Sieve methods and the twin prime conjecture

2019/09/05 by Mbakiso Fix Mothebe, Mothebe, Mbakiso Fix
Mathematics · #11N05 #11N36 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics

paper · pdf · doi:10.48550/arxiv.1909.02205

openalex publication_date 2019/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For n ≥ 3, let p_n denote the n^{\rm th} prime number. Let [ ] denote the floor or greatest integer function. For a positive integer m, let π₂(m) denote the number of twin primes not exceeding m. The twin prime conjecture states that there are infinitely many prime numbers p such that p+2 is also prime. In this paper we state a conjecture to the effect that given any integer a>0 there exists an integer N₂(a) such that [\fracap²_n+12(n+1) ] ≤ π₂(p²_n+1 ) for all n ≥ N₂(a) and prove the conjecture in the case a=1. This, in turn, establishes the twin prime conjecture.

Related

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• A Sieve for Twin Primes
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• A Possible Solution for Hilbert's Unsolved 8th Problem: Twin Prime Conjecture
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