On existence of infinite primes and infinite twin primes

2015/01/08 by Maurice Margenstern, Margenstern, Maurice, Yaroslav D. Sergeyev +1
Computer Science · Mathematics · #03H15 #11A41 #11N80 #Computability, Logic, AI Algorithms #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics #Mathematical and Theoretical Analysis

paper · pdf · doi:10.48550/arxiv.1501.03051

openalex publication_date 2015/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The twin primes conjecture is a very old problem. Tacitly it is supposed that the primes it deals with are finite. In the present paper we consider three problems that are not related to finite primes but deal with infinite integers. The main tool of our investigation is a numeral system proposed recently that allows one to express various infinities and infinitesimals easily and by a finite number of symbols. The problems under consideration are the following and for all of them we give affermative answers: (i) do infinite primes exist? (ii) do infinite twin primes exist? (ii) is the set of infinite twin primes infinite? Examples of these three kinds of objects are given.

Related

• Infinitely many twin primes
• An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes
• An equivalent problem to the Twin Prime Conjecture
• Proof of the Twin Prime Conjecture (Together with the Proof of Polignac's Conjecture for Cousin Primes)
• A Proof There Exists Infinitely Many Primes with a Gap of Exactly 2