Solved: Adapt the proof in the text that there are

Chapter 4, Problem 55E

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QUESTION:

Problem 55E

Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer. [Hint: Suppose that there are only finitely many such primes q1,q2,···,qn, and consider the number 4q1q2···qn – 1.]

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QUESTION:

Problem 55E

Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 4k + 3, where k is a nonnegative integer. [Hint: Suppose that there are only finitely many such primes q1,q2,···,qn, and consider the number 4q1q2···qn – 1.]

ANSWER:

Solution

Step 1

 

Let us suppose that  has only finite primes , where j is a non-negative  integer

Now, assume that  there are n of them

So,

Let us consider now

It is in the form of

Let

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