The Origin and Development of the Goldbach Conjecture

The story begins with a childhood anecdote about a student named Xiao Hui asking his elementary teacher a question, followed by a series of illustrative images.

It then introduces the origin of the Goldbach conjecture, mentioning the amateur mathematician Christian Goldbach and the great mathematician Leonhard Euler.

Prime numbers are defined as natural numbers greater than 1 that have no divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11, 13, 17, 19, …).

Goldbach observed that many positive integers can be expressed as the sum of three primes, providing examples such as 9 = 2+2+5, 16 = 2+7+7, and 30 = 2+11+17.

He asked whether every integer greater than 5 can be written as a sum of three primes; Euler reformulated the problem as: every even integer greater than 2 can be expressed as the sum of two primes.

Examples of the even‑case are given: 6 = 3+3, 18 = 5+13, 24 = 5+19.

The article explains the equivalence of the two statements by noting that adding 2 or 3 to an even number written as a sum of two primes yields a representation of any integer >5 as a sum of three primes.

The concept of an "almost prime" (殆素数) is introduced: a positive integer whose number of prime factors does not exceed a fixed constant. For example, 15 = 3×5 is a 2‑almost‑prime, and 45 = 3×3×5 is a 3‑almost‑prime.

Since directly proving the full Goldbach conjecture is difficult, mathematicians have proved weaker versions: every even number >2 can be expressed as the sum of two k‑almost‑primes for various k. Historical milestones include the "9+9" result (1920), "7+7" (1924), "6+6" (1932), "5+7" and "4+9" (1937), "5+5" (1938), "4+4" (1940), "3+4", "3+3", "2+3" (1956), "1+5", "1+4" (1962), "1+3" (1965), and finally "1+2" proved by Chen Jingrun in 1966.

Chen's theorem states that every even integer >2 can be written as either (prime A + prime B) × prime C or simply prime A + prime B, bringing the problem one step closer to the original conjecture.

Despite these advances, the ultimate goal—proving that every even number >2 is the sum of two primes (the "1+1" case)—remains unresolved after more than fifty years.

The article concludes by acknowledging the enthusiasm of many amateur mathematicians who continue to explore the conjecture, while emphasizing the need for solid mathematical foundations to avoid erroneous proofs.