Chapter 348

Inspiration always comes so unexpectedly!

The corners of Cheng Nuo's mouth hooked slightly, and he turned the page back to the original page.

Since Chebyshev's Bertrand hypothesis is so complex, let's challenge ourselves and see if we can prove Bertrand's hypothesis in simpler mathematical language.

By the way, let's verify how far my ability has come this year's in-depth study.

A simple proof method for Bertrand's hypothesis.

This dissertation title alone is enough to be called a district-level dissertation. Of course, the premise is that Cheng Nuo can really explore the simple solution.

Just as Cheng Nuo had assumed before. The process of proving every conjecture or hypothesis in mathematics is a process from the beginning to the end, some of which are tortuous and some are straight.

Perhaps, what Chebyshev found was a more tortuous route, while Cheng Nuo needed to open up a simpler road on the basis of his predecessors.

But this is simpler than proving the Bertrand hypothesis alone.

After all, he is standing on the shoulders of giants to look at the problem, and with the proof scheme proposed by Chebyshev, the "pioneer", Cheng Nuo can more or less learn something from it and have a unique understanding.

Do what you think!

Cheng Nuo is not such a hesitant person. Anyway, there is plenty of time, and Cheng Nuo found that "this road is not working" and looked for another direction for his dissertation.

If you want to come up with a simpler solution, you must first thoroughly understand the proof ideas proposed by our predecessors.

Instead of rushing straight to his own delving, he lowered his head and read the dozen or so pages of the book from beginning to end.

Two hours later, Cheng Nuo closed the book.

After closing his eyes and savoring for a few seconds, he took out a stack of blank scratch paper from his school bag, picked up the black carbon pen on the table, and began his deduction attentively:

To prove Bertrand's hypothesis, several auxiliary propositions must be proved.

Lemma 1: [Lemma 1: Let n be a natural number and p be a prime number, then the highest power of p divisible by n! is: s =Σi≥1floor(n/pi) (where floor(x) is the largest integer not greater than x)]

Here, all (n) natural numbers from 1 to n need to be arranged in a straight line, and a column of si symbols is superimposed on each number, and obviously the total number of symbols is s.

The relation s =Σ1≤i≤n si means that the number of symbols (i.e., si) of each column is calculated and then summed, and the resulting relation is lemma 1.

Lemma 2: [Let n be a natural number and p be a prime number, then Πp≤n p 2), let's prove the case of n = N.

If N is even, then Πp≤N p =Πp≤N-1 p, and the lemma clearly holds.

If N is odd, let N = 2m + 1 (m ≥ 1). Note that all prime numbers of m + 1 < p ≤ 2m + 1 are combined numbers (2m+1)!/m! (m+1)!, on the other hand, the number of combinations (2m+1)!/m! (m+1)! occurs twice in the binomial expansion (1+1)2m+1, so that (2m+1)!/m! (m+1)!≤(1+1)2m+1 / 2 = 4m.

In this way, you will be able to ......

Cheng Nuo's thinking was smooth, and it took almost no effort to prove these two auxiliary propositions in his own way.

Of course, this is just the first step.

According to Chebyshev's idea, these two theorems need to be introduced into the proof step of Bertrand's hypothesis.

Chebyshev's method is hard to make, yes, it's hard to make!

Through the continuous conversion between formulas, one of the Bertrand hypotheses, or several sufficient and necessary conditions, is converted into the form of lemma 1 or lemma 2, and the solution is simplified and integrated.

Of course, Cheng Nuo definitely can't do this.

Because with this kind of verification scheme, let alone Cheng Nuo, even if Hilbert is asked to come, I am afraid that the proof steps will not be much simpler than Chebyshev. Therefore, it is necessary to change the way of thinking.

But what exactly is a conversion method......

Belch...... Cheng Nuo hasn't thought about it yet.

Seeing that the sun was setting in the west, and it was time to finish eating, Cheng Nuo thought in his mind while strolling towards the cafeteria.

............

At the same time, the country of rice on the other side of the ocean.

Inventiones mathematicae magazine is headquartered in Los Angeles, USA.

As one of the top SCI journals in the field of mathematics, they receive tens of thousands of submissions from mathematicians across the country every year.

However, less than 200 papers have the opportunity to be published.

Moreover, almost four-fifths of the 200 academic papers were occupied by the world's top mathematicians.

Such as Peter Scholze in the field of algebraic geometry.

Richard Hamilton in the field of differential geometry.

Jean Bourgain in the field of mathematical analysis.

And so on and so forth......

Therefore, when reviewing manuscripts, review editors do not review manuscripts in the order of submission, but according to the academic criticism of the authors as the standard.

After all, the higher the academic level of the author, the more likely it is to meet the journal inclusion criteria. The number of papers included in each issue of the journal is roughly a value that fluctuates up and down, but it does not fluctuate much.

In this way, you can greatly save the time of the reviewer editor.

Being able to work as a review editor in such a top journal in mathematics is not unknown.

For example, one of the review editors of Inventiones mathematicae, Rafi-Peter, is a well-known mathematician who has won the Ramanujan Prize.

Currently, in addition to being a review editor for the journal, he is also a visiting professor at the University of California, Los Angeles, where he specializes in analytic number theory.

As a math guru with many titles, he can't stay in the office from 9 to 5 every day to review manuscripts as if he were at work.

In general, he spends one or two mornings a week in his apartment reviewing submissions from top mathematicians and submissions from lesser-known mathematicians who are considered qualified to be included.

However, in most cases, due to the low level of mathematics of ordinary reviewers, only a small number of selected emails meet the journal's inclusion standards.

Eight o'clock in the morning.

Professor Peter leisurely brewed a cup of coffee and sat on the balcony, sipping leisurely as he reviewed the submissions displayed on his laptop.

"The world of mathematics has been a little quiet lately!" said Raphael, sighing softly as he closed a paper.

In recent months, as the ABC conjecture controversy has come to an end, the entire mathematical community has fallen into a lull. Perhaps, when the Philippine Awards are presented in November this year, it will be lively again.

Slowly, the time came to eleven o'clock.

He has reviewed all seven papers submitted by several leading mathematicians. Among them, five papers were above the standard line for inclusion. Peterl marked a few places and asked his subordinates to contact the author for minor repairs.

I was going to end today's work like this, but I remembered that I had someone to treat me at noon today, so I didn't have to rush to make lunch.

In that case, let's read a few more articles.

Peter manipulated the mouse and clicked on the next email.

The title of the paper: "Proof of the weak BSD conjecture when the analytic rank is 1"!