🔫 🗞️ 👂🏽 Patterns in the distribution of prime numbers 💀 🍰 👨🏿‍⚖️

Introduction

A prime number is a natural number that has exactly two different natural divisors - one and itself. Such numbers are of great interest. The fact is that no one has been able to fully understand and describe the pattern by which prime numbers are located in a row of natural numbers.

Even before our era, Euclid formulated and proved the first theorems on prime numbers. Since then, mathematicians, among them Gauss, Fermat, Riemann, Euler, have continued their research and we must pay tribute to them have made significant progress. Many interesting properties of prime numbers have been discovered, many assumptions have been made, some of which have been proven. However, many hypotheses related to prime numbers still remain unfounded.

Distribution of prime numbers

The primary task, the solution of which would automatically lead to the solution of most questions related to prime numbers, is as follows:

Get a recurring formula for the next prime number

$p_ {n + 1} = f (n, p_1, p_2, ..., p_n),$

p _n - n -th prime number ( p ₁ = 2 , p ₂ = 3 , p ₃ = 5 , ...)

There is a related problem about the number of primes not exceeding a given value:

Find a function p (x) whose value at the point x is equal to the number of primes on the segment [ 1, x ] . Where x is any real number not less than one.

The function $\ pi (x)$ is called the prime number distribution function.

There are many approaches to solving the above problems. Let's consider some of them.

, ( , ).

, , , , .

p₁ =2. 2, 2k+1, k – . — .

p₂ = 3. 3m+1, 3m+2, m – . , . , 2k+1.

$\ begin {array} {} {2k + 1 = 3m + 1, \\ 2k + 1 = 2m + 2,} \ end {array}$

k m , p₃ p = 6t + 1 , p = 6t + 5 , t – .

, :

$\ begin {array} {} {5 = 6 * 0 +5, \\ 7 = 6 * 1 + 1, \\ 11 = 6 * 1 + 5, \\ 13 = 6 * 2 + 1.} \ end { array}$

, 6t+1 6t+5 . , 25 = 6 * 4 + 1 .

p₃ = 5. , , 5, p₁ = 2 p₂ = 3, , p₄

$\ begin {array} {} {p = 30t + 1, \; \; \; \; \; \; p = 30t + 11, \\ p = 30t + 7, \; \; \; \; \; \; p = 30t + 17, \\ p = 30t + 13, \; \; \; \; p = 30t + 23, \\ p = 30t + 19, \; \; \; \; p = 30t + 29} \ end {array}$

p₄, p₅ .. , , .

, . , . , , .

, . F(x) , x p₁, p₂, …, p_n. ? ( ), p₁, p₂, … , p_n - p_n+1 ( ). , F(p_n+1 -1) = 1 ( — ), F(p_n+1) = 2 ( p_n+1). , F(x) , p_n+1.

, F(x)? . , p₁, p₂, …, p_n?

p₁ = 2. , $\ frac {1} {2}$ p₁.

3. , $\ frac {1} {3}$ p₂. , 2 3 .

, 2, 3

$1 - \ frac {1} {p_1} - \ frac {1} {p_2} + \ frac {1} {p_1 * p_2} = 1 - \ frac {1} {2} - \ frac {1} {3} + \ frac {1} {2 * 3}.$

, :

$(1- \ frac {1} {p_1}) (1- \ frac {1} {p_2})$

, p₁, p₂, …, p_n,

$1 - \ frac {1} {p_1} - \ frac {1} {p_2} -...- \ frac {1} {p_n} + \ frac {1} {p_1 * p_2} + \ frac {1} { p_1 * p_3} + ... + \ frac {1} {p_ {n-1} * p_ {n}} - \ frac {1} {p_1 * p_2 * p_3} -... + (- 1) ^ n \ frac {1} {p_1 * p_2 * ... * p_n}.$

$P (n) = (1- \ frac {1} {p_1}) (1- \ frac {1} {p_2}) (1- \ frac {1} {p_3}) ... (1- \ frac { 1} {p_n}) \ qquad \ qquad (1)$

P(n). , (n→∞), .

, F (x) = x * P (n) . , P(n) n . n 1 N, N - , P(n), .

? (1), , , p_n, $\ frac {1} {p_n}$ . . , 1,2, 3,4,5,6,7,8,9. 4 9 . , $\ frac {4} {9}$ $\ frac {1} {2}$ . , , .

. , (, ) p_n+1- . , — . , , .

$n * ln (n) + n * ln (ln (n)) - \ frac {3} {2} n <p_n <n * ln (n) + n * ln (ln (n))$

n, 6.

$\ frac {x} {ln (x)} <\ pi (x) <1.25506 \ frac {x} {ln (x)}$

$\ pi (x)$ , - . , , . , .

. , , , , . - , .

. ( ). :

, 2, ?

2. -

p , p + 2 ?

, ?

p .

, 2020 . .

1.

: () ().

: , 5, .

2013 . 133 .

: , , .

, .

, . , . . 11 . .

: , , , ? . N, , .

$K \ geq N$ . p₁ p₂, K = p_1 + p_2 . , , , . p₁ – . — 2. , $2 + p_2 = K \ rightarrow p_2 = K-2$ . , K-2 ( K ) . N, , , N-2, . . , $\ pi (n) \ sim \ frac {n} {2}$ n→ ∞. , $\ pi (n) \ sim n * ln (n)$ n→ ∞.