135: Interview with 张益唐: A 70-Year-Old Mathematician Chasing a Second Lightning Strike

Summary

• 张益唐 sees the Landau–Siegel zero conjecture as the “second time lightning struck” in his life: he says the fundamental breakthrough was completed in 2022, and the challenge now is to correct, simplify and write up the proof so that peers can accept it. He admits that his 2022 paper was “unfinished”: some calculations were wrong and key passages were not fully developed. The host noted that the paper’s exponent on log d was 2, while some believe the method cannot bring it down to 1; 张益唐 said that was “not a fundamental issue.” For long-cycle research and development, it is an extreme example of the gap between discovery, a finished product and external validation.
• The first lightning strike left a verifiable record: 张益唐 remembers figuring out the key method on July 3 or 4, 2012, then publishing a paper in 2013 proving the existence of infinitely many pairs of primes less than 70 million apart, moving the problem from “infinity” toward “finiteness.” When the mainstream route appeared to be “one hair’s breadth” away yet no one could cross it, he approached the problem through the distribution of primes in arithmetic progressions. Staying away from mainstream exchanges preserved a different approach: “If you follow everyone else completely, you probably won’t cross it either.”
• His research portfolio is close to an all-in bet on a small number of ultra-long-horizon, high-failure-rate problems. The host said he does not seem particularly concerned with publishing frequently; 张益唐 said he places little weight on material and worldly pursuits and accepts that he might go his entire life without a result. When he was away from academic positions from 1992 to 1999, he continued working with libraries, paper, a pen and his own mind. Even if he “couldn’t do it in a lifetime,” he would not regret it. He kept a line from a 2014 fortune cookie: “Others don’t believe you can do it, but you did.”
• 张益唐’s view of AI is not dismissive, but draws a clear line between a powerful tool and an autonomous original researcher: AI is powerful, but it cannot solve the Riemann hypothesis independently without humans. IMO problems have answers, are timed by the hour and can be approached through training patterns; research faces an as-yet unanswered future and requires long-term exploration. AI’s more realistic advantage is rereading hundreds of years of papers, processing material no individual could cover and helping find mathematical explanations for black-box models.
• He remains unusually restrained about the Riemann hypothesis: he tried it 20 or 30 years ago, but “doesn’t see a route now,” so he has neither announced a project nor promised a result. The host called it one of the hardest and most important problems in number theory and said many mathematicians were quietly working on it. 张益唐 only replied, “I won’t talk about it, and I don’t even dare to talk about it.” That stands in clear contrast to his categorical statement that the Landau–Siegel zero problem “has already been done.”
• His return to China in 2025 was a choice of research environment: 张益唐 values China’s investment in basic science and the Sun Yat-sen University Institute for Advanced Study’s low-administration, long-horizon autonomy. He also believes China has many smart young people and a large population base, both favorable to scientific development. In his view, the institute’s core is first “a group of world-class scholars,” and second, not allowing teaching to carve up research time. He does not want to be a leader; his immediate priority remains finishing the paper, after which he may consider supervising students.
• On talent, he puts sustained focus ahead of early competition labels: strong Olympiad results raise the probability of future achievement, but are not a prerequisite for becoming a mathematician. 王虹’s move from aerospace and geodesy into mathematics, without an early Olympiad halo, led him to stress that talent must be observed over long-term change. At the introductory stage, children should first experience the pleasure of “not being able to solve it, and then finally solving it”; at the research stage, interest, sensitivity and sustained concentration matter.
• The ultimate payoff of this high-risk research style is neither fame, position nor “immortality,” but what 张益唐 repeatedly calls “the joy of doing it for its own sake.” He does not calculate the cost of immersing himself in mathematics and does not believe he can retire professionally. Life is finite, and there may be only a few important problems one can truly solve, but as long as the interest remains, he will keep thinking.

Deep dive

1. The Second Lightning Strike Has Happened; the Challenge Is Making the Proof Clear

• In spring 2014, while visiting the Institute for Advanced Study in Princeton, 张益唐 received a fortune-cookie slip at a Chinese restaurant called Xinghuacun. Its message was roughly to do what others believed impossible. He thought it “fit me quite well” and kept it, sometimes carrying it in his wallet.
• He describes bounded gaps between primes as the first breakthrough already completed, and the Landau–Siegel zero conjecture as the second. A colleague at the University of California, Santa Barbara once said that if 张益唐 managed to do it again, it would mean he had been “struck by lightning twice in one lifetime.”
• This was not a research persona constructed after the fact. The host said he does not seem particularly focused on publishing frequently; 张益唐 explained that this was simply his personality, and that he places little value on material and worldly pursuits. What he genuinely enjoys is “doing big, highly unusual things.” He believes the second breakthrough is already done; the current challenge is to make the proof clear and get others to accept it.

2. The Absence of an Academic Position Did Not Change His Ultra-Long-Horizon Bet

• While away from academic work from 1992 to 1999, 张益唐 continued thinking about mathematics as much as possible. In the United States, he could at least use libraries to check references. Theoretical mathematics requires no laboratory equipment: “All you need is a brain, a piece of paper and a pen.”
• 曼琪 asked whether this approach could leave him with no important achievement after a lifetime. His answer did not romanticize failure: “I’ve thought about it, but even if I couldn’t do it in my lifetime, I wouldn’t feel regret.” He had little interest in other directions or material life, and barely considered switching fields.
• Once, he helped a college classmate working at Intel solve an encryption problem and obtained a patent. In his telling, it was merely “a small episode.” His friend was surprised that he had retained his sensitivity after years away from mathematics; 张益唐 did not treat the applied result as his main line of work.
• He acknowledges that his early job search came at an “unfortunate time,” but refuses to write an alternative timeline for history. If he had entered academia smoothly, “maybe things would have been better, maybe I would have done things faster.” But he later concluded that if he had followed everyone else completely, he might not have crossed those obstacles either.

3. The First Breakthrough Came from Avoiding That “Hair’s Breadth”

• The key facts recalled on the program are these: 张益唐 remembers figuring out the method on July 3 or 4, 2012; in April 2013 he published “Bounded Gaps Between Primes,” proving that infinitely many pairs of primes exist with gaps below 70 million.
• 70 million was still far from the 2 required by the twin prime conjecture, but the work completed the move “from infinity toward finiteness.” 张益唐 also stated clearly that his method could not reach 2.
• After 2000, this line of work became a hot topic, and the mathematical community felt it was only one step away. Some described the gap as “just one hair’s breadth,” but 张益唐’s judgment was: “In mathematics, being one hair’s breadth short is still no good,” because everyone advancing along the same route would get stuck at the same point.
• An expert meeting in 2008 was highly pessimistic about the prospects; he did not learn it had taken place until 2 years later. His entry point was “the distribution of primes in arithmetic progressions.” At first he did not even know the problem was related to twin primes, only later realizing that one more step could make it useful for bounded gaps.

4. The Value of Independent Research Is Preserving an Entry Point Different from Consensus

• 张益唐 reframes his personal experience as unforeseeable luck: working independently for a long time may add detours and time costs, but it also meant he did not completely follow the mainstream approach. “When everyone else is stuck there, you’ll be stuck there too,” whereas a different entry point may make it possible to cross the barrier.
• On which problems are worth pursuing, he believes little subjective judgment is needed. Number-theory problems that have remained open for decades or centuries and resisted many mathematicians are, by definition, recognized as important. The real filter is not importance, but whether a researcher can bear the probability of failing.
• Choosing a method depends on intuition—not only sensing that a route might be right, but also “thinking of places others have not thought of.” Many ideas that seem original ultimately prove useless or can be replaced by existing methods. Without a sustained pursuit of genuinely different ideas, it is difficult to produce anything new.

5. Intuition Is Not a Single Epiphany but Decades of Accumulation

• 张益唐 says mathematical intuition is “probably innate,” and it is difficult to provide a formula for cultivating it later. But the probability of inspiration rises with prolonged thought: “The more you consider something, the greater the possibility that you’ll have an insight.” Intuition is therefore also a cumulative process.
• His lack of interest in communication may have led him down more detours, but that is part of how he thinks. The twin prime problem and the Landau–Siegel zero problem are both examples of his trying to “think of places others have not thought of,” but neither was guaranteed simply because he chose to pursue it.
• 曼琪 cited Hardy’s view that mathematicians over 60 rarely make original contributions. 张益唐 does not fully accept it. He believes the issue may not be that intuition “declines,” but that no new intuition appears. At 70, he still gives a direct answer about himself: “I think I can still do it.”

6. The Landau–Siegel Zero Breakthrough Is Past; the Delivery Challenge Remains

• 张益唐 began thinking about the Landau–Siegel zero problem around the late 1990s and published a weaker paper in 2007, which he now calls “of no significance.” Over more than 20 years, he was not working only on this problem; several problems continued to circulate in his mind at once.
• The fundamental breakthrough came in 2022. It took roughly half a year to 1 year to work out completely, without a dramatic moment of sudden revelation: “I just thought about it and felt that this could work.” He finds it difficult to disentangle which of the previous decades’ accumulated insights converged at that point.
• The current work is simplifying, filling in details and rewriting. He explicitly acknowledges that the 2022 paper was “unfinished”: some ideas were proposed without all the details being written out, and some calculations were wrong. The hardest sections have dragged on from 2022 to the present, leaving him “incredibly annoyed.”
• The host remembered that the original paper had an exponent of 2 on log d and mentioned doubts that the method could bring it down to 1. 张益唐 responded that his method might not reach 1, but compared it with the bounded-gap result’s inability to reach 2: it is not the final optimal value, but in his judgment “not a fundamental issue.”

7. The Scarce Resource Is Figuring It Out; the Most Time-Consuming Part May Be Making It Clear

• Looking back over his life, 张益唐 recognizes only 2 moments when he was “struck by lightning”; most research time was simply spent “thinking, thinking, thinking.” He becomes especially immersed as progress nears a breakthrough and relaxes somewhat when things are not moving. The intensity of his work varies with the problem’s stage.
• The Landau–Siegel zero problem now has no fundamental breakthrough left to find. He works on it a little every day, comparing different simplifications and rewrites. The difficulty is that mathematical certainty cannot substitute for checkable exposition: “I need to organize it properly, write it out, and make it acceptable to everyone.”
• His typical day starts around 6 or 7 a.m. He walks alone first, then spends a long stretch in the office, eats in the cafeteria at noon, and walks on the lawn or along tree-lined paths when tired. Most of the materials he needs are already on his computer or in his head; parts of the current work amount to rewriting rather than extensive literature review.

8. Working Alone Is a Habit, but the Door to Collaboration Is Not Fully Closed

• The GPY and BFI work came from collaborations among multiple people, yet 张益唐 is accustomed to confronting problems alone. He believes that once research reaches a certain depth, “you can’t communicate with people, and communication may be difficult.” For him, thinking independently is simply a habit.
• He mentioned Wiles’s experience of secretly working on Fermat’s Last Theorem. 曼琪 added that Wiles made a mistake midway and nearly gave up, later crossing the obstacle through collaboration with his former student Richard Taylor.
• Asked whether he might take a similar approach, 张益唐 did not turn independence into a doctrine: “Maybe, it’s possible. I haven’t decided yet.” The current bottleneck has shifted from generating the core idea to organizing a complex, difficult-to-write proof that others must be able to accept.

9. AI Can Expand a Mathematician’s Field of Vision but Cannot Replace Exploration of the Unknown

• 张益唐 believes AI will promote mathematics because “many of the problems AI faces today are, I believe, mathematics at bottom.” Explaining the black-box mechanisms of large models could itself be an interesting use of mathematics, but he says he might be interested in that “in the future”; it is not his current focus.
• On claims that OpenAI and Google models have reached IMO gold-medal level, he refuses to judge without knowing the details. Even if true, competition problems have known answers, are timed by the hour and can be trained through patterns; research faces “the future and the unknown,” with entirely different cycles and exploratory demands.
• His clear judgment is that without humans, AI cannot solve the Riemann hypothesis independently. Computer-assisted proofs of the four-color theorem once led some to predict that traditional mathematics would be replaced. Decades later, computers have played a major role, but the main research still depends on people. AI is likewise “very powerful and very useful,” but he cannot see it fully replacing the human brain.
• If he were setting a problem for AI, he would give it hundreds of years of mathematical papers from the world’s major libraries and ask it to regroup and reassess them, looking for connections people had missed. That would exploit the machine’s ability to process vast amounts of material, rather than pretending it already possessed the full originality of a research mathematician.

10. There Is Still No Road Map to the Riemann Hypothesis

• 曼琪 asked whether his future ambitions included the Riemann hypothesis. 张益唐 first answered, “I haven’t thought about it,” then acknowledged that he had tried it 20 or 30 years ago. There is no contradiction: he still “hasn’t seen a route,” and is nowhere near the stage of formally declaring a campaign.
• The host called the Riemann hypothesis one of the hardest and most important problems in number theory and said many people were challenging it quietly. 张益唐 only replied, “I won’t talk about it, and I don’t even dare to talk about it.” His caution draws a clear boundary against his statement that the Landau–Siegel zero problem “has already been done.”
• Asked why the twin prime problem, the Landau–Siegel zero problem and the Riemann hypothesis are so compelling even if they may have no direct practical use, he offered no utilitarian explanation: “I can’t explain it. Maybe I was born this way. I just like this stuff.” He also noted that some basic number theory has already been applied to coding theory, computing and cryptography.

11. Mathematical Talent Cannot Be Priced Once by Olympiads and Early Precocity

• At around 9, 张益唐 read about Fermat’s Last Theorem and Goldbach’s conjecture in The Ten Thousand Whys and became interested in number theory. That edition did not even mention the twin prime conjecture. His interest was not mapped out by parents or teachers: “There simply wasn’t anyone specifically inspiring me.”
• He never attended high school and, strictly speaking, barely attended middle school. He spent time at a May Seventh Cadre School and worked in a lock factory around Beijing’s Baiwanzhuang. He entered Peking University’s mathematics department in 1978; the host said he should have been 23 at the time. He was first admitted to computational mathematics, then shifted his full attention to theoretical mathematics and number theory after his third year.
• 王虹’s move from aerospace and geodesy into mathematics, without an early Olympiad halo yet with meaningful achievements, led 张益唐 to stress that assessing talent “is not a short-term matter.” Strong Olympiad performance means a higher probability of doing well later, but it does not mean every talented person must show it early.
• He believes the most important trait for a mathematician is focus: “Don’t keep changing direction or wavering all day.” Focus itself is a talent and is connected to intelligence. In early education, children should first enjoy the satisfaction of “a problem I couldn’t solve, and then I finally solved it,” before gradually developing further.

12. The Core of Returning to China Was Giving Researchers Their Time Back

• From leaving for the United States to pursue a PhD in 1985 to returning to China in 2025, exactly 40 years had passed. 张益唐 believes the domestic environment for basic science is “very good,” with national investment and encouragement for exploration. After invitations from multiple institutions, he ultimately chose Sun Yat-sen University, also considering the location, climate and his preference for warmer regions.
• In his view, the core of the Institute for Advanced Study in Princeton is that it “brings together a group of the world’s first-rate scholars.” Its institutional value is that it does not require teaching, giving researchers all their time. More than a month after returning, he said Sun Yat-sen University had provided “quite a good” amount of space, with little routine administration.
• He stated clearly that he “cannot be a leader,” being neither suited nor adapted to administration. His near-term plan is simply to finish the paper first. He may later consider supervising students; there will soon be seminars and opportunities to interact with students, but at least for now he will not teach large lecture courses.
• In the United States he supervised only 2 doctoral students. After returning, he “might set a higher bar” and hopes to train several genuinely strong students. The criteria are not résumé labels but interest and mathematical sensitivity, which must be assessed gradually through discussions of specific problems. If a student eventually changes fields, he would not want that, but would respect the choice.

13. Fame, Immortality and Public Expression Are Not in His Objective Function

• If someone writes his biography in the future, 张益唐 hopes its core will not be his personal story but the twin prime problem itself: “What is this problem about?” How the history developed and how it was ultimately solved. He confirmed that he had seen Gödel’s grave in Princeton and searched for traces of Gödel at the University of Vienna but found nothing. He does not treat his own immortality as a research goal: “I don’t pursue it. I don’t care about that sort of thing.”
• When asked about aspects of his experience that people rarely ask about, he sometimes draws the line directly: when asked whether there was something he had always wanted to say, he replied, “I don’t want to bring it up anymore” and “I don’t want to say anything.” On not returning to China for 25 years, he offered only the brief explanation that there had been many “messy things” at the time and that he had not wanted to return yet.
• He once said that without mathematical talent he “might” have lived more happily by ordinary standards, but insisted this was an unverifiable hypothetical. Even during periods that outsiders considered low points, he was mostly content. Professionally, he also “cannot retire,” because the interest remains.
• Life is finite, and the number of important problems one can complete is “actually just a few.” He therefore does not manufacture a sense of urgency or calculate the price of immersing himself in mathematics. The final answer is simply: “That is the joy of doing it for its own sake… You find it interesting, what you are doing is interesting, and that is enough.”