Powers Which Differ by a Value of One

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The discussion centers on the equation Ax + 1 = By, where A and B are natural numbers and x and y are integers greater than 2. Participants explore the implications of this equation in relation to Catalan's conjecture, which asserts that the only solution in natural numbers for the equation x^a - y^b = 1 occurs for a = 3, b = 2, x = 2, and y = 3. The conversation highlights the rarity of powerful primes differing by one and the significance of recent proofs related to Catalan's conjecture.

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CarsonAdams
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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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