Overview
This lecture explores Goldbach's Conjecture, a centuries-old unsolved problem in number theory, tracing its history, key contributors, and partial progress toward a solution.
The Goldbach Conjecture: Introduction and Statement
- Goldbach's Conjecture asks if every even number greater than 2 can be written as the sum of two prime numbers.
- Example: 6 = 3 + 3, 10 = 5 + 5 or 7 + 3, 42 = 37 + 5.
- The conjecture appears simple, but has resisted proof for nearly 300 years.
Historical Background and Formulation
- Christian Goldbach was a Prussian mathematician who proposed the conjecture in a 1742 letter to Euler.
- Euler reformulated it into two conjectures:
- Weak form: every odd number > 5 is the sum of three primes.
- Strong form: every even number > 2 is the sum of two primes.
- Proving the strong form would imply the weak one, but not vice versa.
Developments and Heuristic Arguments
- Hilbert listed Goldbach’s Conjecture as a major problem in 1900.
- Mathematicians also study H(n): the number of ways an even number n can be written as the sum of two primes.
- Hardy and Littlewood estimated H(n) ≈ n / (ln n)^2 using probability ideas from the prime number theorem, but this is not a proof.
Progress on the Weak Goldbach Conjecture
- Hardy, Littlewood, and Ramanujan developed the circle method to estimate how many ways a number can be written as the sum of primes.
- Vinogradov proved in 1937 that every sufficiently large odd number can be written as the sum of three primes, without assuming the Riemann Hypothesis, but didn’t specify a practical bound.
- Successive mathematicians lowered this bound until Helfgott, in 2013, proved the weak Goldbach conjecture for all integers greater than 5.
Approaches to the Strong Goldbach Conjecture
- Chen Jingrun proved in 1966 that every sufficiently large even number is the sum of a prime and a semiprime (product of two primes).
- Despite massive computational verifications (up to 4 quintillion), a proof for the strong conjecture remains elusive.
Legacy and Mathematical Motivation
- Goldbach's Conjecture is one of the oldest unsolved problems in mathematics and has inspired major developments in number theory.
- Its solution has no known practical applications, but is pursued out of intellectual curiosity and passion for mathematics.
Key Terms & Definitions
- Prime Number — A natural number greater than 1 with no divisors other than 1 and itself.
- Goldbach's Conjecture — The claim that every even number > 2 is the sum of two primes.
- Weak Goldbach Conjecture — Every odd number > 5 is the sum of three primes.
- Strong Goldbach Conjecture — Every even number > 2 is the sum of two primes.
- Circle Method — An analytic technique to estimate ways numbers can be written as sums of primes.
- H(n) — The number of ways n can be written as the sum of two primes.
- Semiprime — A number that is the product of exactly two prime numbers.
Action Items / Next Steps
- Review definitions and main results for strong/weak Goldbach conjectures.
- Study the outlined circle method and sieve methods for number theory exams.
- If assigned, read further about Chen Jingrun and Harald Helfgott’s proofs.