The victory over the biggest opponent, the No. 1 Middle School Team, is equivalent to removing the biggest obstacle on the way to winning the championship, and everyone in the Huai'an Fu Squadron is in a good mood and feels that the future is suddenly bright. Jiang Shuiyuan was also relaxed, and felt that he could stop for a while in the next few weeks, read books, and write. Who knew that pressing the gourd and scooping it up, within a few days, Ge Juntian summoned him and Zhang Jin, and when they met, he asked with a face: "What do you think of the book I gave you?"

Jiang Shuiyuan's eyes widened: "Didn't you say give us a month? If I'm not mistaken, it's only been less than two weeks since that day!"

Ge Juntian didn't feel a little ashamed, but said vigorously: "That's right, but later I found out that I overestimated your ability, not to mention giving you a month, even giving you a semester or an academic year, you may not be able to see what the way is." And the longer you look at it, the greater the blow it may hit you, which will affect your confidence and determination to devote yourself to mathematical research. It is precisely in view of this that I have decided to give you a general introduction to what the Hilbert problem is and the most promising research directions for breakthroughs, so as to give you appropriate help and guidance. Of course, before I do that, I'm going to see what you know about the Hilbert issue.

Zhang Jin was already stuttering, but now he was nervous and stammered even more, staring at his toes and stumbling: "I, I haven't even finished reading the first 'continuum hypothesis' question......"

"Do you understand?"

"Finished, completely seen, incomprehensible......"

Ge Juntian was slightly disappointed, turned his head and asked Jiang Shuiyuan: "What about Jiang Shuiyuan, should you go to participate in that national studies contest again, haven't you read my book?"

"I read the book you gave, but like Zhang Jin, I didn't understand it at all. "In fact, Jiang Shuiyuan took the time to finish the book in the first week after he got it, but it was only finished. This thing is completely different from the "Thirteen Classics" and "Twenty-four History" and other classics of Chinese culture, the latter can basically understand seven, seven, eight or eight after reading it, even if you don't understand the tenths and two, you can also put it in your heart to compare with other classics, chew carefully, once you realize it, it is an excellent paper;

Jiang Shuiyuan's memory is unparalleled, and he has studied high school mathematics textbooks by himself, and read a lot of introductory books on advanced mathematics with Ge Juntian, but after reading this "Hilbert Problem and Research Progress", he remembers a word shadow, and the content is incomprehensible at all, and his brain hurts a lot when he thinks about it.

"Oh, is it?" said Ge Juntian, picking up the chalk and quickly writing down the Arabic numerals from 1 to 23 on the blackboard, and then continued to ask, "Then what can you tell me that has been solved so far?"

Jiang Shuiyuan rolled his eyes: "The second problem, 'non-contradiction of arithmetic axiom system', was properly solved by the American-Czech mathematician Kurt Gödel in 1931 and then the German mathematician Genz using the out-of-limit induction method in 1936 to prove the non-contradiction of the arithmetic axiom system. ”

"That's good!" Ge Juntian erased the word "2" from the blackboard, "What else?"

Jiang Yuanyuan said: "The third problem, 'There are two tetrahedra of equal height, which cannot be decomposed into a finite number of small tetrahedra, so that these two sets of tetrahedra are congruent with each other', Hilbert's student Max Dern proved this as early as 1900 with a counterexample, and this problem has been solved. ”

"Exactly!" Ge Juntian erased the word "3" again and asked, "Go on." ”

……

It wasn't until Jiang Shuiyuan had gone through all the problems that had been solved in the Hilbert problem that Ge Jun was satisfied enough to let Jiang Shuiyuan sit down, his eyes full of approval: "Very good, very good! Of course, Zhang Jin, don't be discouraged, the talent between people is incomparable, and your advantage lies in 'sharpening a sword in ten years'. Just like building a house, you need a talented designer, but also a builder who can turn the blueprint into a real thing. In the same way, there is a need for an ascetic like you in the study of mathematics!"

It's rare for Ge Juntian to comfort others once, but how can Zhang Jin hear how he thinks he is scolding himself in a different way?

Ge Juntian pointed to the remaining numbers on the blackboard and said loudly: "As you can see, although there are 23 problems in the Hilbert problem, after nearly a hundred years of continuous efforts by mathematicians around the world, more than half of the problems have been properly solved, and a series of important research results have been achieved in the remaining problems. However, it is worth noting that among the 12 problems solved, our neighboring Japan has made important contributions, including the fifth problem "Are all continuous groups differentiable", the Japanese mathematician Hidehiko Yamabe gave a completely positive result in 1953, and the Japanese mathematician Masashi Nagata gave a negative solution in 1959 with a beautiful counterexample on the fourteenth problem "Proving the finitude of some complete systems of functions". In addition, the Japanese mathematician Sadaharu Takagi also gave partial answers to the ninth and twelfth problems.

"Compared with the outstanding achievements of the Japanese mathematical community, the contributions made by our Chinese mathematicians are lackluster, mainly in the eighth problem 'prime number distribution problem' and the sixteenth problem 'topological research on algebraic curves and surfaces'. But this is not enough, because the status of Chinese mathematics in the international academic community is far from being comparable to physics, chemistry, electronics, biology, astronomy and other disciplines!

"We all know that mathematics is the mother of science, and if mathematics lags behind, the development of other disciplines will definitely be affected to a greater or lesser extent, after all, all problems are ultimately mathematical problems. Therefore, in recent years, domestic scientific research institutions headed by the Chinese Academy of Sciences and Jingshi University have put forward a grand '20-year plan', which is to mobilize the strength of the whole country to concentrate the best mathematicians to tackle the remaining Hilbert problems, and strive to make breakthroughs in one or two problems in 20 years, and achieve important results in three to five problems. Therefore, you should take participating in the Olympiad and winning the gold medal as the direction of your efforts in the next one or two years, and even more so, you must take solving the Hilbert problem as the goal of your struggle in the next one or two decades or even for the rest of your life!"

Zhang Jin was aroused by this great plan, and stammered and sighed: "It will take twenty years!"

"Yes, twenty years!" Ge Juntian was also full of emotion, "Ten years of life gathering, ten years of lessons, and twenty years should be enough time to train a large number of the world's top mathematicians, so those of you who are now in college and high school are blessed, as long as your talent is not bad, and you calm down and seize the opportunity, you may not be able to solve a Hilbert problem and win the Sun Yuanqi International Youth Science Award! As for the group of mathematics researchers who are now in their thirties and forties, they are miserable, and they have missed a good opportunity, and they are destined to only be a stepping stone for their juniors." - Of course, around the age of thirty is the golden age of mathematicians, and they are now unknown, and I guess that's the end of their lives!"

"Twenty years!" Jiang Shuiyuan also sighed, but his emotion was a little more sentimental: for others, twenty years may be a period of light in life. And for him, twenty years is the length of all his remaining lives!

Ge Juntian looked at the two students with some dissatisfaction: "What's wrong with twenty years? If you can solve a Hilbert problem, it will be worth spending twice as much time!"

"The point is that even if it takes twice as much time, it may not be able to solve a sub-problem of Hilbert's problem in the end!" Jiang Shuiyuan put aside his sentimentality and retorted sharply, "Just as Academician Hui Chengze wrote in his book, Riemann famously proposed ' in his paper as early as 1859." In 1900, forty-one years after the publication of the paper, Hilbert included him in the Hilbert problem, and three years later, the Danish mathematician Gramm calculated the value of 15 zeros, which was the first time that people had peeked into the specific existence of zeros.

Since then, people have been on the arduous journey of finding zeros and counterexamples, from a dozen to more than a hundred, to more than a thousand, from the discovery of the Riemann-Siegel formula to the introduction of electronic computers to distributed computing. It has been more than 100 years since the conjecture was put forward, and more than 10 trillion zero points have been found, but what can be done? 10 trillion pieces of evidence are not as good as one proof. Now the Riemann conjecture still exists majestically!"

Ge Juntian suddenly gasped as if he had a toothache: "Riemann conjecture! That is a human problem that has existed as long as cancer and cannot be solved!"

The Riemann conjecture is indeed difficult. Hilbert once said that if he woke up after 1,000 years of sleep, the first question he would ask was: Has the Riemann conjecture been proven? However, such a difficult Riemann conjecture is only a sub-problem of Hilbert's eighth problem, "Prime Number Distribution Problem", and its similar sub-problems also include the Goldbach conjecture, which is the favorite to solve in China, and the twin prime conjecture, which has become famous not long ago. The difficulty of Hilbert's problem can be imagined!

Ge Juntian's words changed sharply: "It is precisely because there are such hard bones that mathematics is so wonderful and charming! It is really because there are such hard bones in the Hilbert problem that I have come to you for guidance: Do you plan to know that there are tigers in the mountains and prefer to go to the tiger mountains, or are you ready to pick and pinch persimmons softly? Of course, the hardness of soft persimmons in Hilbert's problem is also diamond-level, otherwise they would have been pinched out of the paste by others!"