primes in this context. Catalan's conjecture, which asserts the non-existence of two perfect powers differing by one, is referenced as a significant point in the discussion. The conversation highlights the mathematical curiosity surrounding powerful numbers and their relationships. Overall, the thread emphasizes the complexity of finding solutions to the equation within the specified constraints.
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Are there natural solutions for A and B which satisfy the equation

Ax+1=By

where x and y are integers greater than 2?
(I only include a greater than 2 stipulation because I can see a few obvious solutions such as 23+1=32)

This has just sort of come to mind as I was thinking about the interesting differences in the prime composition of numbers differing by one, and then wondered if there were any powerful primes that differed by only one, or any powerful numbers whatsoever that differ by 1.
 
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