Goldbach's Conjecture

Biodata

Background

Goldbach's Conjecture is one of ten problems the system requires Jiang Ye to solve before moving from mathematics to other disciplines. He treats these objectives as preparation for acquiring the knowledge needed to build a spaceship 9.

A young scholar places it alongside the Four Color Problem and Fermat's Last Theorem in the second tier of mathematical problems, below the Millennium Prize Problems. At that point, Goldbach's Conjecture is the only unsolved problem among those three 13.

The novel traces its history to a letter sent from Moscow to Berlin in 1742. During discussion of Jiang Ye's attempt, viewers identify Chen Jingrun's 1966 “1 + 2” result as its latest major advance, using the conjecture's long history to argue against a twenty-year-old solving it 22 27.

Jiang Ye's Proof

Initial Attempt

After completing the Beal Conjecture, Jiang Ye immediately begins working on Goldbach's Conjecture at the library. Mathematicians who recognize his new objective react with disbelief, while specialists worldwide join the livestream—many intending to locate errors 20 21.

Harald Helfgott finds no obvious problems in the developing argument. He praises Jiang Ye's new method and recognizes traces of the circle method, contour integrals, and the residue theorem, but withholds judgment until he can consult other mathematicians 22.

Jiang Ye completes a manuscript of more than fifty A4 sheets and receives the system's confirmation that it is entirely correct. His Ultimate Learning State then shuts down because his mental energy is nearly exhausted 23.

Independent Review

The International Mathematical Union compiles images of the manuscript and appoints six leading professors to review it independently. The named reviewers include Faltings, Helfgott, and Iwaniec; no completion date is initially given 24.

The Union also requests a presentation on behalf of more than two hundred mathematicians. The Ten Conglomerates instruct the production team to postpone it and find a way to stop Jiang Ye from continually solving mathematical problems 22.

Recorded Presentation and Acceptance

Believing humanity has disappeared, Jiang Ye independently decides to record his proof as evidence of human civilization's intellectual achievements. He takes his research to his university, finds a camera with a removable memory card, and sets it up in a classroom 24 25.

His presentation explains both the mathematical tool he developed and its application to the conjecture. Although addressed to a future intelligent civilization, its detailed explanations allow the watching mathematicians—including the review panel—to assess the argument much faster 26 27.

James Maynard, who had publicly dismissed the manuscript and burned his own research in anger, becomes convinced while listening. He recognizes the use of Bombieri's Theorem and acknowledges that Jiang Ye has succeeded 27.

Minutes after the presentation ends, the International Mathematical Union announces that the panel has completed its review ahead of schedule. It declares the proof completely correct and without flaws 28.

Mathematical Methods

Group Construction Method

Jiang Ye introduces the Group Construction Method as the method of studying the overall structure of group theory, with cyclic groups as its central concept 26.

• Presentation structure: He first explains the method itself, then applies it to Goldbach's Conjecture 26 27.
• Supporting theorem: Maynard identifies Bombieri's Theorem as a decisive component of the argument 27.
• Final derivation: The presentation concludes by deriving theorem 1 from equation (30) and lemmas 8, 9, and 10 27.
• Wider value: Princeton's Institute for Advanced Study considers the method an outstanding theoretical tool for analytic number theory, independently of the conjecture's solution 28.

Formulations Shown

• Andrew Granville recognizes Jiang Ye's written statement, “Any integer greater than two can be expressed as the sum of three prime numbers,” as Euler's statement of Goldbach's Conjecture 21.
• During the recorded proof, Jiang Ye introduces a sufficiently large even number (N) and considers representations of the form (N=p_1+p_2) 27.

Impact

Academic Recognition

• International Mathematical Union: Calls the solution one of the most important academic achievements since the Poincaré Conjecture 28.
• University recruitment: Columbia offers Jiang Ye a full professorship even before the proof's final confirmation; other institutions also announce offers 24.
• Jiang Ye's university: Announces early completion of his studies, the award of graduate qualifications through a doctorate, and an invitation to teach 28.
• Princeton's Institute for Advanced Study: States that it is considering inviting him to head its mathematics department 28.

Conflict with the Survival Show

• Change of focus: Viewers complain that the livestream has become a mathematics program rather than a test of survival and national character 21.
• Production backlash: The directors initially refrain from intervening when Jiang Ye leaves the library, then become furious when he resumes mathematics through his university presentation 29.
• Competing demands: The International Mathematical Union wants him kept on the program without interference, while Dragon Country officials seek his return for safety 28 29.
• Fraud allegations: A later campaign claims that his solved conjectures were supplied by the government. A senior official rejects this reasoning, arguing that their honor, money, and status would make them too valuable to surrender to a stand-in 31.

Notable Quotes

“That concludes my proof of Goldbach's Conjecture. I believe I have proven this proposition.”
— Jiang Ye, ending his recorded presentation 27

“I am Jiang Ye, a Dragon Country citizen and an insignificant member of humanity. Thank you!”
— Jiang Ye, addressing the camera after completing the proof 27