CHAPTER 18

Decisive solution to the problem of infinity many twin prime numbers and method of generating them and definition of twin prime numbers set by twin prime numbers formula

 

 


18.1. Generation of twin prime numbers

†According to theorem of prime numbers production:

"Set of prime numbers is formed from cycles in below form:

We show easily that every new prime twin pair is produced from previous prime twin pairs.

For example, new prime twin pairs are produced from base prime twin pairs namely (5,7) by adding the constant number:

1.

2.

3.

4.

It sounds that three new greater prime twin pairs are produced from pair of base prime twin (5,7):

(11,13), (17,19), (29,31)

Therefore, in first cycle of prime numbers, there is one prime twin pair and the second cycle of prime numbers has three prime twin pairs that can be written as below:

Third cycle of prime numbers is in "" form that the next prime twin pairs will be produced with below method:

4.

††

5.

(In the first pack of third cycle, two prime twin pairs are produced)

6.

(Second pack, one twin pair)

7.

8.

(Third pack, two twin pairs)

9.

10.

(Forth pack, two twin pairs)

11.

(Fifth pack, one twin pair)

12.

13.

†(Sixth pack, two twin pairs)

Here, numeral sequences of prime twin pairs are produced regularly from three pairs of prime twin:

(11,13) : (30+11,30+13),( 60+11,60+13),(90+11,90+13),(180+11,180+13)

(17,19) : (90+17,90+19),(120+17, 120+19),( 180+17,180+19)

(29,31) : (30+29,30+31),(120+29, 120+31),(150+29, 150+31)

Therefore, ten new prime twin pairs are produced that every one can be a generator for new twins. Attend that after the cycle , the twin pairs (5, 7) has no role in producing the new twins, because =1, and also twin pair ( 11,13) from the† next cycle will have no role in producing the new twins, becauseand in the same method, base twin pairs produce new twins and themselves become deleted in next cycle. In fact every base twin is substituted by some new twin pairs.

Same method, in forth cycle, base twin pairs produce new prime twins as below:

(From the first pack to tenth, five new prime twin pairs)

(From first pack to tenth, five new prime twin pairs)

(From the first pack to tenth, four new prime twin pairs)

††††††††††††††

(From the first pack to tenth, seven new prime twin pairs)

(From the first pack to tenth, three new prime twin pairs)

(From the first pack to tenth, only one new prime twin)

(From the first pack to tenth, three new prime twin pairs)

(From the first pack to tenth, two new prime twin pairs)

(From the first pack to tenth, four new prime twin pairs)

(From the first pack to tenth, two new prime twin pairs)

(From the first pack to tenth, four new prime twin pairs)

(From the first pack to tenth, five new prime twin pairs)

(From the first pack to tenth, five new prime twin pairs)

Therefore, obtained twins from third cycle are:

 

1. (11,13) :(431,433), (641,643) (1061,1063),(1481,1483) , (2111,2113)

2. (17,19) : (227,229), (857,859), (1277,1279), (1487,1489), (1697,1699)

3. (29,31) : (239,241), (659,661), (1289,1291), (2129,2131)

4. (41,43) : (461,463), (881,883), (1091,1093), (1301,1303), (1721,1723),

†(1931,1933), (2141,2143)†

5. (59.61): (269,271), (1319, 1321), (1949,1951)

6. (71,73): (281,283)

7. (101,103): (311,313), (521,523), (1151,1153)

8. (107,109): (1787,1789), (1997,1999)

9. (137,139): (347,349), (1607,1609), (2027,2029), (2237, 2239)

10. (149,151): (569,571), (1619,1621)

11. (179,181): (599,601), (809,811), (1019,1021), (1229,1231)

12. (191,193): (821, 823), (1031, 1033), (1451, 1453), (1871,1873),

††††† (2081, 2083)

13.(197,199):(617,619),(827,829),(1667, 1669),(1877,1879),(2087,2089)

18.1.1. Remark

Some of the number of new twins is in form:

18.1.2. Remark

Some new prime twins are produced from successive numbers pair, for example (209,211).

According to the definition, we call these pairs of successive numbers that produce prime twin pairs, as "twin pairs maker".

†††††

Twin pair maker (209, 211), produces three new prime twin pairs:

(209,211): (419, 421), (1049, 1051), (2309, 2311)

Twin pair maker (167,169) produces two new prime twin pairs:

Therefore, the number of prime twins in interval (210, 2311] is equal to "54" that "5" pairs of them have been produced from two twin pair makers.

18.1.3. Result

Growing of twins in any cycle is very slower than the previous cycle. Because from prime twin pairs of the first cycle produces three prime twin pairs and from three prime twin pairs of second cycle produce ten pairs of twin and from three prime twin pairs with addition ten prime twin pairs of third cycle produce "49" prime twin pairs. Ratio of twins of one cycle to the number of previous twins that are produced from them is:

It sounds that as we search more about prime twins in new cycles, with slow growing, the number of prime twins in very far cycles and in very extensive intervals tends to a very great number and it tends to infinite in infinite far cycle.

For example, the ratio of prime twins smaller than and (1224, 8164, 152892) is:

18.1.4. Final Result

Since every prime twin pair with a cycle coefficient †are located in intervals that include some prime numbers, It is obvious that every prime twin pair produces one pair of prime twin or at least one twin maker. So in every cycle, some twin pairs or twin pair makers will be produced at least. Since this phenomenon is repeated in every cycle, it is obvious that every prime twin pair are produced unlimitedly and set of twins is infinite. We prove this assertion in follow.

18.2. There is infinity many twin prime numbers

Proof. We know that every odd prime number greater than "3" is in general form, because residue "1" or "5" will be obtained from dividing of every prime number by "6". It means every odd prime number is in "" or "" forms. On other side:

According to the table of composite numbers, it is clear that odd composite numbers are obtained from below numeral progressions:

†††††††††††††† (1)

It is obvious that if numbers in forms of "" arenít in any form of above progressions, always a prime number will be obtained. Therefore, If we delete all odd composite numbers in form (1) from numbers in general form, number p (n) always is prime. In order to this, it is enough to solve Diophantine equations in below general form and determine "n" with respect to "k" and "p".

†††††††††††††† (2)

For example, we solve some equations and then we obtain general rule and sufficient conditions for solving the problem.

18.2.1. Example

†General equation (2) makes the conditions ""for "n" in case .

Because:

Therefore, in order to the numbers in form are not multiples of "5", it must be "".

18.2.2. Example

General equation (2) makes the conditions for "n" in case p=7. Because:

Therefore, in order to the numbers in form are not multiples of "7", it must be "".

18.2.3. Example

†General equation (2) makes the conditions †for "n" in case ; because:

Therefore, in order to the numbers in form aren't multiples of "11", it must be †as the same method.

18.2.4. Result

In order to the numbers in form of †arenít multiples of prime "p", it must be:

†††††††††††††††††††† (3)

18.3. Set of prime twin pairs

Since prime twin numbers are in †form except of (3,5), therefore, with conditions (3), set of prime twin numbers greater than "3" are infinite and is in below form:

(Set of prime twin pairs) T

So:

T

18.3.1. "H.M" Theorem.

If and

then †is twin number.