Question

Do you think that there are infinitely many primes of the form N² - 1?

Answer 1

Пошаговое объяснение:

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Answer 2

The question of whether there are infinitely many prime numbers of a specific form, such as N² - 1, is a famous open problem in number theory. In this case, the form N² - 1 can be rewritten as (N + 1)(N - 1). So, you are essentially asking whether there are infinitely many prime numbers that can be expressed as the product of two consecutive positive integers plus one.

This particular question is related to a broader class of questions about prime numbers called "twin primes." Twin primes are pairs of prime numbers that have a difference of 2, such as (3, 5), (5, 7), (11, 13), and so on. In your case, you are looking at numbers of the form (N - 1) and (N + 1), which are consecutive integers.

As of my last knowledge update in January 2022, it is not known whether there are infinitely many prime numbers of the form N² - 1, or equivalently, whether there are infinitely many twin prime pairs of the form (N - 1, N + 1). This problem remains an open question in number theory, and mathematicians have been actively researching it for many years.

The question is closely related to the more general question of whether there are infinitely many twin primes. While progress has been made on the twin prime conjecture, proving that there are infinitely many twin primes is still an unsolved problem. Consequently, the question of whether there are infinitely many prime numbers of the form N² - 1 is also unresolved.

Mathematicians continue to work on this and related questions in number theory, and new insights and techniques may eventually lead to a solution. However, as of my last update, the question remains an open and intriguing problem in mathematics.