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

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.
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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?
 
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1. What is Mihăilescu's Theorem (Catalan's Conjecture)?

Mihăilescu's Theorem, also known as Catalan's Conjecture, is a mathematical theorem that states that the only solution to the equation xa - yb = 1, where x and y are positive integers and a and b are positive integers greater than 1, is x = 3, a = 2, y = 2, and b = 3. It was first conjectured by French mathematician Eugène Charles Catalan in 1844 and was proven by Romanian mathematician Preda Mihăilescu in 2002.

2. Why is Mihăilescu's Theorem (Catalan's Conjecture) significant?

Mihăilescu's Theorem is significant because it is a fundamental result in number theory and has applications in many other areas of mathematics. It also has connections to other famous conjectures, such as Fermat's Last Theorem and the ABC Conjecture. Additionally, the proof of Mihăilescu's Theorem required the development of new mathematical techniques and has inspired further research in the field.

3. What was the process of proving Mihăilescu's Theorem (Catalan's Conjecture)?

The proof of Mihăilescu's Theorem was a long and complex process that involved many different mathematicians over the course of several decades. It required the use of advanced mathematical techniques, such as elliptic curves, Galois theory, and modular forms. The final proof was completed by Preda Mihăilescu in 2002 and involved a combination of different approaches from various mathematicians.

4. Are there any known exceptions to Mihăilescu's Theorem (Catalan's Conjecture)?

No, there are no known exceptions to Mihăilescu's Theorem. The proof has been rigorously checked and has been accepted by the mathematical community as a valid and complete solution to the conjecture. However, it is always possible that a mistake could be found in the proof in the future.

5. What impact has Mihăilescu's Theorem (Catalan's Conjecture) had on mathematics?

Mihăilescu's Theorem has had a significant impact on mathematics, particularly in the field of number theory. It has provided a deeper understanding of the properties of numbers and has led to the development of new techniques and approaches in mathematical research. The proof of Mihăilescu's Theorem also serves as an example of the power of collaboration and perseverance in solving difficult mathematical problems.

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