Deep on Goldbach's Conjecture

EasyChair Preprint no. 9419, version 1

Versions: 123history
4 pagesDate: December 4, 2022

Abstract

Goldbach's conjecture is one of the most difficult unsolved problems in mathematics. This states that every even natural number greater than 2 is the sum of two prime numbers. In 1973, Chen Jingrun proved that every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). In 2015, Tomohiro Yamada, using the Chen's theorem, showed that every even number $> \exp \exp 36$ can be represented as the sum of a prime and a product of at most two primes. In 2002, Ying Chun Cai proved that every sufficiently large even integer $N$ is equal to $p + P_{2}$, where $P_{2}$ is an almost prime with at most two prime factors and $p \leq N^{0.95}$ is a prime number. In this note, we prove that for every even number $N \geq 32$, if there is a prime $p$ and a natural number $m$ such that $n < p < N - 1$, $p + m = N$, $N \gg \sigma(m)$ and $p$ is coprime with $m$, then $m$ is necessarily a prime number when $\sigma(m)$ is the sum-of-divisors function of $m$, $N = 2 \cdot n$ and $\gg$ means much greater than''. Indeed, this is a trivial and short note very easy to check and understand which is a breakthrough result at the same time.

Keyphrases: Euler's totient function, Goldbach's conjecture, prime numbers, sum-of-divisors function