# Clarification of Mihăilescu's Theorem (Catalan's Conjecture)

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• e2m2a
In summary, Mihăilescu's theorem proves that Catalan's conjecture is true for the specific case of x^a - y^b = 1, where the only solution in natural numbers is x=3, a=2, y=2, and b=3. However, there are certain restrictions for the theorem to hold, such as x and y being prime integers, a and b being prime integers, and x,y > 0 and a.b > 1. These restrictions are accurately described on Wikipedia.

#### e2m2a

TL;DR Summary
I understand Catalan's conjecture was proven by Preda V. Mihăilescu in 2002. However, I am not sure if it is proved for only certain conditions.
Mihăilescu's theorem proves that Catalan's conjecture is true. That is for x^a - y^b = 1, the only possible solution in naturual numbers for this equation is x=3, a=2, y=2, b=3. What is not clear to me is this. Does Mihăilescu's theorem prove that the difference between any other two powers (not the Catalan expression) will never be equal to 1 but only within certain restrictions? Another words, are there conditions that restrict x or y have to be both prime integers or just one of them must be a prime integer or does a or b have to be both prime integers or just one of them must be a prime integer for Mihăilescu's theorem to be true? Or is the only condition necessary is that x,y >0 and a.b >1?