Twin prime distribution table

An interval defined by 6 (6n 2+6n) (any twin number including 3).

Last pair of serial number logarithm

s 1 8 73 7 1

s 2 7 199 197

s 3 8 433 43 1

s 4 7 66 1 659

s 5 9 1063 106 1

s 6 1 1 1489 1487

s 7 1 1 1999 1997

s 8 13 2593 259 1

s 9 10 3 169 3 167

s 10 19 393 1 3929

s 1 1 19 4723 472 1

s 12 14 552 1 55 19

s 13 15 6553 655 1

s 14 20 756 1 7559

s 15 14 8629 8627

s 16 18 9769 9767

s 17 18 10939 10937

s 18 20 12253 1225 1

s 19 1 1 1368 1 13679

s 20 20 14869 14867

s 2 1 20 16633 1663 1

s 22 28 18 133 18 13 1

s 23 19 19843 1984 1

s 24 29 2 160 1 2 1599

s 25 26 2337 1 23369

s 26 16 25 17 1 25 169

s 27 23 27 109 27 107

s 28 28 29209 29207

s 29 23 3 132 1 3 13 19

s 30 32 33349 33347

s 3 1 30 35593 3559 1

s 32 25 37993 3799 1

s 33 23 40 153 40 15 1

s 34 28 4284 1 42839

s 35 28 45343 4534 1

s 36 25 47809 47807

s 37 37 50593 5059 1

s 38 30 5328 1 53279

s 39 26 56 10 1 56099

s 40 34 59023 5902 1

s 4 1 25 6 198 1 6 1979

s 42 27 6492 1 649 19

s 43 3 1 68 1 13 68 1 1 1

s 44 37 7 126 1 7 1263

s 45 32 74509 74507

s 46 33 777 13 777 1 1

s 47 37 8 1 199 8 1 197

s 48 37 8463 1 84629

s 49 38 88003 8800 1

s 50 35 9 1573 9 157 1

s 5 1 43 95443 9544 1

s 52 4 1 99 139 99 137

s 53 34 10293 1 102929

s 54 39 10686 1 106859

s 55 36 1 1088 1 1 10879

s 56 40 1 14799 1 14797

s 57 43 1 18903 1 1890 1

s 58 46 122869 122867

s 59 34 12729 1 127289

s 60 42 13 17 13 13 17 1 1

s 6 1 44 136069 136067

s 62 35 14055 1 140549

s 63 40 145009 145007

s 64 47 14973 1 149729

s 65 5 1 154279 154277

s 66 43 159 193 159 19 1

s 67 46 163993 16399 1

s 68 36 16890 1 168899

s 69 37 173779 173777

s 70 55 178909 178907

s 7 1 56 183973 18397 1

s 72 44 189 15 1 189 149

s 73 46 194269 194267

s 74 46 199753 19975 1

s 75 47 205033 20503 1

s 76 53 2 1060 1 2 10599

s 77 34 2 15983 2 1598 1

s 78 53 22 17 19 22 17 17

s 79 5 1 227473 22747 1

s 80 55 233 16 1 233 159

s 8 1 47 23892 1 2389 19

s 82 42 24486 1 244859

s 83 54 250969 250967

s 84 47 256903 25690 1

s 85 45 262783 26278 1

s 86 65 26922 1 2692 19

s 87 50 275593 27559 1

s 88 5 1 28 1923 28 192 1

s 89 55 28836 1 288359

s 90 46 294649 294647

s 9 1 56 30 1363 30 136 1

s 92 56 307873 30787 1

s 93 59 3 14599 3 14597

s 94 6 1 32 1469 32 1467

s 95 72 328 129 328 127

s 96 59 335 173 335 17 1

s 97 45 342073 34207 1

s 98 56 34908 1 349079

s 99 56 356263 35626 1

s 100 6 1 363439 363437

s 10 1 44 370873 37087 1

s 102 53 378 15 1 378 149

s 103 57 38559 1 385589

s 104 63 393079 393077

s 105 57 40068 1 400679

Elementary proof of the conjecture of twin prime numbers in 20 15

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.

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)

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)

(N M two natural numbers n "= m)

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 it is 6 (x)+1= p 6 (x)-1= q.

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.

Prime numbers-those factors are all their own numbers except 1-are like atoms in algebra. Since Euclid proved that there are infinite prime numbers 2000 years ago, it has fascinated countless mathematicians.

Because prime numbers are fundamentally related to multiplication, it is difficult to understand their properties related to addition. Some of the oldest unsolved mysteries in mathematics are related to prime numbers and addition, one of which is the twin prime number conjecture-a prime number with infinite pair difference of 2. The other is Goldbach conjecture, which holds that all even numbers can be expressed as the sum of two prime numbers.

There are many prime numbers at the beginning of a natural sequence, but as the numbers get bigger, there are fewer and fewer. For example, among the first 10 natural numbers, 40% are prime numbers-2, 3, 5 and 7-but only 4% of all 10 digits are prime numbers. In the past century, mathematicians have mastered the law of prime number reduction: in large numbers, the interval between consecutive prime numbers is about 2.3 times of the number of digits. For example, in a number of 100 bits, the average interval between two prime numbers is about 230.

But this is only the average. Prime numbers usually look closer or farther than the average expected value. Specifically, "twin" prime numbers usually appear together, such as 3 and 5 and 1 1 and 13, and their difference is only 2. Among large numbers, the twin prime numbers never seem to disappear completely (the largest twin prime numbers found so far are 375680 1, 695685×26666669-1and 375680 1, 695685× 2666).

1849, French mathematician Alfan Polignac put forward "Polignac conjecture": for all natural numbers k, there are infinite prime pairs (p, p+2k). When k is equal to 1, it is called twin prime conjecture, and when k is equal to other natural numbers, it is called weak twin prime conjecture (that is, the weakened version of twin prime conjecture). Therefore, some people regard Polignac as the initiator of the conjecture of twin prime numbers.

Since then, the inherent attraction of these conjectures has made them the holy grail of mathematics, although they may have no practical application value. Although many mathematicians are trying to prove this conjecture, it cannot be ruled out that the interval of prime numbers will continue to grow and eventually exceed a certain upper limit.

192 1 year, British mathematicians godfrey hardy and John Littlewood put forward a conjecture similar to Polinak's conjecture, which is usually called "Hardy Littlewood conjecture" or "strong twin prime conjecture" (that is, an enhanced version of the twin prime conjecture). This conjecture not only puts forward that twin prime numbers have infinite pairs, but also gives its asymptotic distribution form.

In May of 20 13, Zhang made a breakthrough in the study of twin prime numbers, and proved a weakened form of the conjecture of twin prime numbers. In the latest research, Zhang did not rely on unconfirmed inference, and found that there are infinitely many prime pairs with a difference of less than 70 million, thus taking a big step forward on the road of the important problem of twin prime conjecture.

Zhang's paper was published online on May 4th/KLOC-0, and officially published on May 2nd1day. On May 28th, this constant dropped to 60 million. Only two days later, on May 3 1, it dropped to 42 million. Three days later, June 2nd,130,000. The next day, 5 million. On June 5, 400 thousand.

In the "Polymath" project initiated by British mathematician Tim Gowers and others, the twin prime number problem has become a typical example of mathematicians around the world using the network to cooperate. People continue to improve Zhang's proof, further narrowing the distance to finally solve the twin prime conjecture. 20/kloc-in February, 2004, the 70 million Zhang Ba has been reduced to 2.46 million.