Question

Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 3k + 2, where k is a nonnegative integer. [Hint: Suppose that there are only finitely many such primes q_1, q_2,...,q_n, and consider the number 3q_1q_2 ··· q_n − 1.]

Answer 1

To prove that there are infinitely many primes of the form 3k + 2, we can adapt the proof that there are infinitely many primes. Suppose there are only finitely many primes of the form 3k + 2, and then consider the number 3q1q2 ··· qn - 1.

Step-by-step explanation:

To prove that there are infinitely many primes of the form 3k + 2, we can adapt the proof that there are infinitely many primes. Let's suppose that there are only finitely many primes of the form 3k + 2, denoted as q1, q2,..., qn. Now, let's consider the number 3q1q2 ··· qn - 1. This number is not divisible by any of the primes we assumed to be the only primes of the form 3k + 2, since it leaves a remainder of 2 when divided by any of them. Therefore, this number must have a prime factor of the form 3k + 2 that is not in our assumed finite set, proving that there are infinitely many primes of the form 3k + 2.