A large even integer number can be expressed as the sum of two prime numbers

A large even integer number can be expressed as the sum of two prime numbers

                     The proof of the Goldbach conjecture is based on Chen Jingrun’s theorem and negative space

Jianhua Li, Feng, Xie , Lee  Xie Li, Ganquan Xie

Dayuling Supercomputing Science Center, CHINA

Wangcheng, Changsha, Hunan, China

GL GEOPHYSICAL LSBORATORY, USA

                                                                                                       August 16, 2025

Abstract

Chen Jingrun proved that a large even integer number X can be expressed as

1. X = p + p₁

or

2. X = p + p₂ p₃

Here p, p₁, p₂ , p ₃ are prime numbers.

By negative space theory, the second option is excluded. We proved that a large even integer number can be expressed as the sum of two prime numbers .

Intrduction

Chen Jingrun’s paper is a monument to Chinese mathematics in the world. He brought the sieve method to its pinnacle of glory.

Chen Jingrun, a true survivor of the Cultural Revolution, completed his proof in January 1966. Professor Min Sihe reviewed it and found no flaws. Pan Chengdong also reviewed it and found no flaws. Professor Guan Zhaozhi, an expert in functional analysis, Wu Wenjun, a topology expert, and Professor Min, a number theory expert, and Professor Min Sihe, a functional theory expert, jointly recommended it for publication in Volume 17 of the Chinese Science Bulletin in September 1966. In 1973, the full text of Chen’s theorem was published in Chinese Science.  From 1973 to his death in 1996, Chen Jingrun worked for another 23 years. He tried every possible way to eliminate his second option possibility: a large even numbers = p + p₂ p₃. This left only option 1, large even numbers = p + p₁, where p, p₁, p₂ , p ₃ are prime numbers. if so, the Goldbach conjecture was proved. However, he tried every possible way to improve the sieve method, but are all fail. He didn’t know about negative space. He also didn’t know that this was a problem that couldn’t be proved in positive space. Chen Jingrun’s extreme efforts could only prove

 1 + 2.

Proof of GoldBach conjecture

Based on Chen Jingrun’s theorem and the theory of Negative Space,

we can prove that if there is option 2,
                                               X = p + p2 p3 ,                                         (1)
then the option 1 must exist. 
            X = p +p1,                                                 (2)
Is hold. Goldbach  conjecture is proved.

A large even integer number can be expressed as the sum of two prime numbers.

Thus, the 283-year-old puzzle was solved.