1) Definition: The positive number spectrum is the special arrangement with infinite rows and six columns in all positive integers put in order of integer numbers.

2) The expression of columns, line and elements of the positive integer number spectrum

The positive integer number spectrum indicates that any number X can be only expressed into the type of 6 ( n − 1 ) + r , ( n ∈ Z + , n ≠ 1 ; r = 1 , 2 , 3 , 4 , 5 , 6 ), n is row ordinal number, r is column ordinal number.

A number of the r-th ( r = 1 , 2 , 3 , 4 , 5 , 6 ) is denoted with q r , the set of all numbers of the r-th column is denoted with Q r .

1) Multiplication law of the positive integer number spectrum

We can easily prove the positive integer number spectrum satisfying the following laws according to it’s formula of general term 6 ( n − 1 ) + r .

Multiplication law:

q 1 ⋅ q 1 ∈ Q 1 , q 1 ⋅ q 2 ∈ Q 2 , q 1 ⋅ q 3 ∈ Q 3 , q 1 ⋅ q 4 ∈ Q 4 , q 1 ⋅ q 5 ∈ Q 5 , q 1 ⋅ q 6 ∈ Q 6

q 2 ⋅ q 2 ∈ Q 4 , q 2 ⋅ q 3 ∈ Q 6 , q 2 ⋅ q 4 ∈ Q 2 , q 2 ⋅ q 5 ∈ Q 4 , q 2 ⋅ q 6 ∈ Q 6

q 3 ⋅ q 3 ∈ Q 3 , q 3 ⋅ q 4 ∈ Q 6 , q 3 ⋅ q 5 ∈ Q 3 , q 3 ⋅ q 6 ∈ Q 6

q 4 ⋅ q 4 ∈ Q 4 , q 4 ⋅ q 5 ∈ Q 2 , q 4 ⋅ q 6 ∈ Q 6

q 5 ⋅ q 5 ∈ Q 1 , q 5 ⋅ q 6 ∈ Q 6

q 6 ⋅ q 6 ∈ Q 6

2) Some properties of the positive integer number spectrum

Property 1: Any composition of the first column can be written into

q 1 ⋅ q 1 or q 5 ⋅ q 5 .

Proof: Denoting the composition of the first column with H₁, then:

H 1 = q 1 ⋅ q 1 or H 1 = q 5 ⋅ q 5 .

Because the composition 6 ( m − 1 ) + 1 can and only can be discomposed into [ 6 ( n − 1 ) + 1 ] [ 6 ( s − 1 ) + 1 ] ( n ≤ s < m ) or [ 6 ( n − 1 ) + 5 ] [ 6 ( s − 1 ) + 5 ] ( n ≤ s < m ); thereby it can not be discomposed into [ 6 ( n − 1 ) + a ] [ 6 ( s − 1 ) + b ] ( a , b = 2 , 3 , 4 , 6 ).

Therefore, H 1 = q 1 ⋅ q 1 or H 1 = q 5 ⋅ q 5 .

Property 2: Any composition of the fifth column can be written into p₁p₅.

Proof: Denoting the composition of the first column is H₅, then H 5 = q 1 ⋅ q 5 .

Because the composition 6 ( m − 1 ) + 5 can and only can be discomposed into [ 6 ( n − 1 ) + 1 ] [ 6 ( s − 1 ) + 5 ] ( n ≤ s < m ); thereby it can not be discomposed into [ 6 ( n − 1 ) + a ] [ 6 ( s − 1 ) + b ] ( a , b = 2 , 3 , 4 , 6 ).

Therefore, H 5 = q 1 ⋅ q 5 .

Part 2

In this part we introduce the π(n) formula.

1) The principle of deducing π(n)

π(n) is equal to the sum of the number of primes of the first column and the number of primes of the fifth column and 2.

The number of primes of the first column is denoted with m₁, the number of primes of the fifth column is denoted with m₅. m₁ is equal to the number of the first column which is less than n abstracting the number of composite number and abstracting 1. m₅ is equal to the number of the fifth column which is less than n abstract the number of composite number.

2) The computing theorem of the π(n) formula

Number of primes no more than any given positive integer n is the sum of three parts, which are prime numbers of first column and fifth column, and the number 2 of number prime 2 and 3.

1) π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2 ( 2 < n < 25 );

2) π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2                     + ∑ i = 1 [ log 5 n ] − 1 ( − 1 ) i ⋅ ∑ ∑ s x = i ( [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R 6 ]                     + [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ′ 6 ] )     ( n ≥ 25 ) . .

where “[ ]” denotes taking integer.

r = 1 , 2 , 3 , 4 , 5 , 6 ; s x = s 1 , s 2 , ⋯ , s j , s h ; s 1 , s 2 , ⋯ , s j , s h = 0 , 1 , 2 , 3 , ⋯

As i ≥ 2 , 2 ≤ s x ≤ i − 1 ( x = 1 , 2 , ⋯ , j , h ).

2) The conditions of the π(n) formula

As i is determined, k takes n and n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ≥ 6 .

The regulations of R , R ′ in the following:

The base number of the power of the denominator of the fraction

n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h are possible to take a part or all of the following

numbers 5 , 7 , ⋯ , ( 6 k − 1 ) , ( 6 k + 1 ) , setting the minimum of the part or the whole obtained numbers is p.

As p is the number of the fifth column, R = P − 6 , R ′ = P − 4 ;

As p is the number of the first column, R = P − 6 , R ′ = P − 2 .

1) As 2 < n < 25, the expression of π(n) formula is:

π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2.

2) As n ≥ 25, the expression of π(n) formula makes up of three parts:

π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2       + ∑ i = 1 [ log 5 n ] − 1 ( − 1 ) i ⋅ ∑ ∑ s x = i ( [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R 6 ]       + [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ′ 6 ] )

The first one is 2 [ ( n − 1 ) / 6 ] + [ ( r + 1 ) / 6 ] , we set its value to Z, the number of first column is n₁, the number of fifth column is n₅, then Z = n 1 + n 5 − 1 .

The second one is the later expression of sum, setting the value of that is H, H is equal to the number of the compositions of first column and the fifth column.

Set n = 6 ( k − 1 ) + r , where k is the row, r is column.

Z = 2 [ ( n − 1 ) / 6 ] + [ ( r + 1 ) / 6 ] = 2 [ ( 6 ( k − 1 ) + r − 1 ) / 6 ] + [ ( r + 1 ) / 6 ] = 2 [ ( k − 1 ) + ( r − 1 ) / 6 ] + [ ( r + 1 ) / 6 ] = 2 ( k − 1 ) + 2 [ ( r − 1 ) / 6 ] + [ ( r + 1 ) / 6 ]

1) As r = 1 , 2 , 3 , 4

[ r − 1 6 ] = 0 , [ r + 1 6 ] = 0 , then Z = 2 ( k − 1 ) .

Because the row is k, the numbers of the first column number is k in all, except1, the plus is k − 1, therefore the numbers of the first column number and the fifth column number which is less or equal to n except 1 is equal to ( k − 1 ) + ( k − 1 ) = 2 ( k − 1 ) , so the value of Z is right.

2) As r = 5 , 6

[ ( r − 1 ) / 6 ] = 0 , [ ( r + 1 ) / 6 ] = 1 , then Z = 2 ( k − 1 ) + 1 = 2 k − 1 .

Because the row is k, the numbers of the first column number is k in all, except 1, the plus is k − 1, the numbers of the fifth column number is k − 1 in all, therefore the numbers of the first column number and the fifth column number which is less or equal to n except 1 is equal to ( k − 1 ) + k = 2 k − 1 , so the value of Z is right.

Let us deduce the value of H in the following.

The expansion of H is:

H = − [ n 5 + 1 6 ] − [ n 5 − 1 6 ] − [ n 7 − 1 6 ] − [ n 7 − 5 6 ] − ⋯ + [ n 5 × 7 + 1 6 ] + [ n 5 × 7 − 1 6 ] + [ n 5 × 11 − 1 6 ] + [ n 5 × 11 − 1 6 ] + ⋯ + [ n 7 × 11 − 1 6 ] + [ n 7 × 11 − 5 6 ] + [ n 7 × 13 − 1 6 ] + [ n 7 × 11 − 5 6 ] + ⋯ + ⋯ − [ n 5 × 5 × 7 + 1 6 ] − [ n 5 × 5 × 7 − 1 6 ] − [ n 5 × 5 × 11 + 1 6 ] − [ n 5 × 5 × 11 − 1 6 ] − ⋯ − ⋯

+ [ n 5 × 5 × 5 × 7 + 1 6 ] + [ n 5 × 5 × 5 × 7 − 1 6 ] + [ n 5 × 5 × 5 × 11 + 1 6 ] + [ n 5 × 5 × 5 × 11 − 1 6 ] + ⋯ + ⋯ − ⋯ ⋯ + ( − 1 ) ( [ log 5 n ] − 1 ) [ n 5 s 1 ⋅ 7 + 1 6 ] + ( − 1 ) ( [ log 5 n ] − 1 ) [ n 5 s 1 ⋅ 7 − 1 6 ] + ⋯ − ⋯ + ( − 1 ) ( [ log 5 n ] − 1 ) [ n ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h + R 6 ] + ( − 1 ) ( [ log 5 n ] − 1 ) [ n ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h + R ′ 6 ] + ⋯

1) As i = 1, (set p ∈ P 5 ),

[ n p − ( p − 6 ) 6 ] + [ n p − ( p − 4 ) 6 ] expresses the number of compositions of the first column and fifth column that is less than or equal to n and contain p and more than or equal to p².

Let [ n p ] = t , the t numbers that are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,

16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, …, include these numbers 5, 11, 17, 23, …, the method of calculating the numbers is, every six numbers is cut into a segment from former to later with startingq, the last number or numbers from 1

to q as a last segment, then there are [ [ n p ] − ( p − 1 ) + 5 6 ] = [ n p − ( p − 6 ) 6 ] segment in all.

Therefore, the first number of every segment is the number of the fifth column, so the numbers of the composition of the fifth column which contain q 1 , q 2 , q 3 , ⋯ , q n and is less than or equal to n and is more than or equal to p² is:

[ n p − ( p − 6 ) 6 ] .

We can calculate the number of compositions of the first column and the fifth column, which is less or equal to n and more than or equal to p² and contain the factor p is:

[ [ n p ] − ( p + 1 ) + 5 6 ] = [ n p − ( p − 4 ) 6 ] in all.

Because of composition:

p ⋅ t ′ ( t ′ ∈ { 1 , 2 , ⋯ , t } )     factor     t ′ ≥ p ,     so   the   composition     p ⋅ t ′ ≥ p 2 .

Because for the factor t, We can prove the following conclusion with same method that [ n p − ( p − 6 ) 6 ] + [ n p − ( p − 2 ) 6 ] expresses the number of compositions

of the first column and fifth column that is less than or equal to n and contain p and more than or equal to p², where p is the number of the first column, and p ≠ 1 .

2) As i ≥ 2 ,

{ p 1 , p 2 , ⋯ , p j , p h } ⊆ { 5 , 7 , 11 , 13 , ⋯ , ( 6 k − 1 ) , ( 6 k + 1 ) }

and:

p 1 < p 2 < ⋯ < p j < p h .

Set p₁ is the number of the fifth column number, then:

[ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 6 ) 6 ] + [ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 4 ) 6 ]

expresses the number of compositions that is less than or equal to n and contain p 1 , p 2 , ⋯ , p j , p h and more than or equal to p 1 s 1 + 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h , where p₁ is the number of the first column.

Set [ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h ] = t , thet numbers that are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, …, include these numbers 5, 11, 17, 23, …, the method of calculating the numbers is: every six numbers is cut into a segment from former to later with starting q, the last number or numbers from 1 to q as a last segment, then there are:

[ t − ( p − 1 ) + 5 6 ] = [ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 6 ) 6 ] segments in all.

Therefore, the first number of every segment is the number of the fifth column, so the numbers of the composition of the fifth column which contain q 1 , q 2 , q 3 , ⋯ , q n , and is less than or equal to n and is more than or equal to p 1 s 1 + 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h is:

[ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 6 ) 6 ] in all.

We can prove the following conclusion with the same method as that:

[ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 6 ) 6 ] + [ n p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h − ( p 1 − 2 ) 6 ]

Express the number of compositions that is less than or equal to n and contain p 1 , p 2 , ⋯ , p j , p h and more than or equal to p 1 s 1 + 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h ,where p₁ is the number of the first column.

In the following we prove: the composition of the first column and the fifth column p 1 s 1 + 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h which is less or equal to n is unique, that is to calculate one time, where

{ p 1 , p 2 , ⋯ , p j , p h } ⊆ { 5 , 7 , 11 , 13 , ⋯ , ( 6 k − 1 ) , ( 6 k + 1 ) }

and

p 1 < p 2 < ⋯ < p j < p h , s 1 , s 2 , ⋯ , s j , s h ∈ N .

Because of 1 ≤ i ≤ [ log 5 n ] − 1 , and as ∑ s x = i ( x = 1 , 2 , ⋯ , j , h ), any composition of the first column and fifth column p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h which satisfy the condition ∑ s x = i and which is less than or equal to n calculated, s 1 , s 2 , ⋯ , s j , s h ∈ N .

And because p 1 < p 2 < ⋯ < p j < p h . as it satisfy ∑ s x = i and the values of s 1 , s 2 , ⋯ , s j , s h are determined, the composition p 1 s 1 ⋅ p 2 s 2 ⋯ p j s j ⋅ p h s h is unique.

We prove the maximum of i is [ log 5 n ] − 1 .

Because n ≥ 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h > 5 s 1 ⋅ 5 s 2 ⋯ 5 s j ⋅ 5 s h = 5 ∑ s x ,

Therefore the maximum of i is [ log 5 n ] − 1 .

We prove the code of every term is ( − 1 ) i in the following:

Because the composition with three factors can be regarded as the product of two factors, at the same cause, the composition with i factors can be regarded as the product of i − 1 factors. Therefore, we should take the composite of three factors into account when calculating a composite number containing two factors, and we should plus the number of composition with three factors. At the same cause, we should take the number of compositions with i factors into account when calculating the number of compositions with i − 1 factors, and we should plus the number of composition with i factors.

Therefore, asi is an odd, the code is negative, as i is an even, the code is positive, we can write them into the union form ( − 1 ) i , so the code of every terms is ( − 1 ) i .

π ( n ) = { 2 [ n − 1 6 ] + [ r + 1 6 ] + 2         ( 2 ≤ n < 25 ) ; 2 [ n − 1 6 ] + [ r + 1 6 ] + 2 + ∑ i = 1 [ log 5 n ] − 1 ( − 1 ) i ⋅ ∑ ∑ s x = i ( [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R 6 ] + [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ′ 6 ] )           ( n ≥ 25 ) .

that is:

1) π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2 ( 2 ≤ n < 25 );

2) π ( n ) = 2 [ n − 1 6 ] + [ r + 1 6 ] + 2                     + ∑ i = 1 [ log 5 n ] − 1 ( − 1 ) i ⋅ ∑ ∑ s x = i ( [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R 6 ]                     + [ n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ′ 6 ] )     ( n ≥ 25 ) .

where “[ ]” denotes taking integer.

r = 1 , 2 , 3 , 4 , 5 , 6 ;

s x = s 1 , s 2 , ⋯ , s j , s h ;

s 1 , s 2 , ⋯ , s j , s h = 0 , 1 , 2 , 3 , ⋯

As i ≥ 2 , 2 ≤ s x ≤ i − 1 ( x = 1 , 2 , ⋯ , j , h ).

As i is determined, k take n and:

n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h − R ≥ 6 .

The regulations of R , R ′ in the following:

The base number of the power of the denominator of the fraction n 5 s 1 ⋅ 7 s 2 ⋯ ( 6 k − 1 ) s j ⋅ ( 6 k + 1 ) s h are possible take a part or all of the following numbers 5 , 7 , ⋯ , ( 6 k − 1 ) , ( 6 k + 1 ) , setting the minimum of the part or whole obtained numbers is p.

As p is the number of the fifth column, R = P − 6 , R ′ = P − 4 ;

As p is the number of the first column, R = P − 6 , R ′ = P − 2 .

Part 3

In this part we introduce some examples of the π(n) formula application.

π ( 100 ) = 2 [ ( 100 − 1 ) / 6 ] + 2 + [ ( 4 + 1 ) / 6 ]   − [ ( 100 / 5 + 1 ) / 6 ] − [ ( 100 / 5 − 1 ) / 6 ]   − [ ( 100 / 7 − 1 ) / 6 ] − [ ( 100 / 7 − 5 ) / 6 ] = 34 − 9 = 25

π ( 200 ) = 2 [ ( 200 − 1 ) / 6 ] + 2 + [ ( 2 + 1 ) / 6 ]   − [ ( 200 / 5 + 1 ) / 6 ] − [ ( 200 / 5 − 1 ) / 6 ]   − [ ( 200 / 7 − 1 ) / 6 ] − [ ( 200 / 7 − 5 ) / 6 ]   − [ ( 200 / 11 − 5 ) / 6 ] − [ ( 200 / 11 − 7 ) / 6 ]

  − [ ( 200 / 13 − 7 ) / 6 ] − [ ( 200 / 13 − 11 ) / 6 ]   + [ ( 200 / 5 × 7 + 1 ) / 6 ] + [ ( 200 / 5 × 7 − 1 ) / 6 ] = 68 − 23 + 1 = 46

π ( 400 ) = 2 [ ( 400 − 1 ) / 6 ] + 2 + [ ( 4 + 1 ) / 6 ]   − [ ( 400 / 5 + 1 ) / 6 ] − [ ( 400 / 5 − 1 ) / 6 ]   − [ ( 400 / 7 − 1 ) / 6 ] − [ ( 400 / 7 − 5 ) / 6 ]   − [ ( 400 / 11 − 5 ) / 6 ] − [ ( 400 / 11 − 7 ) / 6 ]   − [ ( 400 / 13 − 7 ) / 6 ] − [ ( 400 / 13 − 11 ) / 6 ]

  − [ ( 400 / 17 − 11 ) / 6 ] − [ ( 400 / 17 − 13 ) / 6 ]   − [ ( 400 / 19 − 13 ) / 6 ] − [ ( 400 / 19 − 17 ) / 6 ]   + [ ( 400 / 5 × 7 + 1 ) / 6 ] + [ ( 400 / 5 × 7 − 1 ) / 6 ]   + [ ( 400 / 5 × 11 + 1 ) / 6 ] + [ ( 400 / 5 × 11 − 1 ) / 6 ] = 134 − 62 + 6 = 78

π ( 1000 ) = 2 [ ( 1000 − 1 ) / 6 ] + 2 + [ ( 4 + 1 ) / 6 ]   − [ ( 1000 / 5 + 1 ) / 6 ] − [ ( 1000 / 5 − 1 ) / 6 ]   − [ ( 1000 / 7 − 1 ) / 6 ] − [ ( 1000 / 7 − 5 ) / 6 ]   − [ ( 1000 / 11 − 5 ) / 6 ] − [ ( 1000 / 11 − 7 ) / 6 ]   − [ ( 1000 / 13 − 7 ) / 6 ] − [ ( 1000 / 13 − 11 ) / 6 ]

− [ ( 1000 / 17 − 11 ) / 6 ] − [ ( 1000 / 17 − 13 ) / 6 ] − [ ( 1000 / 19 − 13 ) / 6 ] − [ ( 1000 / 19 − 17 ) / 6 ]

− [ ( 1000 / 23 − 17 ) / 6 ] − [ ( 1000 / 23 − 19 ) / 6 ] − [ ( 1000 / 25 − 19 ) / 6 ] − [ ( 1000 / 25 − 23 ) / 6 ] − [ ( 1000 / 29 − 23 ) / 6 ] − [ ( 1000 / 29 − 25 ) / 6 ]

− [ ( 1000 / 31 − 25 ) / 6 ] − [ ( 1000 / 31 − 29 ) / 6 ] + [ ( 1000 / 5 × 7 + 1 ) / 6 ] + [ ( 1000 / 5 × 7 − 1 ) / 6 ] + [ ( 1000 / 5 × 11 + 1 ) / 6 ] + [ ( 1000 / 5 × 11 − 1 ) / 6 ] + [ ( 1000 / 5 × 13 + 1 ) / 6 ] + [ ( 1000 / 5 × 13 − 1 ) / 6 ] + [ ( 1000 / 5 × 17 + 1 ) / 6 ] + [ ( 1000 / 5 × 17 − 1 ) / 6 ]

+ [ ( 1000 / 5 × 19 + 1 ) / 6 ] + [ ( 1000 / 5 × 19 − 1 ) / 6 ] + [ ( 1000 / 5 × 23 + 1 ) / 6 ] + [ ( 1000 / 5 × 23 − 1 ) / 6 ] + [ ( 1000 / 5 × 25 + 1 ) / 6 ] + [ ( 1000 / 5 × 25 − 1 ) / 6 ] + [ ( 1000 / 5 × 29 + 1 ) / 6 ] + [ ( 1000 / 5 × 29 − 1 ) / 6 ] + [ ( 1000 / 5 × 31 + 1 ) / 6 ] + [ ( 1000 / 5 × 31 − 1 ) / 6 ]

+ [ ( 1000 / 5 × 35 + 1 ) / 6 ] + [ ( 1000 / 5 × 35 − 1 ) / 6 ] + [ ( 1000 / 5 × 37 + 1 ) / 6 ] + [ ( 1000 / 5 × 37 − 1 ) / 6 ] + [ ( 1000 / 7 × 11 − 1 ) / 6 ] + [ ( 1000 / 7 × 11 − 5 ) / 6 ] + [ ( 1000 / 7 × 13 − 1 ) / 6 ] + [ ( 1000 / 7 × 13 − 5 ) / 6 ] + [ ( 1000 / 7 × 17 − 1 ) / 6 ] + [ ( 1000 / 7 × 17 − 5 ) / 6 ]

  + [ ( 1000 / 7 × 19 − 1 ) / 6 ] + [ ( 1000 / 7 × 19 − 5 ) / 6 ]   − [ ( 1000 / 5 × 5 × 7 + 1 ) / 6 ] − [ ( 1000 / 5 × 5 × 7 − 1 ) / 6 ] = 334 − 200 + 35 − 1 = 168

π ( 5000 ) = 2 [ ( 5000 − 1 ) / 6 ] + 2 + [ ( 2 + 1 ) / 6 ]   − [ ( 5000 / 5 + 1 ) / 6 ] − [ ( 5000 / 5 − 1 ) / 6 ]   − [ ( 5000 / 7 − 1 ) / 6 ] − [ ( 5000 / 7 − 5 ) / 6 ]   − [ ( 5000 / 11 − 5 ) / 6 ] − [ ( 5000 / 11 − 7 ) / 6 ]   − [ ( 5000 / 13 − 7 ) / 6 ] − [ ( 5000 / 13 − 11 ) / 6 ]

− [ ( 5000 / 17 − 11 ) / 6 ] − [ ( 5000 / 17 − 13 ) / 6 ] − [ ( 5000 / 19 − 13 ) / 6 ] − [ ( 5000 / 19 − 17 ) / 6 ] − [ ( 5000 / 23 − 17 ) / 6 ] − [ ( 5000 / 23 − 19 ) / 6 ] − [ ( 5000 / 25 − 19 ) / 6 ] − [ ( 5000 / 25 − 23 ) / 6 ] − [ ( 5000 / 29 − 23 ) / 6 ] − [ ( 5000 / 29 − 25 ) / 6 ]

− [ ( 5000 / 31 − 25 ) / 6 ] − [ ( 5000 / 31 − 29 ) / 6 ] − [ ( 5000 / 35 − 29 ) / 6 ] − [ ( 5000 / 35 − 31 ) / 6 ]

− [ ( 5000 / 37 − 31 ) / 6 ] − [ ( 5000 / 37 − 35 ) / 6 ] − [ ( 5000 / 41 − 35 ) / 6 ] − [ ( 5000 / 41 − 37 ) / 6 ] − [ ( 5000 / 43 − 37 ) / 6 ] − [ ( 5000 / 43 − 41 ) / 6 ]

− [ ( 5000 / 47 − 41 ) / 6 ] − [ ( 5000 / 47 − 43 ) / 6 ] − [ ( 5000 / 49 − 43 ) / 6 ] − [ ( 5000 / 49 − 47 ) / 6 ] − [ ( 5000 / 53 − 47 ) / 6 ] − [ ( 5000 / 53 − 49 ) / 6 ] − [ ( 5000 / 55 − 49 ) / 6 ] − [ ( 5000 / 55 − 53 ) / 6 ] − [ ( 5000 / 59 − 53 ) / 6 ] − [ ( 5000 / 59 − 55 ) / 6 ]

− [ ( 5000 / 61 − 55 ) / 6 ] − [ ( 5000 / 61 − 59 ) / 6 ] − [ ( 5000 / 65 − 59 ) / 6 ] − [ ( 5000 / 65 − 61 ) / 6 ] − [ ( 5000 / 67 − 61 ) / 6 ] − [ ( 5000 / 67 − 65 ) / 6 ] + [ ( 5000 / 5 × 7 + 1 ) / 6 ] + [ ( 5000 / 5 × 7 − 1 ) / 6 ] + [ ( 5000 / 5 × 11 + 1 ) / 6 ] + [ ( 5000 / 5 × 11 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 13 + 1 ) / 6 ] + [ ( 5000 / 5 × 13 − 1 ) / 6 ] + [ ( 5000 / 5 × 17 + 1 ) / 6 ] + [ ( 5000 / 5 × 17 − 1 ) / 6 ] + [ ( 5000 / 5 × 19 + 1 ) / 6 ] + [ ( 5000 / 5 × 19 − 1 ) / 6 ] + [ ( 5000 / 5 × 23 + 1 ) / 6 ] + [ ( 5000 / 5 × 23 − 1 ) / 6 ] + [ ( 5000 / 5 × 25 + 1 ) / 6 ] + [ ( 5000 / 5 × 25 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 29 + 1 ) / 6 ] + [ ( 5000 / 5 × 29 − 1 ) / 6 ] + [ ( 5000 / 5 × 31 + 1 ) / 6 ] + [ ( 5000 / 5 × 31 − 1 ) / 6 ] + [ ( 5000 / 5 × 35 + 1 ) / 6 ] + [ ( 5000 / 5 × 35 − 1 ) / 6 ] + [ ( 5000 / 5 × 37 + 1 ) / 6 ] + [ ( 5000 / 5 × 37 − 1 ) / 6 ] + [ ( 5000 / 5 × 41 + 1 ) / 6 ] + [ ( 5000 / 5 × 41 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 43 + 1 ) / 6 ] + [ ( 5000 / 5 × 43 − 1 ) / 6 ] + [ ( 5000 / 5 × 47 + 1 ) / 6 ] + [ ( 5000 / 5 × 47 − 1 ) / 6 ] + [ ( 5000 / 5 × 49 + 1 ) / 6 ] + [ ( 5000 / 5 × 49 − 1 ) / 6 ] + [ ( 5000 / 5 × 53 + 1 ) / 6 ] + [ ( 5000 / 5 × 53 − 1 ) / 6 ] + [ ( 5000 / 5 × 55 + 1 ) / 6 ] + [ ( 5000 / 5 × 55 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 59 + 1 ) / 6 ] + [ ( 5000 / 5 × 59 − 1 ) / 6 ] + [ ( 5000 / 5 × 61 + 1 ) / 6 ] + [ ( 5000 / 5 × 61 − 1 ) / 6 ] + [ ( 5000 / 5 × 65 + 1 ) / 6 ] + [ ( 5000 / 5 × 65 − 1 ) / 6 ] + [ ( 5000 / 5 × 67 + 1 ) / 6 ] + [ ( 5000 / 5 × 67 − 1 ) / 6 ] + [ ( 5000 / 5 × 71 + 1 ) / 6 ] + [ ( 5000 / 5 × 71 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 73 + 1 ) / 6 ] + [ ( 5000 / 5 × 73 − 1 ) / 6 ] + [ ( 5000 / 5 × 77 + 1 ) / 6 ] + [ ( 5000 / 5 × 77 − 1 ) / 6 ] + [ ( 5000 / 5 × 77 + 1 ) / 6 ] + [ ( 5000 / 5 × 77 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 79 + 1 ) / 6 ] + [ ( 5000 / 5 × 79 − 1 ) / 6 ] + [ ( 5000 / 5 × 83 + 1 ) / 6 ] + [ ( 5000 / 5 × 83 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 85 + 1 ) / 6 ] + [ ( 5000 / 5 × 85 − 1 ) / 6 ] + [ ( 5000 / 5 × 91 + 1 ) / 6 ] + [ ( 5000 / 5 × 91 − 1 ) / 6 ] + [ ( 5000 / 5 × 95 + 1 ) / 6 ] + [ ( 5000 / 5 × 95 − 1 ) / 6 ] + [ ( 5000 / 5 × 97 + 1 ) / 6 ] + [ ( 5000 / 5 × 97 − 1 ) / 6 ] + [ ( 5000 / 5 × 101 + 1 ) / 6 ] + [ ( 5000 / 5 × 101 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 103 + 1 ) / 6 ] + [ ( 5000 / 5 × 103 − 1 ) / 6 ] + [ ( 5000 / 5 × 107 + 1 ) / 6 ] + [ ( 5000 / 5 × 107 − 1 ) / 6 ] + [ ( 5000 / 5 × 109 + 1 ) / 6 ] + [ ( 5000 / 5 × 109 − 1 ) / 6 ] + [ ( 5000 / 5 × 103 + 1 ) / 6 ] + [ ( 5000 / 5 × 103 − 1 ) / 6 ] + [ ( 5000 / 5 × 107 + 1 ) / 6 ] + [ ( 5000 / 5 × 107 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 109 + 1 ) / 6 ] + [ ( 5000 / 5 × 109 − 1 ) / 6 ] + [ ( 5000 / 5 × 113 + 1 ) / 6 ] + [ ( 5000 / 5 × 113 − 1 ) / 6 ] + [ ( 5000 / 5 × 115 + 1 ) / 6 ] + [ ( 5000 / 5 × 115 − 1 ) / 6 ] + [ ( 5000 / 5 × 119 + 1 ) / 6 ] + [ ( 5000 / 5 × 119 − 1 ) / 6 ] + [ ( 5000 / 5 × 121 + 1 ) / 6 ] + [ ( 5000 / 5 × 121 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 125 + 1 ) / 6 ] + [ ( 5000 / 5 × 125 − 1 ) / 6 ] + [ ( 5000 / 5 × 127 + 1 ) / 6 ] + [ ( 5000 / 5 × 127 − 1 ) / 6 ] + [ ( 5000 / 5 × 131 + 1 ) / 6 ] + [ ( 5000 / 5 × 131 − 1 ) / 6 ] + [ ( 5000 / 5 × 133 + 1 ) / 6 ] + [ ( 5000 / 5 × 133 − 1 ) / 6 ] + [ ( 5000 / 5 × 137 + 1 ) / 6 ] + [ ( 5000 / 5 × 137 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 139 + 1 ) / 6 ] + [ ( 5000 / 5 × 139 − 1 ) / 6 ] + [ ( 5000 / 5 × 143 + 1 ) / 6 ] + [ ( 5000 / 5 × 143 − 1 ) / 6 ] + [ ( 5000 / 5 × 145 + 1 ) / 6 ] + [ ( 5000 / 5 × 145 − 1 ) / 6 ] + [ ( 5000 / 5 × 149 + 1 ) / 6 ] + [ ( 5000 / 5 × 149 − 1 ) / 6 ] + [ ( 5000 / 5 × 151 + 1 ) / 6 ] + [ ( 5000 / 5 × 151 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 155 + 1 ) / 6 ] + [ ( 5000 / 5 × 155 − 1 ) / 6 ] + [ ( 5000 / 5 × 157 + 1 ) / 6 ] + [ ( 5000 / 5 × 157 − 1 ) / 6 ] + [ ( 5000 / 5 × 161 + 1 ) / 6 ] + [ ( 5000 / 5 × 161 − 1 ) / 6 ] + [ ( 5000 / 5 × 163 + 1 ) / 6 ] + [ ( 5000 / 5 × 163 − 1 ) / 6 ] + [ ( 5000 / 5 × 167 + 1 ) / 6 ] + [ ( 5000 / 5 × 167 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 169 + 1 ) / 6 ] + [ ( 5000 / 5 × 169 − 1 ) / 6 ] + [ ( 5000 / 5 × 173 + 1 ) / 6 ] + [ ( 5000 / 5 × 173 − 1 ) / 6 ] + [ ( 5000 / 5 × 175 + 1 ) / 6 ] + [ ( 5000 / 5 × 175 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 179 + 1 ) / 6 ] + [ ( 5000 / 5 × 179 − 1 ) / 6 ] + [ ( 5000 / 5 × 181 + 1 ) / 6 ] + [ ( 5000 / 5 × 181 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 185 + 1 ) / 6 ] + [ ( 5000 / 5 × 185 − 1 ) / 6 ] + [ ( 5000 / 5 × 187 + 1 ) / 6 ] + [ ( 5000 / 5 × 187 − 1 ) / 6 ] + [ ( 5000 / 5 × 191 + 1 ) / 6 ] + [ ( 5000 / 5 × 191 − 1 ) / 6 ] + [ ( 5000 / 5 × 193 + 1 ) / 6 ] + [ ( 5000 / 5 × 193 − 1 ) / 6 ] + [ ( 5000 / 5 × 197 + 1 ) / 6 ] + [ ( 5000 / 5 × 197 − 1 ) / 6 ]

+ [ ( 5000 / 5 × 199 + 1 ) / 6 ] + [ ( 5000 / 5 × 199 − 1 ) / 6 ] + [ ( 5000 / 7 × 11 − 1 ) / 6 ] + [ ( 5000 / 7 × 11 − 5 ) / 6 ] + [ ( 5000 / 7 × 13 − 1 ) / 6 ] + [ ( 5000 / 7 × 13 − 5 ) / 6 ] + [ ( 5000 / 7 × 17 − 1 ) / 6 ] + [ ( 5000 / 7 × 17 − 5 ) / 6 ] + [ ( 5000 / 7 × 19 − 1 ) / 6 ] + [ ( 5000 / 7 × 19 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 23 − 1 ) / 6 ] + [ ( 5000 / 7 × 23 − 5 ) / 6 ] + [ ( 5000 / 7 × 25 − 1 ) / 6 ] + [ ( 5000 / 7 × 25 − 5 ) / 6 ] + [ ( 5000 / 7 × 29 − 1 ) / 6 ] + [ ( 5000 / 7 × 29 − 5 ) / 6 ] + [ ( 5000 / 7 × 31 − 1 ) / 6 ] + [ ( 5000 / 7 × 31 − 5 ) / 6 ] + [ ( 5000 / 7 × 35 − 1 ) / 6 ] + [ ( 5000 / 7 × 35 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 37 − 1 ) / 6 ] + [ ( 5000 / 7 × 37 − 5 ) / 6 ] + [ ( 5000 / 7 × 41 − 1 ) / 6 ] + [ ( 5000 / 7 × 41 − 5 ) / 6 ] + [ ( 5000 / 7 × 43 − 1 ) / 6 ] + [ ( 5000 / 7 × 43 − 5 ) / 6 ] + [ ( 5000 / 7 × 47 − 1 ) / 6 ] + [ ( 5000 / 7 × 47 − 5 ) / 6 ] + [ ( 5000 / 7 × 49 − 1 ) / 6 ] + [ ( 5000 / 7 × 49 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 53 − 1 ) / 6 ] + [ ( 5000 / 7 × 53 − 5 ) / 6 ] + [ ( 5000 / 7 × 55 − 1 ) / 6 ] + [ ( 5000 / 7 × 55 − 5 ) / 6 ] + [ ( 5000 / 7 × 59 − 1 ) / 6 ] + [ ( 5000 / 7 × 59 − 5 ) / 6 ] + [ ( 5000 / 7 × 61 − 1 ) / 6 ] + [ ( 5000 / 7 × 61 − 5 ) / 6 ] + [ ( 5000 / 7 × 65 − 1 ) / 6 ] + [ ( 5000 / 7 × 65 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 67 − 1 ) / 6 ] + [ ( 5000 / 7 × 67 − 5 ) / 6 ] + [ ( 5000 / 7 × 71 − 1 ) / 6 ] + [ ( 5000 / 7 × 71 − 5 ) / 6 ] + [ ( 5000 / 7 × 73 − 1 ) / 6 ] + [ ( 5000 / 7 × 73 − 5 ) / 6 ] + [ ( 5000 / 7 × 77 − 1 ) / 6 ] + [ ( 5000 / 7 × 77 − 5 ) / 6 ] + [ ( 5000 / 7 × 79 − 1 ) / 6 ] + [ ( 5000 / 7 × 79 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 83 − 1 ) / 6 ] + [ ( 5000 / 7 × 83 − 5 ) / 6 ] + [ ( 5000 / 7 × 85 − 1 ) / 6 ] + [ ( 5000 / 7 × 85 − 5 ) / 6 ] + [ ( 5000 / 7 × 89 − 1 ) / 6 ] + [ ( 5000 / 7 × 89 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 91 − 1 ) / 6 ] + [ ( 5000 / 7 × 91 − 5 ) / 6 ] + [ ( 5000 / 7 × 95 − 1 ) / 6 ] + [ ( 5000 / 7 × 95 − 5 ) / 6 ]

+ [ ( 5000 / 7 × 97 − 1 ) / 6 ] + [ ( 5000 / 7 × 97 − 5 ) / 6 ] + [ ( 5000 / 7 × 101 − 1 ) / 6 ] + [ ( 5000 / 7 × 101 − 5 ) / 6 ] + [ ( 5000 / 11 × 13 − 5 ) / 6 ] + [ ( 5000 / 11 × 13 − 7 ) / 6 ] + [ ( 5000 / 11 × 17 − 5 ) / 6 ] + [ ( 5000 / 11 × 17 − 7 ) / 6 ] + [ ( 5000 / 11 × 19 − 5 ) / 6 ] + [ ( 5000 / 11 × 19 − 7 ) / 6 ]

+ [ ( 5000 / 11 × 23 − 5 ) / 6 ] + [ ( 5000 / 11 × 23 − 7 ) / 6 ] + [ ( 5000 / 11 × 25 − 5 ) / 6 ] + [ ( 5000 / 11 × 25 − 7 ) / 6 ] + [ ( 5000 / 11 × 29 − 5 ) / 6 ] + [ ( 5000 / 11 × 29 − 7 ) / 6 ] + [ ( 5000 / 11 × 31 − 5 ) / 6 ] + [ ( 5000 / 11 × 31 − 7 ) / 6 ] + [ ( 5000 / 11 × 35 − 5 ) / 6 ] + [ ( 5000 / 11 × 35 − 7 ) / 6 ]

+ [ ( 5000 / 11 × 37 − 5 ) / 6 ] + [ ( 5000 / 11 × 37 − 7 ) / 6 ] + [ ( 5000 / 11 × 41 − 5 ) / 6 ] + [ ( 5000 / 11 × 41 − 7 ) / 6 ] + [ ( 5000 / 13 × 17 − 7 ) / 6 ] + [ ( 5000 / 13 × 17 − 11 ) / 6 ] + [ ( 5000 / 13 × 19 − 7 ) / 6 ] + [ ( 5000 / 13 × 19 − 11 ) / 6 ] + [ ( 5000 / 13 × 23 − 7 ) / 6 ] + [ ( 5000 / 13 × 23 − 11 ) / 6 ]

+ [ ( 5000 / 13 × 25 − 7 ) / 6 ] + [ ( 5000 / 13 × 25 − 11 ) / 6 ] + [ ( 5000 / 13 × 29 − 7 ) / 6 ] + [ ( 5000 / 13 × 29 − 11 ) / 6 ] − [ ( 5000 / 5 × 5 × 7 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 7 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 11 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 11 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 13 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 13 − 1 ) / 6 ]

− [ ( 5000 / 5 × 5 × 17 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 17 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 19 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 19 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 23 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 23 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 25 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 25 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 29 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 29 − 1 ) / 6 ]

− [ ( 5000 / 5 × 5 × 31 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 31 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 35 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 35 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 37 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 37 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 41 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 41 − 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 7 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 7 − 1 ) / 6 ]

− [ ( 5000 / 5 × 7 × 11 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 11 − 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 13 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 13 − 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 17 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 17 − 1 ) / 6 ]

− [ ( 5000 / 5 × 7 × 19 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 19 − 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 23 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 23 − 1 ) / 6 ]

− [ ( 5000 / 5 × 7 × 25 + 1 ) / 6 ] − [ ( 5000 / 5 × 7 × 25 − 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 11 + 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 11 − 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 13 + 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 13 − 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 17 + 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 17 − 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 19 + 1 ) / 6 ] − [ ( 5000 / 5 × 11 × 19 − 1 ) / 6 ]

− [ ( 5000 / 5 × 13 × 13 + 1 ) / 6 ] − [ ( 5000 / 5 × 13 × 13 − 1 ) / 6 ] − [ ( 5000 / 7 × 7 × 11 + 1 ) / 6 ] − [ ( 5000 / 7 × 7 × 11 − 1 ) / 6 ] − [ ( 5000 / 7 × 7 × 13 + 1 ) / 6 ] − [ ( 5000 / 7 × 7 × 13 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 5 × 7 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 5 × 7 − 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 5 × 11 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 5 × 11 − 1 ) / 6 ]

  − [ ( 5000 / 5 × 5 × 7 × 7 + 1 ) / 6 ] − [ ( 5000 / 5 × 5 × 7 × 7 − 1 ) / 6 ] = 1668 − 1441 + 495 − 54 + 1 = 699