Goldbach's Conjecture: Historical Journey and Mathematical Exploration

Historical Context

• Xiamen, China, 1954: Amidst tensions with artillery bombardments. In this chaos, mathematician Chen Jingrun begins studying the Goldbach Conjecture.
• Goldbach's Conjecture: Proposed by Prussian mathematician Christian Goldbach in 1742 during correspondence with Euler.

The Conjecture

• Statement: "Every even integer greater than 2 can be expressed as the sum of two prime numbers."
• Weak vs. Strong:
  • Weak Conjecture: Every odd number greater than 5 can be written as the sum of three primes.
  • Strong Conjecture: Every even number greater than 2 is the sum of two primes.

Mathematical Developments

• Hardy & Littlewood's Contribution: Developing a function to estimate solutions, relying on the prime number theorem.
• Vinogradov's Progress (1937): Proved the weak conjecture without Riemann Hypothesis assumptions.
• Challenges: Absolute certainty requires identifying a large enough number K where the conjecture holds.

Solving Techniques

• Circle Method: Used by Hardy and Littlewood for weak conjecture by evaluating minor and major arcs.
• Sieve Methods: Chen Jingrun applied these to show sufficiently large even numbers can be represented as a mixture of primes and semi-primes.

Chen Jingrun's Journey

• Cultural Revolution Impact: Chen faced persecution but continued his work secretly.
• Publication and Recognition: Despite political turmoil, Chen published his findings and eventually became revered in China.

Recent Progress

• Helfgott's Achievement (2013): Proved the ternary Goldbach conjecture (weak conjecture) for numbers greater than five.
• Computational Assistance: Exploration up to numbers as high as 4 quintillion found no counterexamples, supporting the conjecture.

Considerations and Perspectives

• Potential Fallacy and Truth: Would be easily disproven if an even number didn't fit the conjecture, yet none has been found.
• Importance in Mathematics: While direct applications are not evident, solving such problems could unveil new mathematical terrain.

Conclusion

Despite centuries of seeming intractability, advancements continue through dedicated efforts, strongly igniting mathematical passion and curiosity towards unsolved problems like Goldbach's Conjecture.