The Obviously True Theorem No One Can Prove

Overview

This episode delves into the history, significance, and ongoing mystery of Goldbach's Conjecture, a seemingly simple yet unproven mathematical problem that has captivated mathematicians for nearly 300 years. It explores the conjecture's origins, the breakthroughs and setbacks in its study, and the lives of the brilliant minds who have pursued its solution.

Notable Quotes

- Mathematics is the queen of the sciences. Number theory is the queen's crown. And the Goldbach's conjecture is the pearl on the crown. – Chen Jingrun's teacher, on the allure of the problem.

- If Goldbach is false, then it's false because there's a counterexample. And if there's a counterexample, you can give it to me, and I can check for myself. – Harald Helfgott, on the simplicity of disproving the conjecture.

- People should do what fires them up. If you do that, you might do something remarkable because of that passion. – Steven Strogatz, on the importance of pursuing what inspires you.

📜 The Origins of Goldbach's Conjecture

- Goldbach's Conjecture, proposed in 1742, posits that every even number greater than 2 can be expressed as the sum of two prime numbers.

- Christian Goldbach first suggested a related idea in a letter to Euler, who refined it into two conjectures: the weak (odd numbers as sums of three primes) and strong (even numbers as sums of two primes) forms.

- Euler, confident in its truth, admitted he could not prove it, setting the stage for centuries of mathematical pursuit.

🧮 Breakthroughs in the Weak Goldbach Conjecture

- Hardy and Littlewood developed the circle method in the early 20th century, providing a framework to estimate the number of ways numbers could be expressed as sums of primes.

- Ivan Vinogradov proved the weak conjecture for sufficiently large numbers in 1937, eliminating reliance on the Riemann Hypothesis.

- Harald Helfgott completed the proof in 2013, showing that every odd number greater than 5 can indeed be written as the sum of three primes.

🌏 Chen Jingrun and the Cultural Revolution

- Chen Jingrun, a Chinese mathematician, made significant progress on the strong conjecture in the 1960s, proving that every large even number is the sum of a prime and a semi-prime (a product of two primes).

- His work was nearly derailed during China's Cultural Revolution, where intellectuals were persecuted. Despite immense personal suffering, Chen secretly continued his research and published his findings in 1973.

- Chen's perseverance turned him into a national hero, symbolizing the triumph of intellectual pursuit over adversity.

🔢 Computational Advances and Goldbach's Comet

- Modern computers have verified the conjecture for all even numbers up to 4 quintillion, with no counterexamples found.

- The visualization of the conjecture, known as Goldbach's Comet, shows that as numbers grow larger, the number of ways they can be expressed as sums of two primes increases steadily.

- Despite this empirical evidence, a formal proof of the strong conjecture remains elusive, as current methods struggle with the complexity of the problem.

💡 The Philosophy of Mathematical Pursuit

- Goldbach's Conjecture, while lacking direct applications, exemplifies the beauty and challenge of pure mathematics.

- Mathematicians like Steven Strogatz argue that pursuing problems that ignite passion, regardless of their perceived importance, can lead to remarkable breakthroughs.

- The conjecture serves as a reminder of the value of curiosity-driven research and the enduring allure of unsolved mysteries.