The Reincarnated Science Whiz

Chapter 48: The Lecture

September 21: The sky was clear and sunny. The freshmen began their classes. In this first semester, they were required to take nine compulsory courses: Mechanics, Advanced Mathematics, Linear Algebra, Introduction to Computing, College Chinese, Military Theory, Ideological and Moral Education, College English, and Physical Education. Military Theory had already been completed during military training, which meant there were nine courses remaining for the first semester. In addition, there were several elective courses, though the elective classes hadn’t officially started yet. Qin Yuanqing and his classmates attended Mechanics together. Originally, Qin Yuanqing had high expectations, thinking that since the instructor was a professor, the lectures would be excellent. However, after listening for about twenty minutes, Qin Yuanqing felt like saying to the professor, “Come on, Professor, we’re not high school students—you could go into more depth.” Qin Yuanqing was deeply disappointed. “Is that it…? I might as well just study on my own! After attending one class for each of these courses, Qin Yuanqing lost interest in attending lectures. Every time class started, he would sit in the back row and read on his own. Four days flew by, and Qin Yuanqing posted a lecture announcement on the bulletin board next to the library: “An academic lecture titled ‘The Twin Prime Conjecture’ will be held tomorrow at 9:00 a. m. in Lecture Hall XX…” As soon as he saw the words “Twin Prime Conjecture,” Qin Yuanqing’s interest was immediately piqued. For the past few days, he had been pouring all his energy into cracking the final hurdle of the Twin Prime Conjecture, so he never expected a mathematician to be coming to the school to give an academic lecture on the very same topic. Interesting! Qin Yuanqing looked intrigued. Since he had no classes tomorrow morning, he could go listen and see what level of research the speaker had achieved on the “Twin Prime Conjecture.” The Twin Prime Conjecture is a famous unsolved conjecture in number theory. It was formally proposed by Hilbert as Problem No. 8 in his report at the 1900 International Congress of Mathematicians and can be described as “there exist infinitely many twin primes.” Twin primes are pairs of prime numbers that differ by 2. For example, 3 and 5, 5 and 7, 11 and 13, …, 10,016,957 and 10,016,959, and so on, are all twin primes. The Prime Number Theorem describes the tendency for prime numbers to become increasingly scarce as the number of primes approaches infinity. Twin primes, like prime numbers, exhibit the same trend, and this trend is even more pronounced than that observed for prime numbers. Therefore, the Twin Prime Conjecture is counterintuitive. Regarding twin primes, there have been two major achievements over the past century. One was in 1920, when the Norwegian mathematician Viggo Brown, using the famous sieve method, proved that 2 can be expressed as the difference of two numbers each having at most 9 prime factors—a result that comes quite close to the Twin Prime Conjecture. If we refine the condition “numbers with at most 9 prime factors” in this proof to “numbers with at most 1 prime factor,” we can prove the Twin Prime Conjecture. The second major result was achieved in 1966 by the Chinese mathematician Chen Jingrun using the sieve method. He proved that there are infinitely many prime numbers p such that p + 2 is either a prime number or the product of two prime numbers. This result is very similar to his result regarding the Goldbach Conjecture. As for the achievements of the subsequent forty years, none have strayed from these two results. “Zhang Yitang?” Qin Yuanqing muttered to himself as he looked at the name of the lecture’s presenter. After doing some research, he discovered that this person was actually quite remarkable. He earned his bachelor’s degree from the Yan’an University Department of Mathematics from 1978 to 1982; from 1982 to 1985, he studied under the renowned mathematician Professor Pan Chengbiao at Yan’an University to earn his master’s degree; in 1992, he graduated from Purdue University in the United States with a Ph. D. ; and he currently teaches in the Department of Mathematics at the University of New Hampshire in the United States. His research focuses on number theory. Qin Yuanqing continues to work on the Twin Prime Conjecture. He has a feeling that a complete proof of the Twin Prime Conjecture is not far off—with just a little more effort, he can achieve it. At 8:30 a. m. , the lecture hall was nearly full. Qin Yuanqing found a seat in the back row and buried his head in a specialized mathematics textbook he had borrowed from the library. By 8:50 a. m. , the lecture hall was packed to capacity; even the aisles were filled with people. Hearing people argue over seats, Qin Yuanqing realized that the audience wasn’t limited to Shuimu University students—students from other universities, such as Yan Da University, had also come to attend the lecture. It was infuriating for Shuimu students to find no seats on their own campus, so naturally they wanted to chase away students from other schools. But those students weren’t pushovers either. “Why shouldn’t we be here? Your school hasn’t banned us from attending,” they argued. If the university isn’t stepping in, who do you think you are? At exactly 9:00 a. m. , the entire lecture hall fell silent. A middle-aged man wearing glasses stepped up to the podium, opened his laptop—which was connected to the screen—while the host introduced the man’s identity and status. The audience listened quietly and attentively, opening their notebooks to take notes. “……As we all know, a prime number is a natural number with exactly two factors. You may have memorized the first 100 prime numbers back in middle school. Twin primes, on the other hand, are pairs of prime numbers whose difference is 2—that is, p and p+2 are both prime. Examples include 3 and 5, 5 and 7, 11 and 13, 17 and 19, and so on. As numbers get larger, the number of observable twin prime pairs becomes increasingly scarce.” “There are 8 twin prime pairs within 100, while in the range from 501 to 600, there are only 2 pairs. As prime numbers grow larger, the next prime should be farther and farther from the previous one. However, a conjecture as famous and important as the Goldbach Conjecture asserts that there are infinitely many pairs of primes that differ by exactly 2—such as 3 and 5, 5 and 7, and even this one…” At this point, Professor Ren wrote a line of numbers on the blackboard. [2003663613 × 2195000 − 1 and 2003663613 × 2195000 + 1] Zhang Yitang continued, “There are infinitely many prime numbers whose difference is 2. This is the famous Twin Prime Conjecture.” Qin Yuanqing observed how Zhang Yitang had gradually introduced the Twin Prime Conjecture, starting from the basics and moving to more advanced concepts—even students not majoring in mathematics could follow along and understand what he was trying to convey. Sure enough, the students—whether from the Department of Mathematics or non-math majors with a passion for the subject—listened intently and with great interest. However, the lecture soon began to delve into more advanced topics. For example, he discussed historical achievements in proving the Twin Prime Conjecture, such as the “Weak Twin Prime Conjecture” proposed in 2005 by mathematician Dan Goldstone and two colleagues, which states that there are infinitely many pairs of prime numbers whose difference is less than 16. In the entire classroom, the vast majority of people looked completely lost, while only a few were able to keep up. “Junior, do you understand this?” a student wearing glasses, sitting right next to Qin Yuanqing, asked in a low voice. “It’s simple!” Qin Yuanqing replied with a smile. “Yingying, don’t listen to him showing off—he’s just a freshman. There’s no way he could possibly understand it,” said the young man sitting next to the girl, glaring at Qin Yuanqing with hostility in his eyes. Qin Yuanqing shrugged indifferently—he was almost finished with the proof of the “Twin Prime Conjecture,” so why would he lie? After the lecture ended, Qin Yuanqing went to the library, quietly pondering the final proof. The “Twin Prime Conjecture” was even more difficult than the “Zhou Conjecture.” He opened his laptop and saw a QQ notification about a new email. Qin Yuanqing clicked on it—it was a reply from *Mathematics Chronicles*. The message stated that his paper had passed the journal’s review and would be published in the upcoming issue. Qin Yuanqing took a look: the issue was due out in just a few days—on September 30—the very day he was scheduled to visit Jing Tian’s home. What a coincidence. “Mathematics is a very rigorous discipline, and it is the foundation of all other disciplines.” “Whether it’s the natural sciences or engineering, mathematics is a must—and you have to study it in depth.” “College entrance exam scores merely represent the culmination of high school; they do not define college. College is an entirely new beginning, and some students must not dwell on past glories. That’s very dangerous!” the math teacher said meaningfully. The students all turned to look at the last seat, where Qin Yuanqing was dozing off; they all knew the math teacher was talking about him. “Dude, wake up, wake up!” The chubby boy sitting right in front of Qin Yuanqing hurriedly reached over and tugged at Qin Yuanqing’s shirt. “What’s going on? Is there an earthquake?” Qin Yuanqing was startled and blurted out. The entire classroom erupted in laughter, while the math teacher glared at Qin Yuanqing with a grim expression, saying with disappointment, “Qin Yuanqing, I know you’re a CMO and IMO gold medalist, and math is your strong suit—but that was in high school. Now that you’re at Shuimu University, there are many CMO and IMO gold medalists among the undergraduates here, and none of them fall asleep in advanced calculus class.” Qin Yuanqing, still half-asleep, replied lazily, “Professor, you’re explaining it too simply. I already know this.” Seeing that the Calculus professor’s face had darkened to the point of looking as if rain might pour from it, Qin Yuanqing shrugged and said, “If you don’t believe me, Professor, just give me a problem to solve. If I can’t solve it, I’ll pay close attention in class from now on.” “You said it—don’t go back on your word!” The calculus teacher wrote a problem directly on the blackboard: “Find the area of the portion of the sphere x² + y² + z² = a² (a > 0) enclosed by the planes z = a/4 and z = a/2.” Qin Yuanqing looked at the problem and grumbled to himself—he’d expected something much harder, but it turned out to be just that simple. Qin Yuanqing stood up and walked to the blackboard. He picked up a piece of chalk and drew the x, y, and z coordinate axes. The sphere had its center at (0, 0, 0) and a radius of a. He then drew the planes z = a/4 and z = a/2. Using the principle of proportionality, he derived the area of the portion of the sphere enclosed by the two planes. He then wrote down a second proof approach beside it, directly using calculus to find the area. The students watched in astonishment as Qin Yuanqing wrote five different calculation methods on the blackboard, filling the entire surface. Aside from the first one, which they could understand, the remaining four proofs left them completely baffled. Holy crap! A genius is truly a genius! Even the calculus professor was speechless—of Qin Yuanqing’s five proofs, the last three were topics typically encountered by graduate students or even doctoral candidates. And now, here they were, coming from Qin Yuanqing, a freshman.