How many prime numbers can a natural number represent at most?

There are infinite prime numbers, also called prime numbers. The definition of prime number is a number with no other factors except 1 and itself among natural numbers greater than 1.

Chinese name

prime number

Foreign name

prime number

Another name

prime number

example

2、3、5、7、 1 1、 13、 17、 19

Scope of discussion

Natural number set

figure

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The amphoteric theorem of prime numbers

The 6 (x)+-1 = (PP) 6th degree complete inequality plus or minus1is a pair of twin prime numbers.

Where 6(X- 1=(P 6 times negative inequality minus 1 equals negative prime number;

6x)+1 = p) 6th degree positive inequality plus1equals positive prime number.

(X=/=6NM+-(M-N) Negative inequality is not equal to negative upper and lower expressions;

X)=/=6NM+-(N+M) positive inequality is not equal to positive upper and lower expressions.

(x)=/=6NM+-(M+-N) Completely unequal number is not equal to the number generated by the upper and lower formulas of Yin and Yang.

(n, m are two natural numbers, and n = "m".

Prime number distribution law

Taking 36N(N+ 1) as the unit, with the increase of n, the number of prime numbers gradually increases in the form of waves.

Twin prime numbers also have the same distribution law.

Statistics of prime numbers and twin prime numbers in the following 15 interval.

S 1 interval 1-72, with 18 prime numbers and 7 pairs of twin prime numbers. (2 and 3 are not counted, and the last number of twins is also counted in the previous interval. )

S2 interval 73-2 16 has 27 prime numbers and 7 pairs of twin prime numbers.

S3 interval 2 17-432 has 36 prime numbers and 8 pairs of twin prime numbers.

S4 interval 433-720 has 45 prime numbers and 7 pairs of twin prime numbers.

S5 interval is 721-1080, with 52 prime numbers and 8 pairs of double prime numbers.

S6 interval1081-1512, with 60 prime numbers and 9 pairs of twin prime numbers.

S7 interval1513—2016,65 prime numbers,1/twin prime numbers.

The S8 interval is 20 17-2592, with 72 prime numbers and 12 pairs of twin prime numbers.

S9 interval is 2593-3240, 80 prime numbers, 10 pairs of twin prime numbers.

S 10 interval 3241-3960, 9 1 prime, 18 pairs of twin prime numbers.

There are 92 prime numbers in the interval S 1 1-4752, and 17 pairs of twin prime numbers.

S 12 interval 4752-5616 has 98 prime numbers, and 13 pairs of twin prime numbers.

S 13 interval 5617-6552 prime 108, twin prime 14 pairs.

S 14 interval 6553-7560 primes 1 13, twin primes 19 pairs.

S 15 interval 7561-8640 prime 1 16, twin prime 14 pairs. (There are no corrections above, and there may be errors. )

With the discovery of the distribution law of prime numbers, many problems of prime numbers have been solved.

The number of prime numbers is infinite. There is a classic proof in Euclid's Elements of Geometry. It uses a common proof method: reduction to absurdity. The concrete proof is as follows: Suppose there are only a limited number of prime numbers, which are arranged as p 1, p2, ..., pn in descending order, and let n = P 1× P2×...× pn, then is pn plus 1 a prime number?

If pn plus 1 is a prime number, then pn plus 1 is greater than p 1, p2, ..., pn, so it is not in those hypothetical prime sets.

If pn plus 1 is a composite number, because any composite number can be decomposed into the product of several prime numbers; The greatest common divisor of n and N+ 1 is 1, so pn plus 1 cannot be divisible by p 1, p2, ..., pn, so the prime factor obtained by this complex number decomposition is definitely not in the assumed prime number set.

Therefore, whether the number is a prime number or a composite number, it means that there are other prime numbers besides the assumed finite number of prime numbers. So the original assumption doesn't hold water. In other words, there are infinitely many prime numbers.

Other mathematicians have given some different proofs. Euler proved by Riemann function that the sum of reciprocal of all prime numbers is divergent, Ernst Cuomo proved more succinctly, and harry Furstenberg proved by topology.

Used to calculate the number of prime numbers in a certain range.

Although the whole prime number is infinite, some people will ask, "How many prime numbers are there below 100000?" "What is the probability that the random number of 100 is a prime number?" . The prime number theorem can answer this question.

With the discovery of the distribution law of prime numbers, many problems of prime numbers have been solved.

A number A greater than 1 must have at least one prime number between it and its two degrees (that is, within the interval (a, 2a)).

There is a prime arithmetic progression of arbitrary length. (Green and Tao Zhexuan, 2004 [1])

An even number can be written as the sum of two composite numbers, and each composite number has at most 9 prime factors. (Norwegian mathematician Brown, 1920)

Even numbers must be written as prime numbers and composite numbers, where the number of factors of composite numbers has an upper bound. (Renee, 1948)

Even numbers must be written as a prime number plus a composite number consisting of at most five factors. Later, some people called this result (1+5) (Pan Chengdong, China, 1968).

A sufficiently large even number must be written as a prime number plus a composite number consisting of at most two prime factors. Abbreviation (1+2) (Chen Jingrun, China) [2]

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Goldbach conjecture: Can every even number greater than 2 be written as the sum of two prime numbers?

Twin prime conjecture: Twin prime numbers are a pair of prime numbers with a difference of 2, such as 1 1 and 13. Are there infinitely many twin prime numbers?

Does Fibonacci sequence have infinite prime numbers?

Is there an infinite number of mersenne prime?

Is there a prime number every n between n2 and (n+ 1)2?

X2+ 1 Is there an infinite number of such prime numbers?

Riemann hypothesis

Proof that twin prime numbers are infinite.

Key words: complete inequality, SN interval, LN interval.

One. The amphoteric theorem of prime numbers

Prime numbers greater than 3 are only distributed in two series: 6n- 1 and 6n+ 1. (n is a non-zero natural number, the same below)

The composite number in 6n- 1 series is called negative composite number, and the prime number is called negative prime number. The composite number in 6n+ 1 sequence is called positive composite number, and the prime number is called positive prime number.

Negative complex number theorem

6 [6 nm+(m-n)]-1= (6n+1) (6m-1) (n = < m is two nonzero natural numbers, n = < m, the same below).

6[6 nm-(m-n)]-1= (6n-1) (6m+1)

In 6n- 1 sequence, only these two composite numbers and the others are negative prime numbers, so there is a negative prime number theorem.

6NM+-(M-N)=/=x (negative inequality)

6x- 1=q (negative prime number)

Positive complex number theorem

6[6 nm+(n+m)]+1= (6n+1) (6m+1)

6[6 nm-(n+m)]+1= (6n-1) (6m-1)

The 6n+ 1 sequence has only these two composite numbers, and the rest are positive prime numbers, so there is a positive prime number theorem.

6NM+-(N+M)=/=X (positive inequality)

6X+ 1=P (positive prime number)

Two. Complete inequality corresponding to twin prime numbers

A completely unequal number (x) is neither equal to a feminine up-and-down expression; Doesn't mean it's positive

(X)=/= 6 nm +-(M+-N)

Then there is 6 (x)+1 = p 6 (x)-1 = q (p minus1is a prime divisible by 6, and q plus1is a prime divisible by 6, the same below).

The negative prime Q and positive prime P produced by completely unequal numbers are a pair of twin prime numbers.

And there is a one-to-one correspondence between completely unequal numbers and twin prime numbers.

Three. A survey of the distribution of yin and yang quartiles in natural sequences

6NM+(M-N)= female equal number 6NM-(M-N)= female inferior number.

6NM+(N+M)= positive equal number 6NM-(N+M)= negative equal number.

In order to find out their distribution in natural numbers, n in the four formulas is called rank factor number and m is called infinite factor number.

The minimum equation of each stage of the four equations is in the range of 6NN+-(N+N).

The distance between two adjacent equal numbers in each stage is 6n+ 1, and the ratio in natural sequence is 1/(6n+ 1). The total ratio of two equal numbers of each stage is 2/(6n+ 1), (but it is actually slightly smaller than this ratio, because there is no equal number of this stage at the bottom of each stage. The same is true of bad numbers. )

The distance between adjacent equal parts of the inferior number of each level is 6n- 1, the ratio in the natural sequence is 1/(6n- 1), and the total ratio of the inferior number of yin and yang at each level is 2/(6n- 1).

The ratio of the four equations of each stage in the natural sequence is 24n/[(6n+1) (6n-1)].

Four. Mutual Infiltration of Four Equal Sequences

The natural sequence includes negative arithmetic progression, negative lower arithmetic progression, positive arithmetic progression and positive lower arithmetic progression. Their levels are infinite, and the number of sequences in each level is infinite. The same sequence with different equal levels is mutually infiltrated and overlapped, and they are strictly overlapped by the product of equal distances of two levels. When calculating several levels of equations, it is just right to use multiplication formula to express the infiltration overlap relationship. There is mutual penetration and overlap between the four equal series, and only the upper and lower series of Yin and Yang at the same level have no penetration. It is enough to prove the osmotic overlap between the four series without calculation.

Five. SN interval that is basically synchronous with prime number distribution

Natural numbers are divided into intervals of 12, 24, 36 ... increase 12. Such an interval is called SN interval. SN interval is synchronized with four equal sign sequences, namely:

12( 1+2+3+……+N)= 6NN+6N

In this interval, there is no arithmetic progression greater than N, including all four arithmetic progression of N and below, which are completely synchronized with the four arithmetic progression, so they are also synchronized with the distribution of prime numbers.

Six. Every interval greater than S8 has more than 8 complete inequalities.

In each SN interval, only four arithmetic progression from 1 to n can determine the proportion of arithmetic progression at each level, which is due to the infiltration of superiors and subordinates. You can use the following formula to calculate at least the number of completely unequal numbers in the S8 interval.

12*8* 1 1/35*95/ 143*25 1/323*479/575*779/899* 1 15 1/ 1295* 1593/ 1763*2 1 1 1/2303=8.2768

Every other SN interval can be calculated by this method.

With the increase of interval, the number of completely unequal numbers will increase, and it will exceed 8 in the future.

Seven. error analysis

Using the strictest rounding error analysis method, the SN interval is limited to LN interval 1, 2, 4, 8, 16...2 (n- 1). In every SN interval greater than S8, there are more than 8 complete inequalities, and in every LN interval, there are 2 n-655.

8*2^(n- 1)-4*(2^n- 1)=4

After the strictest rounding, there are still four completely unequal numbers in the interval greater than L4.

Eight. abstract

According to the above argument, every SN interval greater than S8 has more than 8 complete inequalities.

After strict rounding, there are more than four completely unequal quantities in every LN interval greater than L4.

LN interval is infinite, and there is a one-to-one correspondence between completely unequal numbers and twin prime pairs, so twin prime numbers are infinite.

This proof looks forward to authoritative statements.

nature

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Prime numbers have many unique properties:

The (1) prime p has only two divisors: 1 and p.

(2) Basic theorem of elementary mathematics: Any natural number greater than 1 is either a prime number itself or can be decomposed into the product of several prime numbers, and this decomposition is unique.

(3) The number of prime numbers is infinite.

(4) The number formula of prime numbers is an irreducible function.

(5) If n is a positive integer, then there is at least one prime number between and.

(6) If n is a positive integer greater than or equal to 2, there is at least one prime number between n and.

(7) If the prime number p is the largest prime number not exceeding n (), then.

(8) Among all prime numbers greater than 10, the unit number is only 1, 3, 7, 9.

procedure

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Basic judgment ideas:

Generally speaking, for a positive integer n, if it is divisible by all integers between 2 and 0, then n is a prime number.

Prime numbers greater than or equal to 2 cannot be divisible by numbers other than themselves and 1.

Python code:

Import sqrt from mathematics

def is_prime(n):

If n == 1:

Returns False

For I(2, int(sqrt(n))+ 1) in the range:

If n% i == 0:

Returns False

Return True

Java code:

Common static boolean test prime 2 (int n) (

If (n < = 3) (

return n & gt 1;

}

for(int I = 2； I & ltn；; i++){

If (n%i == 0)

Returns false

}

Return true

}

/* After optimization */

Common static boolean test prime 3 (int n) (

If (n < = 3) (

return n & gt 1;

}

for(int I = 2； I< = math.sqrt (n); i++){

If (n%i == 0)

Returns false

}

Return true

}

Public class Prime {

Public static void main(String[] args) {

int a = 17； //Judge whether 17 is a prime number.

int c = 0；

for(int b = 2； b & lta； b++) {

If (a% b! = 0) {

c++；

}

if (c == a - 2) {

System.out.println(a+ "is a prime number");

} Otherwise {

System.out.println(a+ "is not a prime number");

}

Php code:

Function is prime($ n){// AVP production in Turkey

if($ n & lt； = 3) {

return $ n & gt 1;

} else if ($n % 2 === 0 || $n % 3 === 0) {

Returns false

} Otherwise {

for($ I = 5； $ i * $ i & lt= $ n； $i += 6) {

if($ n % $ I = = = 0 | | $ n %($ I+2)= = 0){

Returns false

}

Return true

}

C# code:

Use the system;

Namespace computing prime number

{

Class plan

{

Static void Main(string[] args)

{

for (int i = 2，j = 1； I<210000000 &; & ampj & lt= 1000; I++)// outputs all prime numbers within 2 1 100 million, and the j control only outputs 1000.

{

if (st(i))

{

Console. WriteLine("{0，- 10}{ 1} "，j，I)；

j++；

}

Static bool st(int n)// Judge whether a number n is a prime number.

{

int m = (int)Math。 sqrt(n)；

for(int I = 2； I<= m；; i++)

{

if(n % I = = 0 & amp； & amp! =n)

Returns false

}

Return true

}

C/C++ code:

# include & ltiostream & gt

# include & lt algorithm & gt

# include & ltcmath & gt

Use namespace std

const long long size = 100000； //Modify the value of size to change the size of the final output.

The book of the dragon [size/2];

Void work(){// main program

zhi Shu[ 1]= 2；

long long k = 2；

for(long long I = 3； I < = size; I++){// Enumerate each number.

bool ok = 1；

for(long long j = 1； j & ltk； J++){// Enumerates the obtained prime numbers.

if(i%zhishu[j]==0){

ok=！ ok；

Break;

}

If (normal) {

zhi Shu[k]= I；

Cout & lt< "count"<<k<<< me<& ltendl

k++；

}

int main(){

freopen("zhishu.out "，" w "，stdout)；

cout & lt& lt" count 1 2 " & lt； & ltendl

work()；

Returns 0;