A conditional proof of Legendre's Conjecture and Andrica's conjecture

2018/09/27 by Madieyna Diouf, Diouf, Madieyna
Mathematics · #11N05 #FOS: Mathematics #General Mathematics (math.GM) #math.GM #msc:11N05

paper · pdf · doi:10.48550/arxiv.1810.02191

In v3, we have reduced a difficult problem, that is the difference between two consecutive primes, to a much simpler problem where the focus is not in the gap or location of these primes, but simply in whether some particular numbers are even or odd

arxiv created 2019/03/04 · arxiv updated 2019/03/05

Abstract

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called the Parity conjecture. Let p_n and p_n+1 be two consecutive odd primes, let m be their midpoint fixed once for all. Conjecture: The largest multiple of p_n not exceeding m_i² is odd for every integer m_i in the interval (p_n, m]. Main result: We prove that the Parity conjecture implies Legendre's conjecture and Andrica's conjecture.

Related

• Weakening the Legendre Conjecture
• Existence of primes between two consecutive squares
• Legendre's conjecture. Theorem on existence of a prime number between m² and (m+1)²
• Legendre Conjecture over Arithmetic Progressions
• Reasoning about Primes (II)