On existence of infinite primes and infinite twin primes

TL;DR

This paper explores the existence of infinite primes and twin primes using a new numeral system that expresses infinities and infinitesimals, providing affirmative answers to these longstanding questions in number theory.

Contribution

It introduces a novel numeral system to analyze infinite integers and demonstrates the existence of infinite primes and twin primes within this framework.

Findings

Infinite primes exist according to the new system.

Infinite twin primes also exist in this framework.

The set of infinite twin primes is infinite.

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.