On the Infinitude of Primes of Certain Types
Aditi S. Phadke *
Modern Education Society’s, Nowrosjee Wadia College, Pune (Affiliated to Savitribai Phule Pune University), Pune-411 001, India.
Pramod N. Shinde
Modern Education Society’s, Nowrosjee Wadia College, Pune (Affiliated to Savitribai Phule Pune University), Pune-411 001, India.
Samina Boxwala
Modern Education Society’s, Nowrosjee Wadia College, Pune (Affiliated to Savitribai Phule Pune University), Pune-411 001, India.
*Author to whom correspondence should be addressed.
Abstract
Prime numbers and their patterns are a very important topic historically as well as in current times with applications to fields such as cryptography. In this paper, we give different proofs than those available in the literature of infinitude of primes of the type 4k + 3, 6k + 5, and 4k + 1. These are all special cases of the Dirichlet prime number theorem. We have used the technique of Saidak as well as divisibility properties, to give a constructive proof to prove infinitude of the primes of the form 4k + 3 and 6k + 5. The infinitude of primes of the type 3k + 2 is a corollary. In literature, these cases are proved by method of contradiction. To prove that there are infinitely many primes of the type 4k + 1, we show that every prime factor of a Fermat number Fn(n ≥ 1) is of the form 4k + 1 using a classical result on quadratic residues. Also any two Fermat numbers are coprime. Combining these two results has enabled us to prove that there are infinitely many primes of the type 4k + 1.
Keywords: Infinitude of primes, Dirichlet prime number theorem, Fermat number, Dirichlet theorem