Question

Explain why you cannot directly adapt the proof that there

are infinitely many primes (Theorem 3 in Section 4.3) to show that are

infinitely many primes in the arithmetic progression 3 k + 1 , k = 1 , 2 , ...........

Short answer

The product of numbers which has form 3 k + 1 , k = 1 , 2 , ........... is also of the form , but numbers of this form might have numbers, not of this form as their only prime factors. Therefore, we cannot directly adapt the proof that there are infinitely many primes to show that are infinitely many primes in the arithmetic progression 2 k + 1 .

Step-by-step solution

Prime number

Prime: an integer greater than 1 with exactly two positive integer divisors 1 and the number itself.

Explanation

Let’s consider an arithmetic progression of the form 3 k + 1 where k = 1,2,......

In this progression if k = 16

Then we can write

3 (16) + 1 = 49

And if we write the prime factorization of 49

That is

$$\begin{gathered} 49=7\times 7 \\ =\left(3\times 2+1\right)\left(3\times 2+1\right) \end{gathered}$$

From above we can see, that the product of numbers which has form 3 k + 1 is also of the form 3 k + 1 , but numbers of this form might have numbers, not of this form as their only prime factors, like 3 (16) + 1 = 49

Therefore, we cannot directly adapt the proof that there are infinitely many primes to show that are infinitely many primes in the arithmetic progression 3 k + 1 .