universal data

Chapter 345 Peter

348 chapter

Inspiration always comes so unexpectedly!

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

Since the proof process of the Bertrand hypothesis given by Chebyshev (Chebyshev) is so complicated, then challenge yourself to see if you can prove the Bertrand hypothesis in a simpler mathematical language.

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

A simple proof method for Bertrand's hypothesis.

The topic of this thesis alone is enough to be called a district-level thesis.Of course, the premise is that Cheng Nuo can really explore that simple solution.

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

And perhaps, what Chebyshev discovered was the relatively tortuous route, and Cheng Nuo needed to open up a simpler route on the basis of his predecessors.

But this is simpler than proving Bertrand's hypothesis alone.

After all, he was standing on the shoulders of giants to look at the problem. With the proof plan proposed by Chebyshev, the "pioneer", Cheng Nuo could more or less learn from it and make a unique understanding.

Do it when you think of it!

Cheng Nuo was not such a hesitant person.Anyway, there was plenty of time for Cheng Nuo to look for another direction for his thesis after discovering that "this road is dead".

If you want to propose a simpler solution, you must first thoroughly understand the proof ideas proposed by the predecessors.

He didn't rush to start his own research directly, but lowered his head and read the dozen or so pages of Bertrand's hypothesis from the beginning to the end.

Two hours later, Cheng Nuo closed the book.

After thinking about it for a few seconds with his eyes closed, he took out a stack of blank draft paper from his schoolbag, picked up the black carbon pen on the table, and started his deduction with concentration:
To prove Bertrand's hypothesis, several auxiliary propositions must be proved.

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

Here, it is necessary to arrange all (n) natural numbers from 1 to n on a straight line, and stack a column of si marks on each number, obviously the total number of marks is s.

The relational expression s =Σ1≤i≤n si means that the number of symbols in each column (namely si) is calculated first and then summed. The resulting relationship is Lemma 1.

Lemma 4: [Assume n is a natural number and p is a prime number, then Πp≤np < [-]n]

Use mathematical induction. The lemma clearly holds for n = 1 and n = 2.Assuming that the lemma holds for n < N (N > 2), let us prove the case for n = N.

If N is an even number, then Πp≤N p =Πp≤N-1 p, the lemma obviously holds.

If N is odd, let N = 2m + 1 (m ≥ 1).Note that all m + 1 < p ≤ 2m + 1 primes are factors of the combination number (2m+1)!/m!(m+1)!, on the other hand the combination number (2m+1)!/m! (m+1)! appears twice in the binomial expansion (1+1)2m+1, so (2m+1)!/m!(m+1)!≤(1+1)2m+1 / 2 = 4m.
In this way, you can...

Cheng Nuo's thinking was smooth, and he used his own method to prove these two auxiliary propositions without much effort.

Of course, this is just the first step.

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

The method used by Chebyshev is to force it, yes, it is to force it!
Through continuous conversion between formulas, one of Bertrand's assumptions, or certain necessary and sufficient conditions, is converted into the form of Lemma [-] or Lemma [-], and the solution is simplified and integrated.

Of course, Cheng Nuo couldn't do that.

Because with this kind of proof scheme, not to mention Cheng Nuo, even if Hilbert were to come, the proof steps would not be much simpler than Chebyshev's.Therefore, it is necessary to change the way of thinking.

But what kind of conversion method...

Uh... Cheng Nuo hasn't made up his mind yet.

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

…………

At the same time, the United States is far across the ocean.

The headquarters of "Inventiones mathematicae" magazine is located in Los Angeles, USA.

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

But in the end, there were less than [-] papers that had the opportunity to be published.

Moreover, among the two hundred academic papers, almost four-fifths are occupied by the top mathematicians in the world.

Such as Peter Scholze in the field of algebraic geometry.

Richard Hamilton in Differential Geometry.

The Jean Bourgain of Mathematical Analysis.

Etc., etc……

Therefore, when reviewing manuscripts, the reviewing editors do not review the manuscripts in the order of submission, but use the academic criticism of the signed author as the standard.

After all, the higher the academic level of the author, the higher the possibility of meeting the journal inclusion standards.The number of papers included in each issue of the journal generally fluctuates up and down, but the fluctuation is not large.

In this way, the time of reviewing and editing can be greatly saved.

Being able to serve as a review editor in such a top journal in the mathematics field is not an unknown person.

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

Currently, in addition to being the review editor of this journal, he also serves as a visiting professor at UCLA, focusing on the field of analytic number theory.

As a mathematician with multiple titles, it is impossible for him to stay in the office from [-] to [-] every day to review manuscripts like going to work.

Generally speaking, he spends one or two mornings a week, staying in his apartment, reviewing those sent by ordinary review editors, several contributions by top mathematicians, and some lesser-known ones. Sent by mathematicians, but considered by them to be qualified for inclusion.

But in most cases, due to the low mathematics level of ordinary review editors, only a small part of the selected emails meet the journal's inclusion standards.

Eight o'clock in the morning.

Professor Peter made a cup of coffee leisurely, sat on the balcony, and sipped leisurely while reviewing the submissions displayed on the laptop.

"The mathematics world has been a little quiet recently!" Raphael closed a paper and sighed softly.

In recent months, with the end of the ABC conjecture controversy, the entire mathematics world has fallen into a calm.Perhaps, it will be lively again when the Philippine Awards are presented in November this year.

Slowly, the time came to eleven o'clock.

He has reviewed all seven papers submitted by several top mathematicians.Among them, the level of five papers is higher than the inclusion standard line.Peter marked a few places and asked his subordinates to contact the author for minor revisions.

I originally planned to end today's work like this, but remembering that there is a treat at noon today, there is no need to rush to make lunch.

If so, let’s read a few more.

Peter controlled the mouse and clicked on the next email.

The title of the paper: "Proof of the Weak BSD Conjecture When Analytical Rank Is 1"!
(End of this chapter)