Key Moments
The Man Who Worked At Subway, Then Solved An "Impossible" Problem
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Key Moments
A former Subway worker solved a math problem deemed 'unattackable' for 100 years, proving primes come much closer than previously imagined.
Key Insights
Yitang Zhang proved in 2013 that there are infinitely many pairs of prime numbers that differ by at most 70 million.
The GPY method (Goldston, Pintz, Yildirim) in 2005 showed that prime gaps can be arbitrarily small fractions of the average gap (log n), but couldn't prove a fixed bound.
Brun's sieve method in 1919 proved that there are infinitely many pairs of numbers, two apart, where each number has at most nine prime factors, a precursor to proving twin primes.
The Polymath project, following Zhang's breakthrough, reduced the bound on prime gaps to 4,680 by refining his method.
James Maynard developed an independent method capable of proving bounded prime gaps, achieving a bound of 600 and later 246 without relying on the 'level of distribution' conjecture.
Assuming the Elliot-Halberstam conjecture, Maynard showed a potential gap of just 12, and the Polymath group demonstrated a gap of 6 under a stronger assumption.
A breakthrough email on an 'impossible' problem
In April 2013, the Annals of Mathematics received an email containing a 50-page proof for one of mathematics' most enduring unsolved problems: the twin prime conjecture. This conjecture posits that there are infinitely many pairs of prime numbers separated by just two (e.g., 11 and 13, 17 and 19). The proof, however, didn't come from a renowned mathematician, but from Yitang Zhang, an obscure figure who had previously worked for seven years at a Subway restaurant. Experts initially expected to quickly find an error, but upon closer examination, they realized Zhang had achieved a monumental breakthrough in number theory. The problem had been deemed 'unattackable' by mathematicians like Edmund Landau.
Early attempts: Brun's sieve and heuristic estimates
The twin prime conjecture has been a persistent challenge for mathematicians. Early approaches involved statistical estimates and sieving methods. Hardy and Littlewood, in 1923, refined the prime number theorem to provide a heuristic (a rule of thumb not a proof) estimating the number of twin primes, which closely matched observed data up to very large numbers. However, heuristics cannot definitively prove the conjecture because they don't rule out a 'conspiracy' where primes might conspire to avoid forming pairs. Viggo Brun, in the early 20th century, adapted the 'sieve of Eratosthenes' to tackle the problem. By modifying the sieve to count pairs (n, n+2) where both are prime, he proved that there are infinitely many pairs of numbers two apart where each number has at most nine prime factors. This was a significant step, showing progress towards the conjecture by weakening the conditions on the numbers involved, but it didn't directly prove the existence of infinite twin primes.
The GPY method: Arbitrarily small gaps
A major shift occurred in 2005 with the work of Goldston, Pintz, and Yildirim (GPY). They developed a new method that proved, astonishingly, that there are infinitely many pairs of prime numbers whose gaps are not just small, but arbitrarily small fractions of the average gap between primes. For instance, if the average gap around a certain large number is, say, 100, GPY showed that primes occur infinitely often within gaps much smaller than 100 – perhaps 1/10th, 1/100th, or even 1/1,000,000th of the average gap. This was a stunning result, suggesting that primes come much closer together than previously thought possible. However, their method, while proving these arbitrarily small gaps, crucially did not prove a fixed, absolute bounded gap between consecutive primes. It fell short of proving that primes are sometimes only 2 apart.
The problem with 'level of distribution'
The GPY method's limitation stemmed from a mathematical concept known as the 'level of distribution' (theta, θ), related to how evenly primes are distributed in arithmetic progressions. Their proof relied on this level of distribution being greater than 1/2. While mathematicians believed it was likely true for values greater than 1/2, it had never been rigorously proven for values beyond a certain point (e.g., up to √x). Zhang's breakthrough in 2013 was a direct assault on this barrier. He managed to push this level of distribution just beyond 1/2, to approximately 1/2 + 1/584. This tiny increment, achieved by cleverly reorganizing error terms in the calculations using special sets of step sizes, was enough to overcome the obstacle that had stymied mathematicians, proving that there exist infinitely many pairs of primes with a gap of at most 70 million.
Refining Zhang's result and Maynard's independent breakthrough
Following Zhang's groundbreaking work in 2013, which established a bounded gap of 70 million, mathematicians flocked to refine his methods. The Polymath Project, led by Terence Tao, used Zhang's techniques to systematically lower the bound. Within months and years, the upper limit on the gap between prime pairs was reduced significantly, eventually reaching 4,680. Simultaneously, a young mathematician named James Maynard developed a completely independent and more powerful method. Maynard's approach did not rely on the problematic 'level of distribution' conjecture. His method allowed him to prove much stronger results, showing it was possible to find infinitely many pairs of primes with a gap of 600, and later, that prime pairs could be found within a gap of 246.
Conditional results and the future of the conjecture
While the current record stands at a gap of 246, mathematicians have explored hypothetical scenarios. If one assumes the Riemann Hypothesis or the Elliot-Halberstam conjecture (which states primes are evenly distributed in arithmetic progressions with a level of distribution of 1), the potential gap between primes drops dramatically. Under those high assumptions, Maynard showed a gap of 12, and the Polymath group further reduced it to 6. However, these results are conditional on unproven conjectures. The general consensus among mathematicians is that the twin prime conjecture is true and will eventually be proven, though it may require a fundamental new idea. Yitang Zhang's work, and the subsequent rapid progress, demonstrated that even seemingly impossible problems can be solved by approaching them from unexpected angles.
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Common Questions
The twin prime conjecture states that there are infinitely many pairs of prime numbers that differ by exactly two, such as 11 and 13, or 17 and 19.
Topics
Mentioned in this video
A mathematician who described the twin prime conjecture as 'unattackable'.
The mathematician who worked at Subway and ultimately proved the existence of infinitely many pairs of prime numbers with a gap of at most 70 million.
A mathematician who developed an orthogonal approach to the bounded gap problem, significantly reducing the gap and earning a Fields Medal.
A mathematician who was James Maynard's advisor in Montreal and recognized the parallel development of Maynard's and Tao's ideas.
The first person to break the 4-minute mile, serving as an analogy for how breaking perceived barriers can lead to further progress.
A theorem stating that the probability of a large number n being prime is roughly 1 over the natural logarithm of n.
A fundamental hypothesis in number theory concerning the distribution of prime numbers, referenced in relation to theorems about prime distribution in arithmetic progressions.
An ancient algorithm for finding all prime numbers up to a specified integer by iteratively marking as composite the multiples of each prime.
A revolutionary personal tutor from Brilliant that makes one-on-one learning accessible, using AI to explain concepts and guide users through problems.
The organization that convened a 2005 meeting of experts to discuss and attempt to prove bounded gaps between primes.
The university where Yitang Zhang worked as a lecturer, allowing him to dedicate time to mathematics.
An online collaboration group that Terry Tao spearheaded to optimize Yitang Zhang's methods, progressively lowering the bounded gap.
One of the institutions whose experts designed the courses for Brilliant's Cooji tutor.
One of the institutions whose experts designed the courses for Brilliant's Cooji tutor.
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