Mihailescu's theorem, also known as the Catalan's conjecture, is a mathematical theorem that states that the only consecutive powers that can be both perfect powers (numbers that can be expressed as an integer raised to a positive integer power) are 8 and 9.
Mihailescu's theorem does not explicitly use the infinitude of primes. However, the proof of this theorem relies on various properties of prime numbers, such as the unique factorization theorem and the Euclidean algorithm.
The infinitude of primes is important in mathematics because it is a fundamental concept that has been studied for centuries and has many applications in different areas of mathematics, such as number theory, algebra, and cryptography. It also plays a crucial role in the proof of many theorems, including Mihailescu's theorem.
No, Mihailescu's theorem cannot be used to generate prime numbers. This theorem is a result that proves the non-existence of certain types of consecutive perfect powers, it does not provide a method for generating prime numbers.
No, Mihailescu's theorem was first conjectured by the Swiss mathematician Eugène Charles Catalan in 1844. However, it was not until 2002 that Romanian mathematician Preda Mihăilescu provided a proof for this conjecture, making it a theorem.