Chapter Summaries

Xiao Yi completed the paperwork to miss his exams and traveled to Princeton with Academician Xun Yong ahead of his Twin Prime Conjecture report. On January 8, as the crowded conference awaited him, reporters swarmed Xiao Yi when he arrived.

• Xiao Yi asked Professor Liu Bin for leave and an exemption from the Mathematical Analysis exam, then solved three difficult questions to secure Liu Bin’s approval.
• Xiao Yi submitted his exemption forms and flew to Shangjing, where he met Academician Xun Yong and discussed mathematical research with him.
• Xiao Yi and Xun Yong traveled to Princeton, while Math and Physics Events posted about the upcoming report conference and the enormous public interest in it.
• On January 8, more than five hundred people gathered in Princeton University’s largest lecture hall as reporters waited outside for Xiao Yi.
• Xiao Yi arrived by car, and the reporters rushed toward him.

Xiao Yi faced reporters and began his Princeton lecture on the Twin Prime Conjecture, explaining the parity problem behind sieve methods. He reached the crucial section of his proof, where the audience waited to see how he classified numbers by the parity of their prime factors.

• Xiao Yi told reporters he trusted his proof but left its judgment to the mathematics community, and credited Professor Gerd Faltings with advising him to study the Elliott-Halberstam Conjecture.
• Schultz and Faltings met Xiao Yi backstage, where Faltings praised his paper and Schultz teased Faltings about his advice.
• More than a thousand people gathered at Princeton for Xiao Yi's lecture, while Deligne, Bombieri, Professor Sanak, and other luminaries encouraged him backstage.
• Xiao Yi opened his lecture by stating the Twin Prime Conjecture and reviewing its history and the role of sieve methods.
• Xiao Yi explained that avoiding the parity problem had stalled progress, then wrote expressions introducing his classification method as the mathematicians focused on the key part of his proof.

Xiao Yi explained the crucial steps of his proof of the Twin Prime Conjecture, showing how automorphic forms, the trace formula, duality, and Poisson summation overcame a key obstacle in sieve methods. He concluded that prime pairs differing by 2 occur infinitely often, and the audience rose in applause.

• Xiao Yi explained how automorphic theory and the trace formula helped bound the sums needed in his proof.
• Dan Goldston and Henrik Iwaniec recognized that Xiao Yi’s method could suppress the parity problem, while Terence Tao and Maynard discussed inviting him to collaborate.
• Zhang Yitang considered asking Xiao Yi to work with him on the Landau-Siegel zero conjecture and resolved to approach him after the presentation.
• Xiao Yi completed his proof and announced that prime pairs differing by 2 would continue to appear among the primes.
• Faltings began applauding, and the audience stood to applaud Xiao Yi’s conclusion.

Xiao Yi received acclaim after presenting a proof of the Twin Prime Conjecture, which the mathematicians accepted as correct. As the audience celebrated and Guo Xiaoming livestreamed the event, Wen Chenghua admitted he could not understand the presentation.

• The audience and mathematicians praised Xiao Yi's clear presentation and inventive techniques, and Deligne and Peter Sarnak expected him to win the Best Speaker Award.
• Faltings and Schultz discussed Xiao Yi's mathematical ability and joked about Schultz's past decision to repeat a year for the IMO.
• Students celebrated the proof, while Ivanets and Goldstone reacted emotionally to the resolution of a problem they had pursued for years.
• Guo Xiaoming stopped his text livestream and streamed video of Xiao Yi, telling viewers he would post photos later.
• Guo Xiaoming interviewed Wen Chenghua, who said he could not understand Xiao Yi's presentation.

Wen Chenghua praised Xiao Yi’s presentation proving the Twin Prime Conjecture, and the achievement quickly became a worldwide sensation. The chapter ended with people across China and abroad celebrating and discussing Xiao Yi’s proof.

• Wen Chenghua explained why most people could not follow Xiao Yi’s presentation and praised the rigor and originality of his proof.
• Wen Chenghua predicted that the proof would draw attention from the entire world because Xiao Yi had solved the Twin Prime Conjecture.
• CCTV News reported that Xiao Yi had proved the long-standing conjecture, and Academician Huang Ting described his work as a milestone for mathematics.
• Chinese social media users celebrated Xiao Yi’s achievement, shared photos and videos of him, and praised his talent and appearance.
• Foreign netizens and major international media discussed the proof, while Science and Nature featured it on their websites.

At Princeton’s mathematics conference banquet, Xiao Yi helped Wen Chenghua improve his dissertation paper and agreed to join Terence Tao and James Maynard’s research on the x²+1 prime problem, as well as Zhang Yitang’s Landau-Siegel Zero Conjecture project. The chapter ended with mathematicians congratulating Xiao Yi on winning the Cole Algebra Prize, leaving him surprised.

• Xiao Yi met Guo Xiaoming and Wen Chenghua, learned Wen’s paper had used his theory, and wrote a note suggesting a way to strengthen a proof.
• Wen Chenghua recognized Xiao Yi’s suggestion as a solution to the paper’s weakest section and hurried off with Guo Xiaoming to revise it after taking a photo with Xiao Yi.
• Xiao Yi met Terence Tao and James Maynard, who invited him to join their research on the x²+1 prime problem, and he accepted.
• Zhang Yitang arrived to invite Xiao Yi to work on the Landau-Siegel Zero Conjecture, and Xiao Yi agreed to join his project too.
• A group of mathematicians congratulated Xiao Yi on winning the Cole Algebra Prize, leaving him dumbfounded.

Xiao Yi was stunned when Professor Sanak and AMS vice president Jesus Loera told him he had won the Cole Algebra Prize for his work on the Elliott-Halberstam Conjecture. As the news spread, journal editors fought to publish his Twin Prime Conjecture proof, while Xiao Yi and his collaborators began discussing their future work.

• Professor Sanak and Jesus Loera told Xiao Yi that the AMS had announced him as the Cole Algebra Prize recipient, recognizing his work on the Elliott-Halberstam Conjecture.
• Professor Loera explained that Xiao Yi qualified through his papers published in Annals of Mathematics, and Professor Sanak offered to discuss submitting his Twin Prime Conjecture proof there.
• Professor Loera and Edgar Barton each tried to persuade Xiao Yi to submit the proof to their journals, and they and Professor Sanak got into a fight over it.
• Terence Tao remarked on the commotion, while Zhang Yitang and Maynard suggested continuing their discussion; Xiao Yi and the others then began planning their collaboration.
• News of Xiao Yi's prize spread through the media and online, and USTC hung new banners celebrating his award alongside its banners about the Twin Prime Conjecture.

Xiao Yi received the Best Presenter award at the Princeton mathematics conference, and Peter Sarnak announced that the Twin Prime Conjecture paper would be published in the Annals of Mathematics. After the conference, Xiao Yi traveled to UCLA with Terence Tao and James Maynard to begin research discussions, while preparing to receive the Cole Algebra Prize.

• Peter Sarnak praised Xiao Yi and the Twin Prime Theorem during the conference's closing ceremony.
• Xiao Yi received the Best Presenter award and thanked Peter Sarnak.
• Peter Sarnak announced that Xiao Yi had signed a copyright transfer agreement with the Annals of Mathematics for his proof.
• Xun Yong asked Xiao Yi to stay with Terence Tao and Zhang Yitang and avoid unfamiliar places during his stay in the United States.
• Xiao Yi boarded a flight to Los Angeles with Terence Tao and James Maynard, while Zhang Yitang flew to Santa Barbara.
• Terence Tao settled Xiao Yi into a UCLA dormitory and arranged to begin project discussions the next morning.

Xiao Yi worked with Terence Tao and Maynard on a sieve method, then learned that his Parity Check Classification Sieve had raised concerns among banks about RSA encryption. Leonard Kleinrock asked the group to help address the security risk.

• Xiao Yi, Terence Tao, and Maynard discussed how to refine their sieve approach and made progress before pausing their work.
• Leonard Kleinrock arrived at Terence Tao's office and greeted Xiao Yi, the young mathematician who had proved the Twin Prime Conjecture.
• Kleinrock invited Xiao Yi to attend his computer science classes, and Xiao Yi said he would go if he had time.
• Kleinrock explained that banks wanted help limiting the damage Xiao Yi's Parity Check Classification Sieve could do to RSA encryption.
• Xiao Yi reacted with an innocent expression as the others looked at him.

Professor Kleinrock introduced the Millionaires' Problem, asking how Alice and Bob could compare their wealth without revealing their assets. Xiao Yi rapidly devised a protocol on the blackboard, leaving Kleinrock and the mathematicians astonished by his solution.

• Maynard and Terence Tao discussed how Xiao Yi's classification sieve could make RSA encryption more vulnerable, while Professor Kleinrock warned about the risk to banking security.
• Kleinrock explained the Millionaires' Problem, in which Alice and Bob needed to compare their wealth without disclosing their exact assets.
• Xiao Yi began writing a protocol using permutations, encrypted values, and a prime modulus to let Alice and Bob compare their assets privately.
• Xiao Yi finished the protocol and said it kept both assets confidential unless they were equal, though he could not write the program himself.
• Kleinrock asked Terence Tao whether mathematicians solved computer science problems so quickly, and Tao called Xiao Yi a rare exception.

Xiao Yi joined Kleinrock, Terence Tao, and Maynard on a project to develop secure cross-chain transactions, after demonstrating how multi-party comparisons could be built from secure computation. A week into their research, Xiao Yi considered whether his classification sieve could speed up cryptocurrency mining and help him create a blockchain venture.

• Xiao Yi extended his solution to the Millionaires' Problem to comparisons among n people, and Kleinrock explained the project's goal of creating a secure protocol for atomic transactions across blockchains.
• Kleinrock told Terence Tao, Maynard, and Xiao Yi about the project's funding and invited them to join; Xiao Yi and Maynard agreed despite their limited time and computer-science experience.
• The group added cross-blockchain transaction protocols to its research schedule alongside the x^2+1 prime number problem.
• A week later, Xiao Yi studied how the classification sieve might help crack encrypted transactions and wondered whether it could make Bitcoin mining faster.
• Xiao Yi considered creating mining software or a blockchain of his own, while recognizing that it would need widespread recognition to succeed.

Xiao Yi developed a new complex-domain polynomial expansion that he used to protect cross-chain transactions from attacks by the classification sieve. After demonstrating it to Terence Tao and Maynard, he left them impressed and Kleinrock alarmed by something happening offscreen.

• Xiao Yi considered how to secure cross-blockchain transactions against the classification sieve while preserving RSA compatibility.
• Xiao Yi used étale fundamental groups and L-functions to derive a method that removed the classification sieve’s threat to cross-chain transactions.
• Xiao Yi realized his new technique might also help solve Zhang Yitang's Zero Conjecture and potentially the Riemann Hypothesis.
• The next morning, Xiao Yi presented his derivation to Terence Tao, Maynard, and Kleinrock, explaining how a Fourier Transform could preserve RSA decryption while hiding the original information.
• Terence Tao recognized Xiao Yi’s new polynomial expansion as a potentially useful method in complex analysis, and Maynard praised it.
• Kleinrock suddenly cried out, “Don't jump!”

Xiao Yi and the mathematicians confirmed that his polynomial could strengthen RSA against classical and quantum attacks, then used it to outline a secure multiparty protocol for cross-chain transactions. The group concluded that the design could prevent platform embezzlement and greatly reduce hacking risks, though protocol development still remained.

• After Kleinrock climbed onto the table and was talked down, Terence Tao explained that Xiao Yi's polynomial raised RSA's cracking complexity to exponential levels, even against Shor's algorithm.
• Terence Tao said the polynomial would be easier to learn than ECC, and Kleinrock asked why he could not understand it; Xiao Yi, Terence Tao, and Maynard each took the blame for not explaining it clearly.
• Terence Tao shifted the discussion to protocol design, and Xiao Yi announced that he had ideas for the cross-chain protocol's multiparty security problem.
• Xiao Yi described a protocol in which buyers and sellers generated polynomials from matching codes and shared randomly selected terms processed with a Fourier Transform.
• Terence Tao said the design could prevent platform embezzlement and make it extremely difficult for hackers to intercept both parties' information, while Kleinrock praised it as a genius design.

Xiao Yi used his new polynomial expansion to raise the proportion of Riemann zeta-function zeros proved to lie on the critical line to 50%, then posted his paper on arXiv. The paper appeared soon after, and Faltings noticed it in Germany and was stunned to see Xiao Yi’s name.

• Xiao Yi and the professors agreed that he would receive six million dollars from the World Bank protocol reward, while Terence Tao, Kleinrock, and Maynard would divide the rest and handle the remaining work.
• Xiao Yi set aside the protocol and organized a paper introducing Xiao’s Polynomial Expansion.
• After studying Conrey’s method and later work on the Critical Line Theorem, Xiao Yi combined it with his expansion and calculated a new result of 50%.
• Xiao Yi organized the paper and uploaded it to arXiv, where it appeared shortly after he went to sleep.
• Faltings found Xiao Yi’s paper at the Max Planck Institute for Mathematics and was shocked by its title and author.

Faltings and Schultz read Xiao Yi’s new paper and realized its polynomial expansion could be more significant than its advance in approximating the Riemann Hypothesis’s critical line. As more mathematicians discovered the paper, they too were stunned by the method.

• Faltings read Xiao Yi’s paper and found that it had pushed the Critical Line Theorem to 50%.
• After carefully checking the paper’s first eight pages, Faltings recognized Xiao’s expansion as a powerful new tool for complex analysis and other mathematical problems.
• Faltings considered applying Xiao’s expansion to his own research and thought it could help Zhang Yitang solve the Siegel-Zero Conjecture.
• Schultz arrived with coffee, learned Xiao Yi had written the paper, and began reading it after Faltings left for lunch.
• Schultz agreed that the expansion could become an enduring method in the Complex Domain and might play a major role in a future proof of the Riemann Hypothesis.
• More mathematicians discovered Xiao Yi’s paper and were left astonished by its first eight pages.

Xiao Yi's Polynomial Expansion drew growing attention as mathematicians used it to advance the Riemann Hypothesis's Critical Line Theorem to 55%. At a group meeting, Kleinrock offered Xiao Yi up to $150,000 in project funds to spend, leaving Xiao Yi unsure what to buy.

• MathOverflow users debated Xiao Yi's expansion and its Riemann Hypothesis result, while a Cambridge professor urged readers to study the new method.
• Two new papers pushed the critical line to 53% and 55%, prompting mathematicians to hope that proving the Riemann Hypothesis might be within reach.
• Terence Tao and James Maynard confronted Xiao Yi about his rapid breakthrough, while Kleinrock reassured them that Xiao Yi was a rare exception.
• Kleinrock said the cross-chain protocol paper was nearly complete and that most rival teams were still struggling with the classification sieve problem.
• Kleinrock offered Xiao Yi up to $150,000 from the project budget for related purchases, and Xiao Yi repeatedly declined.

Xiao Yi agreed to use part of his funding to buy a server for building a blockchain and accepted an invitation to present his Xiao's Polynomial Expansion at the Joint Mathematics Meeting. Four days later, the cross-chain transaction protocol passed simulations with tens of millions of attacks and no data leaks, and Kleinrock prepared to send the program and paper to the World Bank.

• Xiao Yi decided to use some of the project funds to buy a server and explore building a blockchain and developing a mining algorithm.
• Professor Doggs invited Xiao Yi to present his Xiao's Polynomial Expansion at the Joint Mathematics Meeting in San Jose, and Xiao Yi agreed.
• Xiao Yi and the researchers completed their cross-chain transaction protocol and tested it using computers and a UCLA server.
• The simulations withstood tens of millions of attacks without leaking data or allowing funds to be intercepted.
• Kleinrock told the team to package the program and paper and prepare them for submission to the World Bank.

World Bank CIO Brian Winston reviewed Xiao Yi’s cross-chain transaction protocol and watched it pass the bank’s simulations without leaks. After recognizing Xiao’s mathematical insight in the protocol, Winston decided to attend Xiao Yi’s upcoming lecture at the Joint Mathematics Meeting.

• Brian Winston learned that a team led by Professor Kleinrock had submitted a cross-chain protocol proposal featuring seventeen-year-old Xiao Yi.
• Winston watched the laboratory simulation record up to one hundred million attacks with zero leaks.
• Winston read the protocol and realized Xiao Yi had applied Xiao’s Polynomial Expansion to computer security and developed it while addressing Classification Sieve attacks.
• Winston reviewed the final report, saw that the protocol passed every test, and planned to inform the banks and companies that it was ready.
• Winston learned Xiao Yi would lecture on Xiao’s expansion at the Joint Mathematics Meeting and asked his assistant to put the lecture on his schedule.

As the Joint Mathematics Meeting drew near, mathematicians and computer scientists flocked to San Jose for Xiao Yi’s lecture, leaving the organizers baffled by the surge in registrations. Jones Charles revealed that Xiao Yi had already solved the Parity Check Classification Sieve attack, though the reason behind the computer scientists’ interest remained only partly explained.

• Mathematicians on MathOverflow complained about making another trip to hear Xiao Yi, and an Oxford professor scolded them for attending lectures they could not understand.
• Another Oxford professor exposed Professor Max’s complaints as jealousy over missing a seat at the Princeton conference, and Max deleted his comment.
• Hales Carl and Adam Douglas discussed how Xiao Yi’s lecture had increased attendance and overwhelmed the Joint Mathematics Meeting organizers with extra registrations.
• Hales Carl questioned why nearly two hundred computer science experts, including cryptographers and technology executives, had applied to attend Xiao Yi’s lecture.
• Jones Charles told Hales Carl that some computer scientists were coming because of the Parity Check Classification Sieve attack, but Xiao Yi had already solved it.

At the 2021 Joint Mathematics Meeting, Xiao Yi prepared to present his Polynomial Expansion to a packed audience of mathematicians and computer scientists. Before the talk began, attendees discussed his solution to classification sieve attacks and gathered to hear what new insights he would offer.

• Jones Charles told Hales Carl and the organizing committee that Xiao Yi had devised a solution to classification sieve attacks during a blockchain security project.
• Hales Carl closed the conference application channel after another fifty people applied to attend.
• The 2021 Joint Mathematics Meeting opened in San Jose, and Xiao Yi's talk was scheduled for the evening so all attendees could attend.
• Brian Winston, Silvio Micali, and Prabhakar Raghavan discussed their interest in Xiao Yi's work and his computer security breakthrough.
• Hales Carl introduced Xiao Yi's Polynomial Expansion and its progress on the Riemann Hypothesis, and Xiao Yi walked onstage to begin his presentation.