Omnipotent Data

Chapter 348

Inspiration is always so unprepared!

Cheng Nuo ticked the corner of his mouth slightly and turned the page back to the original page.

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

By the way, let’s verify what level of abilities my own ability has reached after this year’s in-depth study.

Simple proof method of Bertrand hypothesis.

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

Just as Cheng Nuo had assumed before. The process of proving every conjecture or hypothesis in mathematics is a process from the starting point to the ending point. Some routes are tortuous and some are straight.

Perhaps, what Chebyshev discovered was the tortuous route, while Cheng Nuo needed to blaze a simpler path on the basis of his predecessors.

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

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

Just do it!

Cheng Nuo is not so hesitant. Anyway, there is plenty of time, Rong De Cheng Nuo, after discovering that "this road is not working", looked for another direction for thesis.

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

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

Two hours later, Cheng Nuo closed the book.

After closing his eyes for a few seconds, he took out a stack of blank draft paper from his schoolbag, picked up the black carbon pen on the desktop, and started his own deduction with concentration:

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 that can divide n! is: s=Σi≥1floor(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≤nsi means that the number of tokens in each column (i.e. si) is calculated first and then summed. The relation obtained from this is Lemma 1.

Lemma 2: [Suppose n is a natural number and p is a prime number, then Πp≤np2), let us prove the case of n=N.

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

If N is an odd number, set N=2m+1 (m≥1). Note that all prime numbers with m+1p≤2m+1 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 thoughts were smooth and it didn't take much effort, so he used his own method to prove these two auxiliary propositions.

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 hypothesis later.

Chebyshev’s method is hard to make, yes, hard to make!

Through continuous conversion between formulas, one of Bertrand's hypothesis, or certain necessary and sufficient conditions, is converted into the form of Lemma 1 or Lemma 2, and the solution is simplified and integrated.

Of course, Cheng Nuo certainly cannot do this.

Because with this kind of verification plan, let alone Cheng Nuo, even if Hilbert is allowed to come, the proof steps will not be much simpler than Chebyshev. Therefore, we must change our thinking.

But what a conversion method...

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

Seeing that the sun was slanting westward and it was time to finish eating again, Cheng Nuo wandered towards the cafeteria while thinking in his mind.

…………

At the same time, the United States is on the other side of the ocean.

The headquarters of "I Women" 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 from all over the country every year.

But in the end, only less than two hundred papers were given a chance to get published.

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

Such as Peteholze in the field of algebraic geometry.

Rihard Hamilton in the field of differential geometry.

Jean Bourgain in the field of mathematical analysis.

Etc., etc……

Therefore, when reviewers review manuscripts, they do not review the manuscripts in the order of submission, but follow the academic criticism of the signed author as the standard.

After all, authors with higher academic levels are more likely to meet journal inclusion standards. The number of papers included in each issue of the journal is generally a floating value, but the fluctuation is not large.

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

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

For example, one of the reviewing editors of "I Woman", Rafi Peterel, is a well-known mathematician who thought he had won the Ramanu Gold Medal.

Currently, in addition to being the review editor of this journal, he is also a visiting professor at the University of California, Los Angeles, focusing on analytical number theory.

As a mathematician with multiple titles, it is impossible for him to stay in the office from nine to five every day to review manuscripts like he is at work.

Generally speaking, he spends one or two mornings a week, staying in his apartment, reviewing those submitted by ordinary reviewing editors, several submissions by top mathematicians, and some lesser-known ones. A mathematician sent it, but they considered it to be qualified for inclusion.

But in most cases, due to the low level of mathematics of ordinary reviewers, only a few of the selected emails meet the journal's inclusion criteria.

Eight o'clock in the morning.

Professor Peterle took a leisurely cup of coffee and sat on the balcony, while reviewing the submissions displayed on the laptop ~www.readwn.com~ while taking a leisurely sip.

"The mathematics world has been a bit calm these days!" Raphael closed a paper and sighed softly.

In recent months, as the controversy over the ABC conjecture has ended, the entire mathematics world has fallen into peace. 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 top mathematicians. Among them, the level of five papers is higher than the acceptance standard. Peterel 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 I remembered that someone had a treat at noon today, so I don’t have to worry about making lunch.

That being the case, let's read a few more articles.

Peter controlled the mouse and clicked on the next email.

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