Chen Jingrun, China's famous mathematician: devastated by brain injuries on the doorstep to solving a fundamental mathematical puzzle

Chen Jingrun (1933–1996), perhaps the most prodigious mathematician of his time, focused on the field of analytical number theory. His work on Waring's problem, Legendre's conjecture, and Goldbach's conjecture led to progress in analytical number theory in the form of “Chen's Theorem,” which he published in 1966 and 1973. His early life was ravaged by the Second Sino-Japanese War and the Chinese Cultural Revolution. On the verge of solving Goldbach's conjecture in 1984, Chen was struck by a bicyclist while also bicycling and suffered severe brain trauma. During his hospitalization, he was also found to have Parkinson's disease. Chen suffered another serious brain concussion after a fall only a few months after recovering from the bicycle crash. With significant deficits, he remained hospitalized for several years without making progress while receiving modern Western medical therapies. In 1988 traditional Chinese medicine experts were called in to assist with his treatment. After a year of acupuncture and oxygen therapy, Chen could control his basic bowel and bladder functions, he could walk slowly, and his swallowing and speech improved. When Chen was unable to produce complex work or finish his final work on Goldbach's conjecture, his mathematical pursuits were taken up vigorously by his dedicated students. He was able to publish Youth Math, a mathematics book that became an inspiration in Chinese education. Although he died in 1996 at the age of 63 after surviving brutal political repression, being deprived of neurological function at the very peak of his genius, and having to be supported by his wife, Chen ironically became a symbol of dedication, perseverance, and motivation to his students and associates, to Chinese youth, to a nation, and to mathematicians and scientists worldwide.

A GENERALIZATION OF CHEN'S THEOREM ON THE ERDŐS–TURÁN CONJECTURE

Let ℕ be the set of all nonnegative integers and k ≥ 2 be a fixed integer. For a set A ⊆ ℕ, let rk(A, n) denote the number of solutions of a1 + a2 + ⋯ + ak = n with a1, a2, …, ak ∈ A. In this paper, we prove that there is a set A ⊆ ℕ such that rk(A, n) ≥ 1 for all integers n ≥ 0 and the set of n with rk(A, n) = k! has density one. This generalizes a recent result of Chen.