Chapter 285

Can Chen's theorem be applied to the study of the conjecture of equal difference primes?

Many mathematicians throughout the ages have given a negative answer to this question.

When conducting research on the conjecture of equal primes, Konstantin also took it for granted.

The inertia of thinking made Konstantin not think about trying to use Chen's theorem from beginning to end.

But now, Konstantin realizes that he may have made a huge mistake.

Chen's theorem may really be the key to opening the door to the conjecture of equal difference primes.

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

"The content of the conjecture of equal difference prime refers to the existence of a series of equal difference numbers of prime numbers of any length. ”

"One thing to note here is that it's an equal-difference series of arbitrary length, not an infinite length of equal-difference series. ”

"There is a big difference between the terms arbitrary length and infinite length. ”

Take the equal-difference prime conjecture as the simplest example. ”

Speaking of this, Gu Lu held a marker pen and wrote a few symbols on the blackboard behind him.

"First, let's assume that the first term of a prime-like difference series is N and the tolerance is D, so what is the N+1 term of the difference-like series?"

"It's N+ND. Gu Lu asked himself and then circled the formula, "And N+ND must be a multiple of the first term N, obviously, in this case, N+ND is not a prime number." To put it simply, this series of difference is not a series of difference numbers of prime numbers made up entirely of prime numbers!"

"Therefore!" Gu Lu tapped on the blackboard and highlighted the emphasis, "For the conjecture of equal difference primes, we can only say that there is a series of equal difference numbers of any length of the prime number, but we cannot say that there is a series of equal difference numbers of infinite length. ”

Mathematicians in the field of algebraic geometry have long known these things.

The reason why Gu Lu said it again was to popularize a little bit of relevant knowledge to the group of mathematicians in other fields in the conference room, so as to avoid talking about it later, leaving them in a state of confusion.

"So, with regard to the conjecture of equal difference primes, our goal is clear. That is, to prove that a series of equal differences made up of prime numbers can be arbitrarily long and have any number of groups. ”

Here, we introduce the concept of K value, which refers to the number of prime numbers in a sequence of equal difference composed entirely of prime numbers. ”

When K is an even number, the problem of the conjecture of equal difference primes has been discussed and proved by Professor Constantine a few days ago, and I will not repeat it too much here. ”

When he said this, Gu Lu glanced at Constantine, who was holding his arm and had a gloomy expression, and then continued to speak, "Next, I will directly explain the proof of the conjecture of equal difference prime when K is an odd number!"

Gu Lu's proof officially began.

Everyone in the audience sat upright, pricked up their ears, and their notebooks were at hand, ready to record at any time, for fear of missing any details.

Just like yesterday, without the aid of any electronic equipment, Gu Lu directly deduced the proof process of the deductive conjecture of equal difference prime numbers step by step on the blackboard.

As for the conjecture of equal difference primes, Gu Lu had just proved successful yesterday afternoon.

But every detail, every step, has long been imprinted in Gu Lu's mind.

What Gu Lu needs to do now is to present it in front of everyone.

In the conference room, several cameras pointed at Gu Lu at the same time to film the whole process of Gu Lu's proof.

For the mathematical community, this is destined to be a valuable visual material.

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

“...... We first let P(1,2) be the number of prime numbers p that fit the following conditions, x-p=p1 or x-p=p1p2. Among them, p1, p2, and p3 are all prime numbers. ”

Next, we denote a sufficiently large even number with x, and the order Cx=Π(p2)(1-1/(p-1)^2). For any given even number h, and sufficiently large xp, xh(1,2) is used to denote the number of prime ps that satisfy the following conditions: p≤x, p+h=p1 or p+h=p2p3. Here, p1, p2, p3 also represent prime numbers. ”

“...... After that, we get two theorems, which are:

Theorem 1: [(1,2) and Px(1,2)≥0.67xCx/(logx)^2.]

Theorem 2: For any even number h, there are an infinite number of prime numbers p, so that the number of prime factors of p+h does not exceed 2 and xh(1,2)≥0.67xCx/(logx)^2.]"

Gu Lu has been talking for five minutes.

There are four blackboards, and nearly two of them are almost filled up by the formula written by Gu Li.

The method of proving the conjecture of equal difference prime adopted by Gu Law has begun to take shape with the continuous elaboration of Gu Law.

Konstantin, in particular, can be said to have seen it most thoroughly.

The proof process of Gu Law does use Chen's theorem.

However, unlike Constantine's guess, Gu Lu did not quote the specific content of Chen's theorem, but some of the methods and theories used by Academician Chen in the process of deriving Chen's theorem.

For example, when Gu Lu constructed the three prime numbers p1, p2, and p3, it was exactly the same as the construction method of Academician Chen.

There are also even numbers and the derivation of two key theorems, and the shadow of Academician Chen's paper flows between the lines.

Even if Konstantin does not have a good impression of Gu Lu, he has to admit that Gu Lu's operation is enough to be called a stroke of genius.

Not only Konstantin, but the rest of the mathematicians in the room who understood it were also amazed.

What a fanciful idea!

Everyone couldn't help but admire.

Although the idea is wild, I have to admit that Gu Lu's operation can be said to link the conjecture of equal difference prime with Chen's theorem without any obstacles.

Let everyone see the hope of successfully proving the conjecture of equal difference primes.

"But, just these words are obviously not enough!" Konstantin looked at Gu Lu's derivation steps on the blackboard and muttered softly.

Konstantin saw it more thoroughly than everyone else.

Gu Lu's genius stroke, although it is amazing enough, is not enough to be the last straw to press the conjecture of equal primes.

If Gu Lu really only has this ability, then I'm afraid this is the end of today.

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

Will Gu Lu stop here?

Apparently not.

It is obvious that Gu Lu will never fight an unprepared battle.

Since Gu Lu chose to report on stage, it shows that he has full confidence and confidence in his own proof process.

I saw Gu Lu smile slightly, pull down a blank blackboard, and elaborate while writing.

"Next, we need to construct a few more lemmas. ”

"Lemma 1: Suppose y≥0 and [logx] denotes the integer part of logx, x>1, φ(y)=1/2πi∫(2+i∞,2-i∞)ydw/w(1+w/(logx)^l)^[logx]+1."

"Lemma 2: Let c(α)=e^2πiα,S(α)=∑ane(na),Z=......"

"Lemma 3: ......"

The three lemma are constructed.

Gu Lu smiled and said, "Next, we need to introduce another formula to combine with these three lemmas. ”

After speaking, Gu Lu wrote a string of formulas on the blackboard.

∑(m1^2+m2^2+m3^2≤x)1=4π/3*x^1.5+O(x^2/3)！

This formula is......

The prime distribution formula for the whole point problem in the sphere!

Many mathematicians looked at this familiar formula, and their pupils shrank suddenly.