Goldbach Conjecture Explained: Helfgott's Proof and the Circle Method's Journey

Veritasium explores one of math's oldest mysteries: can every even number be written as the sum of two primes? The video threads together the historical legacies of Goldbach, Euler, Ramanujan and Hardy with a modern breakthrough by Harald Helfgott. It explains the circle method used to count representations of numbers as sums of primes and shows how major and minor arcs guide the analysis, revealing why the weak Goldbach conjecture was eventually resolved while the strong form remains open. Interwoven are personal stories of the mathematicians, the political upheavals that affected their work, and the humbling beauty of a problem that for centuries resisted a complete proof. The tale ends with reflections on passion, perseverance and the human side of deep math.

Introduction: The Goldbach Conjecture

Goldbach’s conjecture, posed in the 18th century, asks whether every even number greater than two can be written as the sum of two primes. For centuries the question lived as a tantalizing open problem, simple to state and astonishing in its depth. The video introduces the distinction between the strong form, which demands two primes for every even number, and the weak form, which involves three primes and odd numbers. Though the problem is deceptively accessible, its proof required techniques far beyond classroom number theory.

The Circle Method and Hardy–Littlewood

Hardy and Littlewood reframed the challenge using a powerful analytic tool called the circle method. By examining exponential sums over primes and integrating on the unit circle in the complex plane, they constructed a framework to estimate the number of representations of N as sums of primes. The method splits the analysis into major arcs, where primes align in a constructive way, and minor arcs, where cancellations yield small contributions. This leads to a main-term prediction for how many representations an integer should have, with an error term that must be controlled for a proof to hold.

Ramanujan, Hardy, and Littlewood: A Human Story

The narrative follows the remarkable collaboration between Hardy and Ramanujan, a mathematician whose intuition and dream-guided results challenged conventional proofs. Ramanujan’s ideas, even when unproven, inspired Hardy and Littlewood to push the circle method further. The video frames this as a human story about genius, collaboration, and the boundaries of mathematical knowledge, set against personal and historical upheavals that shaped their work.

Vinogradov and the Quest for Proof

Moving to the early 20th century, Vinogradov provided an important independence-free path toward proving the weak conjecture, showing that every sufficiently large odd number can be expressed as the sum of three odd primes. His work laid the groundwork for later, more precise estimates and opened the door for later improvements that did not rely on unproven hypotheses about zeros of the Riemann zeta function.

Helfgott’s Breakthrough and Computational Feats

Harold Helfgott’s decisive advance, published in the early 2010s, proved the weak Goldbach conjecture: every odd number greater than five can be written as the sum of three primes. This also implies that every even number greater than two is the sum of at most four primes. Helfgott achieved this through enhanced analytic estimates and rigorous bounds in the circle method, along with massive computer-assisted verifications that pushed the computational frontier far beyond previous checks.

Chen Jing Run and the Cultural Revolution

The story highlights Chen Jing Run, a Chinese mathematician who made a close approach to the strong conjecture using sieve methods. His work, conducted under extremely difficult conditions during a turbulent era, represented one of the most significant near-misses in the history of the problem. The narrative links his perseverance to the broader human cost of political upheaval on scientific progress.

Reflections: Why This Matters

Beyond a single theorem, the tale illustrates how deep mathematical ideas develop over centuries, driven by curiosity and stubborn persistence. The video frames Goldbach as a lens into the culture of mathematics, showing how progress often comes in stages, with breakthroughs that are incremental yet transformative. The final message invites readers to pursue what they love and let the broader mathematical landscape reveal its own connections and surprises.