Primes of the Form 4k + 3

In this document, we will use Euclid's idea to prove the following result.

Theorem: There are infinitely many primes of the form 4k + 3.

Proof: We proceed by contradiction. So, we start by assuming that we have a finite, complete list of all primes of this form, given by

P₁ = 3, P₂, ..., P_M.

Now consider the integer

N = P₂P₃...P_M + 3.

Now 3 cannot divide N, since it would then divide the product of the remaining primes, which is impossible. Moreover, none of the other primes can divide N, since it would then also have to divide 3. Finally, every integer of the form 4k + 3 has a prime factor of the same form.

Once again, we're not going to quit simply because we're at the end of the proof. There are some questions that are just begging to be asked.

Problem: Can you use Euclid's method to show thay there are infinitely many primes of the form 6k + 5? What about primes of the form 6k + 1?

Problem: Are there any other remainders r upon division by 6 such that 6k + r could yield infinitely many primes as k varies?

Problem: Can you find conditions on d and r that ensure that there are only finitely many primes of the form dk + r as k varies?

Problem: Are there any other numbers d and r such that Euclid's technique shows that there are infinitely many primes of the form dk + r as k varies?

[Home Page] [Bioinformatics] [Mathematics] [Creative Efforts]

[HOME] [PREVIOUS] [NEXT] [UP] [DOWN]