The Obviously True Theorem No One Can Prove

Key Concepts

• Goldbach's Conjecture (Strong and Weak)
• Prime Numbers
• Even Numbers
• Number Theory
• Hardy-Littlewood Circle Method
• Sieve Methods
• Riemann Hypothesis
• Cultural Revolution (China)
• Heuristic Arguments
• Asymptotic Behavior

Goldbach's Conjecture: The Pearl on the Crown

• The Problem: Can every even number greater than 2 be written as the sum of two prime numbers?
• Examples:
  • 6 = 3 + 3
  • 10 = 5 + 5 = 7 + 3
  • 42 = 37 + 5
• Visual Representation: A pyramid of prime numbers is constructed, where diagonal lines are drawn from each prime, and their intersections represent the sums of those primes. This visually demonstrates how even numbers appear frequently as sums of primes.
• Origin: The conjecture originated from a letter written by Christian Goldbach to Leonard Euler in 1742. Euler refined Goldbach's idea into two separate conjectures:
  • Weak Goldbach Conjecture: Every odd number greater than 5 can be written as the sum of three primes.
  • Strong Goldbach Conjecture: Every even number greater than 2 can be written as the sum of two primes.
• Relationship between Strong and Weak: If the strong conjecture is proven, the weak conjecture follows automatically. The reverse is not true.
• Euler's Belief: Euler believed the conjecture to be "a completely certain theorem, although I cannot prove it."

The Hardy-Littlewood Circle Method

• Hilbert's Influence: In 1900, David Hilbert included Goldbach's conjecture in his list of 23 important problems for the 20th century, reigniting interest in the problem.
• New Approach: Mathematicians shifted focus from simply proving the conjecture to determining the number of ways an even number can be written as the sum of two primes, denoted as H(N).
• Hardy and Littlewood's Estimation: G.H. Hardy and John Littlewood attempted to estimate H(N) using the prime number theorem.
• Prime Number Theorem: States that a large number N has approximately a 1/ln(N) chance of being prime.
• Hardy-Littlewood's Approach (Simplified):
  1. Consider a large even number 2N.
  2. Split it into two numbers A and B, such that A + B = 2N.
  3. A = N - C and B = N + C, where C is a small difference.
  4. The probability of A being prime is approximately 1/ln(N-C) ≈ 1/ln(N).
  5. The probability of B being prime is approximately 1/ln(N+C) ≈ 1/ln(N).
  6. The probability of both A and B being prime is (1/ln(N)) * (1/ln(N)) = 1/(ln(N))^2.
  7. Since there are approximately N possible pairs, the total expected number of prime pairs is N / (ln(N))^2.
• Hardy and Littlewood's Refinement: They refined this estimate with a correction factor, but the core concept remained the same.
• Limitation: This was just an estimate, not a proof.

Ramanujan and the Weak Goldbach Conjecture

• Ramanujan's Letter: In 1913, Srinivasa Ramanujan, an unknown mathematician from India, sent a letter to G.H. Hardy containing numerous unproven theorems.
• Hardy's Reaction: Hardy was astounded by Ramanujan's work, recognizing his genius despite the lack of rigorous proofs.
• Collaboration: Hardy invited Ramanujan to Cambridge, where they collaborated on various problems in number theory.
• Circle Method Invention: Around 1917, Hardy and Ramanujan invented the circle method, which Hardy and Littlewood later developed further.
• Applying the Circle Method to the Weak Conjecture:
  1. The goal is to prove that any large odd number N can be written as the sum of three primes: N = P1 + P2 + P3.
  2. Imagine a "counting machine" that checks all possible combinations of three primes smaller than N.
  3. For each combination, it adds the primes and checks if the sum equals N.
  4. If the sum equals N, a counter is incremented.
• Mathematical Representation:
  1. Start with the function E^(iθ), which traces a unit circle in the complex plane.
  2. Restrict θ to be a multiple M of 2π.
  3. Multiply the angle by α (a number between 0 and 1): E^(i2παM).
  4. Average all possible values of α by integrating from 0 to 1: ∫[0,1] E^(i2παM) dα.
  5. This integral equals 1 if M = 0, and 0 otherwise.
  6. Replace M with the expression P1 + P2 + P3 - N.
  7. Sum this expression over all possible combinations of primes to get H(N), the number of ways to write N as the sum of three primes.
• Reframing the Problem: Hardy and Littlewood reframed the problem by taking the sum inside the integral, analyzing the collective behavior of all primes at once.
• The Function S(α, N): Defined as the sum of exponentials, each governed by a prime number: S(α, N) = Σ E^(i2παP), where P is a prime number less than N.
• Clock Analogy: Each exponential can be visualized as a clock, with its prime number determining how fast it spins.
• Constructive Interference: For most values of α, the clocks cancel each other out. However, at specific points (small rational fractions), the clocks interfere constructively, resulting in a large resultant value.
• Major and Minor Arcs: The circle is divided into major arcs (regions where constructive interference occurs) and minor arcs (regions where destructive interference occurs).
• Result: Hardy and Littlewood showed that if the generalized Riemann hypothesis is true, the main term (from the major arcs) grows faster than the error term (from the minor arcs), proving the weak Goldbach conjecture for sufficiently large numbers.
• Limitations: The generalized Riemann hypothesis was assumed to be true, and the specific value of "sufficiently large" was not determined.

Vinogradov and Helfgott: Closing in on the Weak Conjecture

• Vinogradov's Proof: In 1937, Ivan Vinogradov proved the weak Goldbach conjecture without relying on the generalized Riemann hypothesis. However, he still didn't specify the lower bound for "sufficiently large" numbers.
• Subsequent Reductions: Over the following decades, mathematicians gradually reduced the lower bound, but it remained astronomically large.
• Helfgott's Breakthrough: Harald Helfgott, in 2013, finally proved the weak Goldbach conjecture by:
  • Extensively checking numbers up to 8.8 x 10^30 by computer (with David Platt).
  • Refining the mathematical estimates to lower the constant K to 10^27, which was below the computationally verified limit.
• Helfgott's Theorem: Every odd number greater than 5 can be written as the sum of three primes. Consequently, every even number greater than 2 can be written as the sum of at most four primes.

The Strong Goldbach Conjecture: An Unyielding Challenge

• Lack of Real-World Application: The strong Goldbach conjecture, like the weak one, doesn't have any direct applications to the real world.
• Helfgott's Pessimism: Helfgott believes that solving the strong Goldbach conjecture is "hopeless for the moment."
• Limitations of the Circle Method: The circle method, successful for the weak conjecture, doesn't work for the strong conjecture because the main term doesn't grow faster than the error term.
• Chen Jingrun's Theorem: In 1966, Chen Jingrun proved that every sufficiently large even number is the sum of a prime and a number that is either a prime or the product of exactly two primes (a semiprime).
• Cultural Revolution's Impact: Chen's work was interrupted by the Cultural Revolution in China, during which he faced persecution and hardship.
• Chen's Resilience: Despite the challenges, Chen secretly continued working on his math.
• Publication and Recognition: In 1973, Chen published his theorem, and after the Cultural Revolution, he was celebrated as a national hero.
• Counterexamples: If the strong Goldbach conjecture is false, it can be disproven by finding a single counterexample (an even number that cannot be written as the sum of two primes).
• Computational Verification: Computers have checked all numbers up to four quintillion without finding a counterexample.
• Goldbach's Comet: A visualization of the number of ways to write even numbers as the sum of two primes, resembling a comet, showing an increasing trend as numbers get larger.
• The Importance of Passion: The video concludes by emphasizing the importance of pursuing problems that one is passionate about, regardless of their perceived importance or applicability.

Conclusion

The video provides a detailed exploration of Goldbach's conjecture, tracing its history, the mathematical techniques used to tackle it, and the individuals who have made significant contributions. While the weak Goldbach conjecture has been proven, the strong Goldbach conjecture remains one of the most elusive unsolved problems in mathematics, requiring potentially new and innovative approaches. The video highlights the perseverance and dedication of mathematicians like Chen Jingrun and Harald Helfgott, and emphasizes the importance of pursuing one's passion in the face of seemingly insurmountable challenges.