PRIME NUMBER LAW. DEPENDENCE OF PRIME NUMBERS ON THEIR ORDINAL NUMBERS AND GOLDBACH – EULER BINARY PROBLEM USING COMPUTER
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Abstract (English):
The article considers the methods of defining and finding the distribution of composite numbers CN, prime numbers PN, twins of prime numbers Tw and twins of composite numbers TwCN that do not have divisors 2 and 3 in the set of natural numbers - ℕ based on a set of numbers like Θ = {6∙κ ± 1, κ ∈ ℕ}, which is a semigroup in relation to multiplication. There has been proposed a method of obtaining primes by using their ordinal numbers in the set of primes and vice versa, as well as a new algorithm for searching and distributing primes based on a closedness of the elements of the set Θ. It has been shown that a composite number can be presented in the form of products (6x ± 1) (6y ± 1), where x, y ℕ - are positive integer solutions of one of the 4 Diophantine equations: . It has been proved that if there is a parameter λ of prime twins, then none of Diophantine equations P (x, y, λ) = 0 has positive integer solutions. There has been found the new distribution law of prime numbers π(x) in the segment [1 ÷ N]. Any even number is comparable to one of the numbers i.e. . According to the above remainders m, even numbers are divided into 3 types, each type having its own way of representing sums of 2 elements of the set Θ. For any even number in a segment [1 ÷ ν], where ν = (ζ−m) / 6, , there is a parameter of an even number; it is proved that there is always a pair of numbers that are elements of the united sets of parameters of prime twins and parameters of transition numbers , i.e. numbers of the form with the same λ, if the form is a prime number, then the form is a composite number, and vice versa.

Keywords:
prime and composite numbers, parameters of primes, Diophantine equations, binary (strong) Goldbach – Euler, algorithm for solving the binary Goldbach – Euler problem.
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Introduction Gauss was the first who observed the regularity of the arrangement of primes showing that the probability of the appearance of primes in the int is equal to . Then Legendre improved the type and introduced the function The first essential contribution to the study of was made by Euler with his -function, [1] with the real variables s, s > 1. By means of it, we showed that the series diverge, which means that primes are infinite. In 1859 Riemann proposed to consider the variation of the variable s of the Euler function in the complex plane and related the distribution of primes to the nontrivial zeros of the – function B. Riemann formulated the problem now known as the Riemann hypothesis, namely, that all the nontrivial zeros of the – function are in the strip and are symmetric with respect to the critical line , where i. e. all nontrivial zeros of having a real part are complex numbers. In the work of V. Minayev an attempt was made to describe the laws of natural numbers, on the basis of the elements of the set with the introduction of concepts of fundamental negative n-s , positive n-s , composite n-s & primes. But nothing is said about the methods and definitions, i.e. at what values of the parameters the numbers & are composite or prime. In 2013, Itan Chang proved that there are infinitely many pairs of successive primes with a difference and Terens Tao reduced this diffeerence to 246. In recent decades interest in the laws of prime distribution [1] from the theoretical point of view has been increasingly shifting to the practical one. Their use in cryptography is a particularly important example [2], that is why any results that clarify certain features of the laws of prime distribution immediately become the subject of study of the field cryptographers. Special interest in cryptography in a system with public keys (in particular, in the RSA encryption system) raises the question, whether this particular (large) number is prime or not. Interest in Goldbach – Euler binary problem in mathematics, as well as its application in related sciences and technologies are very important in cryptography [1, 2]. In the last decade the significant advances have been made in the field of additive number theory. For example, in 2013 Harald Helfgott proved Goldbach ternary problem. From the properties of even number we can see that are ζ , where are considered and their representations as sums of 2 numbers of the form and . An even number in the segment and , where , is an even number parameter. It has been proved that there always exist a pair of numbers and , where . The aim of the paper is to study the laws of prime distribution and composite numbers CN, to prove the solvability of the Diophantine equation in the domain of integers z > 2 for any even number and the summands . Method for selecting prime numbers When searching for primes in a natural series, a large loss of time is necessary for numbers that are divisible by 2 & 3. If in the set there is a factorization with base 6, we can simplify the search for primes and study their properties. We divide the set into 2 and disjoint subsets Η and Θ i. H ∩ Θ = ∅, ℕ=H ∪ Θ. Let H include 1 (one) and natural numbers, which after division by 6 give the remainders 0, 2, 3, 4. Obviously, the set includes two primes 2 and 3. The set contains numbers which when divided by 6 give the remainders 1 and 5, i.e. numbers of the form, as in the expression Numbers with reminders are composite numbers, since they are divisible by 2 or 3. So, we have embedding the sets P (≥ 5) ⊂ Θ ⊂ ℕ, where P (≥ 5) is a set of primes . It is not difficult to verify directly that the elements of a set Θ = {6∙κ ± 1, κ ∈ ℕ}, are closed under the operation of multiplication, i.e. the set Θ is a semigroup with operation of multiplicationumbers. We have relations: (1) Products of numbers of the forms and are expressed by numbers of the form, and it is obvious that the secret of composite and prime numbers lies here. Algorithms and programs of obtaining and distributing primes and composite numbers, twins of primes and twins of composite numbers were performed using Computer Software and Computing. Definition. For a number the values of the numerical functions ∀ x, y ∈ ℕ represented in (1) will be called the parameters of the number .We note that the problem of a single – valued correspondence between the numbers n and their parameters remains open, i. e several different parametric functions can correspond to the same number . However, from the point of view of later studies, this fact is not significant. We note that the form can be transformed into another form by multiplication (1). Since the set is a semigroup with respect to the operation of multiplication, all its constituent elements ∀ λi ∈ ℕ will be: (2) In the set there are elements , for which the number of factors in (2) is equal to 1, i. e. they do not decompose into products of other numbers in , these numbers are primitive elements of the set , i. e., primes in the set . For each of the expressions (1) we introduce a parameter , then from the link (1) we have: (3) where the “–” sign corresponds to the first two expressions (1) and the “+” sign - to the last two expressions (1). That is, we obtain equations for finding positive integer connecting the numbers (x, y) and parameter λ: (4) If at least one of the equations in (4) has one solution, then the number from (1) is a composite number. Ιf for some λ the Diophantine equations from (4) do not have positive integer solutions, then the number from (1) for this λ is a prime number. It is quite obvious from representations (1) and (2) that any constituent element ∈ will ultimately disintegrate into a product of primitive elements. Whence the main theorem of arithmetic follows, that any positive integer numbers can be represented as a product: n , where are primes and are the numbers of corresponding primes involved in the decompositionumbers. The distribution of composite numbers of the set Θ From (1) one can see that the primes and composite numbers are formed by values of the following functions: (5) – composite numbers of the set of the form (we denote them ) in view of (1) are generated by values of the symmetric and non-one-to-one functions: , , since for unequal values the values of the functions can be equal to and . With allowance for (1) we notice that the numbers of the forms and are composite numbers – composite numbers of set of the form (we denote them ) in view of (1) are generated by the values of the non-symmetric and non-one-to-one functions . It also follows from (1) that the numbers of forms and are composite numbers, ∈ Since the numbers are the composite numbers, then the values of the variables are solutions of the corresponding equations or And in the same way, for composite numbers values of are solutions of the equations or . We also note that for the factorization (taking into account that prime number is the least divisor [3]) of a composite number, the most effective and the best way is to use expression (2) that is the number divided by numbers of the form , where … (see [4]). From (4) it is easy to deduce that the set of parameters of all 4 types of primes and composite numbers of set in the natural series of numbers is infinite. Indeed, for example, if the range of function value from (5) is defined as the set , then for any specific number , the expression Similarly, other functions are also considered as given in (5), we denote the sets as , , , Thus, the set of values of all the functions (5) in Table 1: are infinite as unions of infinite sets. We introduce the notation for the set of parameters of composite numbers of the forms: (6) Thus, if all composite numbers of the set consist of a union then the set of all parameters of the composite numbers will be a union It is obvious that the sets are infinite as unions of infinite sets. To determine and study the parameters of primes and composite numbers of the set , it will be necessary to find all the parametric solutions of the equations (4). However, the solution of Diophantine equations is a complicated problem, therefore, to solve equations (4) we can construct a table of values of the function or functions (5). Then to the number λ in the table there corresponds a composite number n, otherwise the number n is prime numbers. To study the parameters of primes and composite numbers of the set , we set any values for the values of the functions (5) from 1 to where s is the specified table size. Let us construct Table 1 in the dimension . Note that for the same values of the variables in each row of Table 1 we have an increasing sequence of functions: Forming strings in Table 1 and the search for the values of the functions (5) are carried out, according to the principle: We choose the value to demonstrate the algorithms described below, but the constructions described below can be realized for any s. Let us find the values of the functions: , where is composite number, since the values of the variables are known as predetermined solutions of the equations (4). Let then the values of the functions (5) in the following row differ from the values of the previous row to the following value functions: for function , for function for function for function Let . Since the differences , then the functions (5) are increasing and infinite. Example 1. Find the composite numbers of the set Θ in the int 1 ÷ N = 155. We calculate in the specified interval the max parameter: from the Table 1 we write the parameters of composite numbers . As a result, we have: ΠCN = {+4, –6, +8, +9, –11, –13, +14, +15, –16, +19, + –20 –21, +22, + – 24, –26}. Symbols in numbers are obtained, according to sets and . On the basis of the definition of the parameters of the composite numbers from the set we find their values: : 6 ∙ 4 + 1 = 25; 6 ∙ 8 + 1 = 49; 6 ∙ 9 + 1 = 55; 6 ∙ 14 + 1 = 85; 6 ∙ 15 + 1 = 91; 6 ∙ 19 + 1 = 115; 6 ∙ 20 + 1 = 121; 6 ∙ 22 + 1 = 133; 6 ∙ 24 + 1 = 145. : 6 ∙ 6 – 1 = 35; 6 ∙ 11 − 1 = 65; 6 ∙ 13 − 1 = 77; 6 ∙ 16 − 1 = 95; 6 ∙ 20 − 1 = 119; 6 ∙ 21 − 1 = 125; 6 ∙ 24 − 1 = 143; 6 ∙ 26 − 1 = 155. Hence, the complete sequence of composite numbers of + = {25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 155}. Table 1 Formation of the parameters of composite numbers in the set Θ x y f11(x, y) f12 (x, y) f21 (x, y) f22 (x, y) G x y f11 (x, y) f12 (x, y) f21 (x, y) f22 (x, y) G1 1 1 4 8 6 G4 4 4 88 104 96 96 2 9 15 13 11 5 111 129 121 119 3 14 22 20 16 6 134 154 146 142 4 19 29 27 21 7 157 179 171 165 5 24 36 34 26 8 180 204 196 188 6 29 43 41 31 9 203 229 221 211 7 34 50 48 36 10 226 254 246 234 8 39 57 55 41 G5 5 5 140 160 150 150 9 44 64 62 46 6 169 191 181 179 10 49 71 69 51 7 198 222 212 208 G2 2 2 20 28 24 24 8 227 253 243 237 3 31 41 37 35 9 256 284 274 266 4 42 54 50 46 10 285 315 305 295 5 53 67 63 57 G6 6 6 204 228 216 216 6 64 80 76 68 7 239 265 253 251 7 75 93 89 79 8 274 302 290 286 8 86 106 102 90 9 309 339 327 321 9 97 119 115 101 10 344 376 364 356 10 108 132 128 112 G 7 7 7 280 308 294 294 G 3 3 3 48 60 54 54 8 321 351 337 335 4 65 79 73 71 9 362 394 380 376 5 82 98 92 88 10 403 437 423 417 6 99 117 111 105 G 8 8 8 368 400 384 384 7 116 136 130 122 9 415 449 433 431 8 133 155 149 139 10 462 498 482 478 9 150 174 168 156 G 9 9 9 468 504 486 486 10 167 193 187 173 10 521 559 541 539 → → G10 10 10 580 620 600 600 Distribution of the parameters of primes and composite numbers in The distribution of the parameters of primes and composite numbers of the set Θ is the analog of the distribution of primes and composite numbers that do not have divisors 2 and 3 in It will be necessary to find all primes of the form . We describe the algorithms for constructing these numbers. Let in the int [ ) the entries in the file are given according to the structure , where the field are the serial numbers of the records and the fields and take the values Before the beginning of algorithms 1 and 2 below in the interval [1÷[n / 6]) the sign are entered line by line in the fields and [5]. 1. Algorithm distributing primes of a set Θ of the form 6λ − 1. Let vary according to the principle of Table 1. Then, by values of the parameters and of a composite number of the form from a file by direct access to the records in the field F2 the sign “+” changes into the sign “–” and the remaining records at the end of the algorithm in the adjacent field F1 = “+” indicate the presence of primes of type: . We introduce the notation , , then . 2. Algorithm distributing primes of a set Θ of the form Let vary according to the principle of Table 1. Then by values of the parameters and of composite numbers of the form from a file Rπ by direct access to the records in the field F1 the sign “+” changes to the sign “–” and the remaining records at the end of the algorithm in adjacent field indicate the presence of primes of . We introduce the notation ; ; , then So, the primes consist of the union of two sets: . Combining algorithms 1 and 2 into one algorithm [5] we obtain the algorithm for distributing parameters of composite numbers and primes Θ in ℕ. Parameters with the assigned fields F1 and with the value , according algorithms to 1 and 2, are parameters of twins of primes and with the value are the parameters of twins of composite numbers. Theorem 1. At least one of the equations (4) has solutions when the form or is composite number. Necessity. Let a number of the form is a composite number, since is a semigroup, where exist 2 numbers and , with , as According to (1), there are only 2 cases for numbers, i. e. ; , which gives and which gives . In any case, one of the 2 first equations (4) has positive solutionumbers. The same is true for numbers of the form , in which case one of the two last equations (4) has positive solutionumbers. This completes the proof. Adequacy. Suppose, one of the equations (4), for example, equation , has a solutionumbers. This means that there exists a triple of natural numbers (λ, x, y) ∈ ℕ, that is fair . Then from we have , i. e. the number corresponding to parameter λ is a composite number, since both of are natural numbers different from 1, because . The same is true for any of the equations (4). Solutions of Diophantine equations Example 2. 2.1. We know some solution of one of the equations (4). 1. Let are solution of → . Taking (3) into account, we have or . From (1) → → is a composite n. 2. Let is a solution of , which . Taking (3) into account, we have from (1) → , i. e is a composite numbers. 2.2. We know that for some λ one of the corresponding numbers is prime. 1. , hence, and have no solutions. 2. Let → , hence, have no solutions. 2.3. We know that for some parameter λ one of the corresponding numbs is composite. Taking we know that is a composite number. Hence, one of the Diophant equations or has solutions. 1. If we express y in terms of x, than , which must be an integer. So, we search for which values of x in 1 ÷ the above value of y is an integer. Easily found , , 2. If we express y in terms of , we have , which must be an integer. So, we search for which values of x in 1÷(λ – 1) the above value of y is an integer. We found 2 solutions , . Let we know that is a composite numbers Hence, one of or has solutions. 3. We rewrite last relation as , which must be an integer. So, we search for which values of in the above of is an integer. One solution of the triplet is found. 4. We express y in terms of and have , which must be an integer. So, we search for which values of in the above of y is an integer. One solution is found. According to the distribution of the parameters of prime and composite numbers without divisors 2 and 3, we compose a Table 2. Table 2 Distribution of the parameters of primes and CN of the set Θ in the ℕ* Id = λ F1 F2 - 1 + + 41 – – 81 – + 121 – + 161 – + 2 + + 42 + – 82 + – 122 – + 162 + – 3 + + 43 + – 83 – + 123 – + 163 + – 4 + – 44 + – 84 + – 124 + – 164 + – 5 + + 45 + + 85 + – 125 – + 165 – + 6 – + 46 – + 86 – – 126 – + 166 – + 7 + + 47 + 87 + + 127 + – 167 – – 8 + – 48 – – 88 – – 128 – + 168 – + End of table 2 Distribution of the parameters of primes and CN of the set Θ in the ℕ* Id = λ F1 F2 9 + – 49 + – 89 – – 129 + – 169 + – 10 + + 50 – 90 – + 130 – – 170 + + 11 – + 51 – + 91 – + 131 – + 171 – – 12 + + 52 + + 92 – – 132 – – 172 + + 13 – + 53 + – 93 +100 – 133 + – 173 – + 14 + – 54 – – 94 + – 134 – – 174 – – 15 + – 55 – + + + 135 + + 175 + + 16 – + 56 – + 96 – + – – 176 – – 17 + + 57 – – 97 – – 137 + + 177 + + 18 + + 58 + + 98 + – 138 + + 178 – + 19 + – 59 + – 99 + – 139 – – 179 – – 20 – – 60 + – 100 + + 140 + – 180 – – 21 – + 61 – + 101 – + 141 – – 181 – + 22 + – 62 – + 102 – + 142 – + 182 + + 23 + + 63 – + 103 + + 143 + + 183 + – 24 – – 64 + – 104 – – 144 + – 184 + – 25 + + 65 + – 105 – + 145 – – 185 + – 26 – + 66 – + 106 – – 146 – + 186 – + 27 – + 67 + – 107 + + 147 +150 + 187 – + 28 + – 68 – + 108 + – 148 + – 188 – + 29 + – 69 – – 109 + – 149 – – 189 – – 30 + + 70 + + 110 + + 150 – – 190 – – 31 – – 71 – – 111 – – 151 – + 191 – – 32 + + 72 + + 112 – + 152 + – 192 + + 33 + + 73 – + 113 + – 153 – + 193 – – 34 – – 74 + – 114 + – 154 – – 194 + – 35 – + 75 + – 115 – + 155 + – 195 – + 36 – – 76 – + 116 – – 156 – + 196 – – 37 – + 77 + + 117 + – 157 + – 197 + – 38 + + 78 + – 118 – + 158 + – 198 + – 39 + – 79 – – 119 – – 159 + – 199 + – 40 +50 + 80 + – 120 + – 160 – – 200 – + *The values of elements of the set Θ, respectively, over the fields F1: 6λ – 1 and F2: 6λ + 1. By analogy to the filds F1, F2, the “+” sign corresponds to primes and “–” to composite numbers. It is obvious that these subsets of numbers in the natural series are parameters of the corresponding subsets of the set Θ. Ascribing to serial numbers of records signs “+” or “–”, in the fields F1 and F2 form a table of signs (Table 2) and using algorithms 1 and 2, the natural numbers are divided into subsets of numbers according to the combinations of the signs “+” & “–”. Consider the relationship between the equations (4) and the twins of primes. From the definition of twins of primes it is known that these are the with Note that for the same λ the difference of . So, if for this value of the λ the numbers are primes, then the pair of these numbers is generating a twin of primes. Hence, for this value of λ the numbers will be twin primes. Theorem 2. In order that is a parameter of twins of primes, it is necessary and sufficient that for this particular none of the Diofantine equations (4) has solutions. Necessity. Suppose for one and the same all equations (4) have no solutions. Then, by Theorem 1 from equations and it follows that is a prime number and from equations and it follows that is a prime number. Since then, by definition, of twins of primes and this λ is a parameter of twins of primes. Adequacy. Let λ be a parameter of twins of primes, i. e. , are primes by virtue of the definition of the twins of primes . Hence, from Corollary 1, for a prime number it follows that the equations & have no solutions and from Corollary 2 the equations & it follows that they have no solutions for prime . Thus, none of the Diophantine equations (4) has solutions. Definitions of subsets of the set By virtue of the distribution of the parameters of primes and composite numbers of the set Θ in the serial numbers (see Table 2), which are parameters of the following significant subsets (Tw – twins of primes), (TwCN – twins of composite numbers do not have divisors 2 and 3), (UC – unique composite numbers), (PN – unique primes) and ( ), of union unique primes and unique composite numbers). Twin composite numbers (2 composites numbers on the forms with the same parameter λ, having a difference = 2), i. e. , where (7) The set is parameters λ of twins of composite numbers, which lie on non – empty intersections of the values of two functions (5), one of which belongs to and the other to . The values of the fields F1 = “–” and F2 = “–” in Table 2 correspond to the parameters λ, the set of twins of composite numbers. 2. Unique composite numbers (the difference between 2 composite numbers without divisors 2 and 3 is > 2 where from (6) is the set of parameters λ that are representable in the form or for some values and is the set of parameters that are representable in the form or . The parameters of unique composite numbers ΠUC = FN− \ ΠTwCN FN+ \ ΠTwCN = (FN+ FN−) \ (FN+ ∩ FN−) = FN+ ∆ FN− - we have the symmetric difference of sets Obviously, the parameters of all composite numbers will be and . Unique primes Parameters ΠPN = FN− \ ΠTwCN FN+ \ ΠTwCN = (FN+ FN−) \ (FN+ FN−) = FN+ ∆ FN− i. e. the symmetric difference of sets. Since the set of parameters \ are not solutions of equations , then by Corollary 1 number of the form is a prime number and set of parameters \ ΠTwCN are not solutions of equations = 0, then by Corollary 2 number of the form is prime. Twins of primes (part 2): , where ; Parameters of twins of primes are at the intersections of the complements of the sets & i. e. In Table 2 the parameters of twins of primes λ correspond to the values of the fields Then, the parameters of all primes . Transition numbers (Unique Prime Composite) UPC = {6λ ± 1, λ ∈ FN + ∆ FN−} That is, numbers go from composite numbers to primes or vice versa by changing the forms with the same parameter λ.The parameters corresponded in Table 2 to the values of the fields or . Lemma 1. The set of natural series of numbers ℕ and the set are equivalent sets. Proof. For two sets ℕ~Θ to be equivalent, it is necessary and sufficient that there is a one-to-one correspondence between their elements. Note that the distribution of the parameters of the elements of the set by the design of Table 2 sets to each ordinal number id = λ an element from the subsets , and . This means that the elements of set natural series can be fully described by the function (5), i. e. set covers set ℕ. Hence, we have the right to assert that is a parameter of one of the listed subsets of the set So, it holds: . Lemma 2. Parameters and respectively, of the unique composite numbers of the types and then the parameters of unique primes will be numbers of the form and , respectively. Proof. Excluding the values of parameters from Table 1, we have the remaining values of parameters of the composite numbers of the form and values of parameters for composite numbers of the form . Since the values of parameters are in most cases different from the parameters , (because functions (5) are all different from each other), then, by Corollaries 1-2 they are the parameters of unique primes for the form and likewise the values of the parameters are not representable with the parameterss i. e. are the parameters of unique primes for the form . So, the laws of the distribution of unique primes by are the same as for composite numbers, respectively by type and . And, since the and are the infinity, then the infinity of unique primes follows from this. Calculation of primes by their ordinal numbers & vice versa in the set primes P From Table 2 of distribution of parameters of primes & composite numbers of in ℕ it is not difficult to see that between the ordinal numbers of primes in the set of primes P (≥ 5) and parameters λ of primes shown in Table 2, there are dependencies. Let n be the ordinal number of the prime number . in the set of primes P (≥ 5), then the corresponding prime number is found from the formula: where id is the line number, is the index of field , on which the n-th symbol “+” in Table 2. Counting of the signs “+” is carried out according to the following principle of viewing the lines. At the beginning of the algorithm, the values of the variables S1 = S2 = ψ(n) = 0. α. Algorithm for obtaining a prime by its ordinal number in the prime number P. β. Algorithm for obtaining serial numbers by their primes. Example 3. Let the ordinal number n = 10 of a prime number be p ≥ 5, then from Table 2 in the fields F1 and F2, summing up the number of characters “+” starting from the first line id = 1 and on, each time we chek the sum S1 + S2 for equality with n. If equal, the algorithm ends, otherwise continues. If equality occurs when calculating S1, then ψ(n) = 1, otherwise for S2 then ψ(n) = 2. Table 3 Algorithmic way to get a prime number by its ordinal number in the P (≥ 5) id S1 = S1 + (0: F2 = “−”), (1: F2 = “+”) S1 + S2 = n ? ψ(n) S2 = S2 + (0: F2 = “−”), (1: F2 = “+”) S1 + S2 = n ? ψ(n) 1 0 + 1=1 1 + 0 =1 0 0 + 1 = 1 1 + 1 = 2 0 2 1 + 1=2 2 + 1 = 3 0 1 + 1 = 2 2 + 2 = 4 0 3 2 + 1=3 3 + 2 = 5 0 2 + 1 = 3 3 + 3 = 6 0 4 3 + 1=4 4 + 3 = 7 0 3 + 0 = 3 4 + 3 = 7 0 5 4 + 1=5 5 + 3 = 8 0 3 + 1 = 4 5 + 4 = 9 0 6 5 + 0=5 5 + 4 = 9 0 4 + 1 = 5 5 + 5 = 10 2 Hence, this prime number with the ordinal number n = 10 in the table of primes P ≥ 5 will be the number Now, consider the opposite case. Let given a prime number P = 37, find its serial number n. Table 4 Algorithmic way of getting a serial number by the value of its prime in the P(≥5 ) i S1 = S1 + (0: F1 = “−”) (1: F1 = “+”) n = S1 + S2 ? S2 = S2+(0: F2 = “–”) (1:F2 = “+”) n =S2 + S1 6∙i + 1 = p ? 1 0 + 1 = 1 1 + 0 = 1 5 ≠ p 0 + 1 = 1 1 + 1 = 2 7 ≠ p 2 1 + 1 = 2 2 + 1 = 3 11 ≠ p 1 + 1 = 2 2 + 2 = 4 13 ≠ p 3 2 + 1 = 3 3 + 2 = 5 17 ≠ p 2 + 1 = 3 3 + 3 = 6 19 ≠ p 4 3 + 1 = 4 4 + 3 = 7 23 ≠ p 3 + 0 = 3 3 + 4 = 7 25 ≠ p 5 4 + 1 = 5 5 + 3 = 8 29 ≠ p 3 + 1 = 4 4 + 5 = 9 31 ≠ p 6 5 + 0 = 5 5 + 4 = 9 35 ≠ p 4 + 1 = 5 5 + 5 = 10 P = 37 Primes distribution and formula for finding π(x) For the calculation of primes in (1 ÷ x) there are functions of 2 kinds: arithmetic formulas and analytical formulas derived from the works of Riemann and Mangoldt. For example, the known formula non – trivial zeros functions. Since Riemann was a great specialist in the development of complex numbers and was oriented in an imaginary plane better than in real one, the learned people were at an impasse. However, the most accurate and simple formulas are rear. Since we have the distribution of the parameters primes & composite numbers , then the law of distribution of primes π(x) in interval will look like for (8) Example 4. Determine the number of primes, π(x) in the numerical interval, . , it is known that in the However in the same segment, according to the known function of the law of distribution of primes it is easy to see that the result of formula (8) obtained i s closer to the truth. And no matter how large the interval of the value x remains unchanged with deviation of ±1. An algorithm for finding primes p ≥ 5 in the interval [1 ÷ N) Since the set of primes P(≥ 5) ⊂ Θ ⊂ ℕ, it is obvious that the search for primes will go faster in the set Θ than in the set . The most natural way to remove composite numbers of set Θ in interval is by using the properties of closed elements with respect to the multiplication operationumbers Multiplying numbers of the forms by each element of the set Θ, where is easy and simple to achieve the goal. First, the natural numbers are entered line by line in the file Θ = PrmNub1(id. [N]), but in the place the numbers divisible by 2 or 3 are filled with an empty symbol “–”. Then, based on the RasPrm algorithm (Fig.) the elements of the file are deleted, those that are elementwise products of numbers are multiplied by the next (at the begining ) number and deleting for numbers with record numbers where (see Example 5 below). The window of the program that implements the RasPrm algorithm Each successive new element erects squared to avoid repetitions of multiplication operation, and then is multiplied by the . The removal process continues until . If the product of the numbers is greater than n, then performs the next element and the above procedure is repeated. Similarly, it is true for numbers of the form . The method described for the elimination of composite numbers from the set Θ for numbers of the form is easy to use and works more efficiently and faster than such well-known algorithms as the Eratosthenes, the Sundaram and the Atkin sieves. Since in all these algorithms the domain of functioning is the set , the method of deleting composite numbers from the set Θ by numbers of the form allows to obtain the same results as the above mentioned algorithms, but with a much smaller number of multiplication operations. Example 5. Find all prime numbers of the set Θ in the interval . With using numbers of the form , where we form the elements of the file 1. Deleting composite numbers from the file having the form . Let , then . As a new θ1, we find its square: and if then from the Θ by the record number by direct access, this entry is raised and the field value is deleted то . Then θ1 is multiply by the next number of the file Θ and also deleting for numbers with record numbers Since we take the next step i. e. new element the record is deleted When the next step new element and terminates, the process for numbers . 2. Deleting composite numbers from the file Θ having the form . Let , then as a new element θ1, we find its square and it is removed in the same way as in the previous examples. That is, by directly accessing the record the following record numbers are also selected and the numbers are deleted. Since next step Representation of even numbers in the form of sums of 2 elements from the Θ Pair for numbers of forms are called the corresponding pair of parameters of an even number if Lemma 3. The even numbers are comparable with ≡ 0, 2, −2 (mod 6). Evidence. From the type of even numbers, , i. e. the number n has the following form , where ν ∈ ℕ and it is obvious that the residues . Suppose: 1. that is, is divisible by 6 with the remainder 0. 2. that is, is divided by 6 with the remainder 2. 3. or i. e divisible 6 with remain –2. Lemma 4. Any even number is representable by the sums of 2 elements of set Θ. Proof. Types of decomposition of even number comparable with the ≡ (0, 2, −2) (mod 6): 1. Let us have even numbers of the form and let then, if we add and subtract 1 (one), we have an even number ζ = 6λ1 + 6λ2 = (6λ1 + 1) + (6λ2 − 1) = θ1+ + θ2−. 2. Let us have even numbers of the form and let therefore, the even number will be . 3. Let even numbers of the form be and we have where the elements and lie in the set Θ. Since the parameters of twins composite numbers (7) lie on the intersections of the function (5), th en the number of their parameters on the segment ] by the intersection of the sets will be less or equal than to their union the number of images of the functions themselves. Hence, on the segment for the parameters of the subsets: the following inequality is satisfied: (9) consider by analogy with the remainders decomposition of even numbers in the form of sums . To do this, for each type of even numbers of the form we add and subtract the numbers . α) ∀even number of the form ζ = 6ν + 0, where we add and remove the numbers of type have and in the same way for numbers of form then . We denote the parameters of the terms Hence, or (10) is also fulfilled the identity (11) Let the even number then when from link (10) → or . β) ∀ even number of the form ζ = 6ν + 2, where ν = (ζ - 2) / 6 have We denote the parameters of the terms respectively, , which means that even number of the form (12) also fulfills the identity (13) Let then , with we have from link (12) → . γ) ∀even number of the form ζ = 6ν − 2, where the parameter , we have We denote the parameters of the terms: hence species (14) also fulfills the identity (15) Suppose that for we have from link (14) → and . From the properties α, β, γ it follows that ∀ even number can be represented by sums , where and also if m is the remainder of even numbers divided by 6, then the parameter of the even number is found by the formula (16) The binary (strong) Goldbach – Euler problem The decomposition of even numbers into the sum of 2 primes is verified directly, so consider the solution of binary Goldbach problem for . We first introduce the definition and prove a number of lemmas.For example to find the corresponding parameters of the even numbers . Since, , then the parameter of even numbers Hence, 1. , Since, corresponding parameters . 2. then or and Since then the corresponding pair paramets Thus, even ζ = 30 has 2 corresponding parameters Lemma 5. ∀ even numbers in the segments there always exists parameters which belong to the union of the sets and are corresponding pair parameters of even numbers. Evidence. The parameters of twins of composite numbers are increasing in segment since the intersection of the functions (5) is increasing, then in the segments and naturally, they will be have different elements . Since, the natural numbers in the segment begin with the parameters of twins of primes (see Table 6 below or Table 5 in [6]) then all elements of the segment , obviously, can not be parameters of twins of composite numbers. Suppose that all elements of the segment are parameters of twins of composite numbers, then by (9) we have a contradiction, since the total number of elements of on the segm. can’t be greater than the number of elements Hence, the assumption is false, that is, in the segments and , the number of parameters If in segment the number of parameters of twins of composite numbers is equal to and in segment is , then the number of parameters in is and in Let , then it is obvious that . Let the parameters and assume that (to strengthen the assertion of Lemma 5) the corresponding pair parameters are elements of Similar reasoning on the parameters is found in segment . Then, the numbers for parameters in the segment or will be , where shows the number of corresponding pair parameters, one of which the parameter is of twins of composite numbers on the segment . Because the in the segments and the number of parameters of twins of composite numbers of is less than the number of elements , then naturally in and there is always a pair of numbers such and are the corresponding pairs of parameters of an even number . Lemma 6. In segm. the number of elements Evidence. From the contents of the procedure for constructing the (see Table 1) it is obvious that the ele ments of the sets and in each row give rise to 2 elements, that is, in the segment the number of elements . Hence, the following inequalities are true . Theorem 3. Any even number is decomposable into a sum of two primes. Evidence. Since, any even numbers of the form and are considered and following from (16) that the values of the parameters of the even numbers , for the remainders . Then, by analogy with the remainders m, the even numbers are represented by the sums , where and by the types of expansions of even numbers (Lemma 4) or by properties we have the following structures of the corresponding forms of the decomposition of even numbers : . (17) We investigate the elements of the segment for belonging to the sets , and find out for which values of the parameters and from the listed sets the numbers and are primes or for which composite numbers. For elements of the set they are not considered, since is composite numbers. However, by Lemma 5 in segment there are always and such that then ΠTw ∪ (FN+ ∆ FN−). Hence it remains to verify the elements , on belonging to the sets and FN+ ∆ FN− with the removed parameters of set in the segment we have: A. Let , , then the terms of the even numbers are primes, as the constituent elements of twins of primes Hence, according to the corresponding forms in (17) we have and infinity of twins of primes I will soon publish. B. & (λ2 or λ1 ϵ (FN+ ∆ FN−) so the term θ1 is prime numbers as constituent elements of twins of primes . And the 2nd term by Lemma 2 in one of the variants of the forms: is a prime numbers. Then adding to the prime numbers obtained a prime numbers (by analogy with the corresponding from (17), we get and infinity of twins of primes will soon be published. C. Let (λ1, λ2 (FN+ ∆ FN−), then in order that the term of an even numbers depending on the corresponding form defined in (17) and by Lemma 2 be a prime numbers, it is sufficient to establish whether the parameter belongs to one of the sets . For the term the parameter of which on the segment can obviously correspond to the parameters or again the elements from the set (FN + ∆ FN−). If then there are no problems, otherwise the following parameter is determined by the identities (11), (13), (15), respectively, by type of an even numbers , which is obviously an element of either ΠTw or (FN + ∆ FN−). But since by Lemma 6 the number of parameters in segments , then one of the items (A, B) is repeated in . Thus, the decomposition of an even number by the sum of 2 numbers where the numbers of form , whose parameters ( and in any of the listed variants (A, B, C) always lead to the forms and of primes and since the sum of the numbers is even numbers, either or , then the Goldbach – Euler binary problem is solved in any case positively. Example 6. Let an even number ζ = 360. Let us calculate the remainder m from dividing an even numbers ζ by 6. By Lemma 3 we find the type of an even number 6ν + 0, and by (17), respectively, we have the string (17, a). We establish the forms of the terms in the expansion of the even numbers we have numbers of the form , we calculate the even numbers’ parameter by the where the remainder whence . We write out the elements of the sets ПTwCN, ПTw, ПUPC from the corresponding tables on the segment [1 ÷ 60] = {20, 24, 31, 34, 36, 41, 48, 50, 54, 57} (see Table 5 below or Table 6 [3]). = N\FN = {1, 2, 3, 5, 7, 10, 12, 17, 18, 23, 25, 30, 32, 33, 38, 40, 45, 47, 52, 58} (see Table 6 below or Table 5 [3]). = {6, 11, 13, 16,*20, 21,*24, 26, 27,*31,*34, 35,*36, 37, *41, 46, *48, *50, 51,*54, 55, 56, *57} (Table 1). ={4, 8, 9, 14, 15, 19, *20, 22, *24, 28, 29, *31, *34, *36, 39, *41, 42, 43, 44, *48, 49, *50, 53, *54, 55, *57, 59 ,60}. Symbols in numbers according to belonging to sets . We decompose the elements of the corresponding pairs of parameters lieing in and taking into account their belonging to parameter sets, according to Λ1 ={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, −27, +28, +29, 30}. Λ2 ={*31, 32, 33, *34, −35, *36, −37, 38, +39, 40, *41, +42, +43, +44, 45, −46, 47, *48, +49, *50, −51, 52, +53, *54, −55, −56, *57, 58, +59, +60}. The numbers of corresponding pairs of parameters (λ1, λ2) lie in segment and, naturally, for the parameters of the numbers ПTwCN, the corresponding summands are composite numbers, but we are not interested in such. Then, it will be necessary to choose from Λ1, and Λ2 those parameters that belong to the union of the sets ПTw ∪ (FN+ ∆ FN−). For example, suppose the parameter values are: λ1 = 2 ∈ ПTw, λ2 = 58 ∈ ПTw 6λ1 – 1 = 11 ∈ PN −, 6λ 2 + 1 = 349 ∈ PN + 6λ 1 + 1 = 13 ∈ PN +, 6λ2 + 1 = 347 ∈ PN − λ1 = 25 ∈ ПTw, λ2 = 35 ∈ FN − 6λ1 – 1 = 149 ∈ PN −, 6λ 2 + 1 = 211 ∈ PN + 6λ1 + 1 = 151 ∈ PN +, 6λ2 – 1 = 209 ∈ CN − λ1 = 5 ∈ ПTw, λ2 = 55 ∈ FN + 6 λ1 –1 = 29 ∈ PN − 6λ2 + 1 = 331 ∈ CN+ 6λ1 + 1 = 31 ∈ PN +, 6λ2 – 1 = 329 ∈ PN – λ1 = 46 ∈ FN , λ2 = 14 ∈ FN + 6λ1 – 1 = 275 ∈ CN − 6λ2 + 1 = 85 ∈ CN + 6λ1 + 1 = 277 ∈ PN +, 6λ2 – 1 = 83 ∈ PN − λ1 = 39 ∈ FN+, λ2 = 21 ∈ FN − 6λ1 – 1 = 233 ∈ PN −, 6λ2 + 1 = 127 ∈ PN+ 6λ1 + 1 = 235 ∈ CN +, 6λ2 – 1 = 125 ∈ CN − Since the parameters on then the numbers of the form are primes as of twins of primes, i. e. (5–7, 11–13, 29–31, 137–139, 149–151) and the corresponding parameters in one of the variants of the forms or are prime numbers, by Lemma 2. Thus, even number 360 is decomposable into sums of primes 11 + 349, 13 + 347, 149 + 211, 29 + 331, 277 + 83, 233+ 127. Example 7. Let an even number ζ = 362. We compute the remainder m from dividing an even number ζ by 6, and by Lemma 3 we find the type of an even number 6ν + 2 → m = 2, according to (17), respectively, we have the string (17, b). We fix the forms of the terms in the expansion of an even number , we have numbers of the form . We calculate the even numbers’ parameter by the formula (16), where ν = 60. We write out the elements of the subsets ПTwCN, ПTw, ПUPC from the corresponding tables on the segment . It is obvious that the values of the parameters of the sets ПTw, ПTwCN, and remain unchanged, as in Example 6, but since the numbers here are of the same type, in order that θ1, θ2 are primes, it is necessary that the parameters were taken from the sets ПTw or FN− \ ПTwCN : λ1 = 23 ∈ ПTw λ 2 = 37 ∈ 6 λ1 + 1= 139 ∈ 6λ 2+ 1 = 223 ∈ λ1= 27 ∈ λ 2 = 33 ∈ ПTw: 6 λ1 +1 = 163 ∈ 6λ 2+ 1= 199 ∈ λ1= 6 ∈ λ 2 = 14 ∈ 6 λ1 + 1= 277 ∈ 6λ 2+ 1 = 85 ∈ λ1= 47 ∈ ПTw λ 2 = 13 ∈ 6 λ1 + 1= 283 ∈ 6λ 2+ 1 = 79 ∈ Thus, the number 362 is decomposable into sums of 2 primes: 139 + 223, 163 + 199, 283 + 79. Table 5 Parameters of twins of composite pairs (ΠTwCN) in set Θ 1 ÷ 10 000 020, 024, 031, 034, 036, 041, 048, 050, 054, 057, 069, 071,079, 086, 088, 089, 092, 097, 104, 106, 111, 116, 119, 130, 132, 134, 136, 139, 141, 145, 149, 150, 154, 160, 167, 171, 174, 176, 179, 180, 189, 190, 191, 193, 196, 201, 207, 209, 211, 212, 219, 222, 223, 224, 225, 226, 231, 232, 234, 236, 244, 246, 251, 253, 256, 265, 272, 274, 275, 279, 280, 281, 284, 286, 288, 294, 295, 299, 301, 303, 306, 307, 309, 314, 316, 320, 321, 323, 324, 326, 327, 328, 337, 339, 341, 343, 345, 349, 351, 353, 354, 358, 361, 362, 364, 365, 366, 371, 372, 376, 377, 384, 386, 387, 388, 394, 401, 405, 409, 414, 415, 416, 418, 419, 421, 427, 428, 429, 431, 433, 434, 438, 440, 442, 454, 456, 460, 462, 464, 468, 469, 471, 478, 479, 482, 487, 489, 491, 496, 497, 498, 499, 501, 505, 509, 512, 516, 517, 519, 521, 522, 524, 525, 526, 529, 533, 539, 540, 541, 544, 546, 547, 548, 549, 556, 559, 561, 563, 564, 566, 567, 570, 571, 573, 574, 579, 580, 581, 584, 587, 592, 594, 596, 598, 600, 608, 609, 611, 614, 619, 624, 625, 626, 629, 631, 635, 636, 638, 640, 643, 645, 649, 650, 656, 659, 660, 662, 663, 664, 666, 672, 673, 674, 677, 678, 681, 684, 686, 687, 691, 694, 695, 697, 698, 699, 701, 704, 706, 708, 711, 713, 717, 718, 719, 720, 722, 724, 728, 730, 731, 734, 736, 738, 739, 745, 746, 748, 750, 755, 756, 757, 759, 762, 763, 768, 769, 771, 772, 778, 781, 783, 785, 786, 790, 791, 794, 795,796, 801, 804, 806, 807, 808, 809, 811, 814, 816, 819, 821, 827, 830, 836, 838, 839, 841, 844, 845, 849, 854, 855, 856, 857, 860, 864, 867, 869, 870, 874, 875, 876, 878, 881, 882, 886, 888, 890, 893, 894, 895, 896, 904. 909, 910, 911, 915, 916, 919, 923, 924, 925, 931, 933, 934, 935, 936, 938, 939, 944, 946, 951, 954, 955, 959, 960, 961, 962, 966, 972, 979, 981, 982, 985, 986, 989, 991, 993, 994, 995, 996, 999, 1000, 1003, 1004, 1010, 1014, 1016, 1018, 1021, 1023, 1026, 1028,1030, 1031, 1032, 1039, 1040, 1042, 1047, 1049, 1051, 1058, 1064, 1067, 1068, 1069, 1072, 1073, 1074, 1076, 1077, 1081, 1083, 1084, 1085, 1086, 1089, 1090, 1093, 1098, 1099, 1102, 1104, 1105, 1107, 1108, 1111, 1114, 1116, 1119, 1121, 1124, 1125, 1126, 1128, 1129, 1131, 1133, 1135, 1136, 1141, 1142, 1146, 1148, 1149, 1154, 1155, 1156, 1157, 1159, 1168, 1172, 1175, 1177, 1179, 1181, 1182, 1183, 1186, 1189, 1190, 1191, 1194, 1195, 1197, 1200, 1204, 1210, 1211, 1212, 1213, 1215, 1217, 1219, 1221, 1223, 1224, 1226, 1227, 1229, 1230, 1231, 1233, 1234, 1237, 1238, 1240, 1241, 1244, 1245, 1249, 1252, 1259, 1261, 1266, 1269, 1271, 1272, 1276, 1277, 5192, 1285, 1289, 1291,1294, 1295, 1296, 1297, 1300, 1301, 1302, 1306, 1308, 1310, 1315, 1316, 1319, 1324, 1326, 1328, 1329, 1330, 1331, 1333, 1334, 1337, 1338, 1339, 1341, 1344, 1346, 1351, 1355, 1356, 1357, 1359, 1364, 1366, 1367, 1369, 1371, 1375, 1376, 1380, 1384, 1387, 1389, 1390, 1391, 1393, 1397, 1399, 1400, 1401, 1402, 1406, 1409, 1412, 1413, 1414, 1415, 1416, 1418, 1422, 1425, 1426, 1428, 1431,1432, 1434, 1436, 1439, 1442, 1443, 1454, 1461, 1462, 1465, 1466, 1469, 1471, 1474, 1476, 1470,1480, 1483, 1484, 1485, 1486, 1491, 1493, 1496, 1497, 1498, 1499, 1503, 1504, 1506, 1509, 1512, 1513, 1514, 1516, 1519, 1520, 1524, 1528, 1532, 1536, 1539, 1541, 1542, 1544, 1545, 1548, 1550, 1551, 1555, 1559, 1560, 1561, 1564, 1568, 1571, 1574, 1575, 1576, 1581, 1584, 1586, 1588, 1593,1594, 1595, 1596, 1597, 1599, 1601, 1606, 1609, 1611, 1612, 1614, 1617, 1618, 1619, 1621, 1626, 1627, 1629, 1633, 1637, 1641, 1644, 1646, 1649, 1652, 1653, 1656, 1659, 1660, 1663, 1664, 1665, 1666, 1667, 1669, 1670, 1671, 1672, 1674, 1675, 1676, 1679, 1681, 1686, 1687, 1688, 1691, 1698, 1700, 1701, 1703, 1705, 1706, 1713, 1714, 1716, 1718, 1721, 1725, 1727, 1729, 1730, 1731, 1734,1735, 1736, 1737, Table 6 Parameters of twins prime pairs (ΠTw) 1 ÷ 50 000 001, 002, 003, 005, 007, 010, 012, 017, 018, 023, 025, 030, 032, 033, 038, 040, 045, 047, 052, 058, 070, 072, 077, 087, 095, 100, 103, 107, 110, 135, 137, 138, 143, 147, 170, 172, 175, 177, 182, 192, 205, 213, 215, 217, 220, 238, 242, 247, 248, 268, 270, 278, 283, 287, 298, 312, 313, 322, 325, 333, 338, 347, 348, 352, 355, 357, 373, 378, 385, 390, 397, 425, 432, 443, 448, 452, 455, 465, 467, 495, 500, 520, 528, 542, 543, 550, 555, 560, 562, 565, 577, 578, 588, 590, 593, 597, 612, 628, 637, 642, 653, 655, 667, 670, 675, 682, 688, 693, 703, 705, 707, 710, 712, 723, 737, 747, 753, 758, 773, 775, 787, 798, 800, 822, 828, 835, 837, 850, 872, 880, 903, 907, 913, 917, 920, 940, 942, 943, 957, 975, 978, 980, 1015, 1022, 1033, 1045, 1050, 1060, 1075, 1092, 1095, 1110, 1115, 1117, 1127, 1130, 1132, 1138, 1145, 1158, 1160, 1188, 1202, 1218, 1222, 1225, 1243, 1248, 1258, 1260, 1265, 1293, 1313, 1325, 1335, 1348, 1370, 1372, 1382, 1398, 1405, 1423, 1433, 1438, 1470, 1473, 1477, 1495, 1500, 1502, 1507, 1540, 1547, 1557, 1570, 1572, 1573, 1577, 1605, 1613, 1620, 1628, 1643, 1655, 1668, 1673, 1678, 1682, 1690, 1712, 1717, 1722, 1738, 1743, 1750, 1755, 1785, 1810, 1815, 1823, 1843, 1845, 1853, 1860, 1862, 1892, 1915, 1925, 1950, 1953, 1963, 1972, 1990, 1995, 2007, 2012, 2018, 2027, 2040, 2042, 2063, 2090, 2102, 2137, 2153, 2167, 2168, 2203, 2223, 2233, 2280, 2282, 2285, 2287, 2293, 2305, 2313, 2317, 2322, 2333, 2335, 2347, 2375, 2387, 2398, 2408, 2425, 2427, 2432, 2438, 2478, 2523, 2545, 2548, 2555, 2560, 2597, 2607, 2608, 2622, 2623, 2648, 2662, 2677, 2678, 2690, 2698, 2705, 2727, 2742, 2772, 2775, 2782, 2805, 2817, 2830, 2838, 2865, 2882, 2898, 2903, 2915, 2930, 2933, 2943, 2947, 2958, 2965, 2973, 2985, 2987, 2993, 2998, 3007, 3008, 3010, 3020, 3022, 3042, 3048, 3052, 3087, 3090, 3152, 3153, 3180, 3190, 3197, 3202, 3230, 3237, 3238, 3245, 325, 3283, 3292, 3307, 3315, 3332, 3337, 3358, 3372, 3393, 3407, 3413, 3418, 3425, 3440, 3453, 3458, 3462, 3468, 3483, 3497, 3502, 3503, 3510, 3532, 3553, 3563, 3582, 3587, 3593, 3598, 3600, 3602, 3608, 3623, 3640, 3673, 3682, 3685, 3693, 3712, 3713, 3728, 3747, 3757, 3762, 3770, 3773, 3783, 3790, 3810, 3827, 3838, 3840, 3843, 3867, 3882, 3895, 3923, 3927, 3938, 3945, 3948, 3957, 3972, 3985, 4018, 4030, 4062, 4070, 4153, 4163, 4172, 4195, 4217, 4218, 4235, 4245, 4263, 4267, 4300, 4308, 4322, 4333, 4352, 4375, 4377, 4447, 4450, 4452, 4455, 4477, 4480, 4482, 4492, 4510, 4518, 4540, 4547, 4568, 4580, 4588, 4590, 4597, 4615, 4623, 4625, 4632, 4653, 4657, 4683, 4685, 4697, 4713, 4718, 4725, 4735, 4758, 762, 4770, 4777, 4792, 4837, 4855, 4868, 4898, 4900, 4928, 4945, 4960, 4980, 5002, 5015, 5023, 5045, 5065,5078, 5082, 5093, 5140, 5142, 5145, 5180, 5187, 5192, 5197, 5208, 5220, 5232, 5252, 5257, 5287, 5288,5295, 5308, 5338, 5343, 5353, 5357, 5365, 5383, 5387, 5395, 5402, 5407, 5422, 5427, 5435, 5453, 5467, 5472, 5485, 5490, 5495, 5512, 5525, 5530, 5548, 5555, 5558, 5598, 5600, 5603, 5625, 5628, 5635, 5638, 5672, 5688, 5693, 5702, 5710, 5717, 5728, 5745, 5750, 5752, 5765, 5775, 5793, 580, 5808, 5827, 5842, 5847, 5880, 5908, 5918, 5922, 5932, 5955, 5967, 5973, 5983, 6002, 6018, 6057, 6078, 6088, 6130, 6132, 6150, 6155, 6170, 6200, 6218, 6223, 6227, 6258, 6262, 6265, 6282, 6297, 6302, 6332, 6373, 6388, 6408, 6410, 6428, 6435, 6442, 6445, 6452, 6458, 6487, 6507, 6527, 6538, 6540, 6557, 6562, 6585, 6638, 6640, 6673, 6688, 6692, 6738, 6755, 6773, 6783, 6808, 6857, 6863, 6867, 6872, 6898, 6902, 6920, 6935, 6960, 6975, 6993, 6997, 7003, 7012, 7030, 7037, 7047, 7068, 7077, 7095, 7107, 7117, 7140, 7150, 7175, 7220, 7233, 7257, 7263, 7268, 7275, 7297, 7298, 7315, 7327, 7338, 7348, 7355, 7367, 7378, 7380, 7397, 7422, 7437, 7450, 7462, 7520, 7523, 7530, 7553, 7557, 7598, 7637, 7675 ,7682, 7697, 7712, 7718, 7725, 7740, 7765, 7780, 7795, 7803, 7805, 7843, 7858, 7892, 7898, 7903, 7943, 7950, 7952, 7957, 7963, 7968, 8020, 8052, 8068, 8080, 8090, 8108, 8113, 8122, 8130, 8137, 8143, 8145, 8165, 8172, 8187, 8195, 8200, 8213, 8222, 8228, 8232, 8235, 8255, 8258, 8278, 8290, 8298, 8320, 8323, 8332, .... Example 8. Let an even number We compute the remainder m from dividing an even number ζ by 6 and by Lemma 3, we find the type of even numbers 6ν – 2 → m = −2 and by (17), respectively, we have the string (17, c). We fix the forms of the terms in the expansion of an even number , we have numbers of the form and . We calculate the even numbers’ parameter by the formula (16), whence . Then the values of the parameters of the sets ПTw, ПTwCN, and remain almost unchanged, as in Example 6, only the parameters 61 are added, that is, . Since here are again the numbers of one kind, then, in order for numbers of the form terms θ1, θ2 to be primes obviously by Lemma 3 the parameters must belong to one of the sets: ПTw or FN− \ ПTwCN: λ2 = 2 ∈ ПTW λ2 – 1 = 59 ∈ FN+ 6λ2 – 1 = 11 ∈ PN− 6λ2 – 1 = 353 ∈ PN− λ2 = 19 ∈ ПTW λ1 – 1 = 42 ∈ FN+ 6λ1 – 1 = 113 ∈ PN− 6λ1 – 1 = 25 ∈ PN− λ2 = 46 ∈ FN– λ2 – 1 = 15 ∈ FN+ 6 λ1 – 1 = 275 ∈ CN − 6λ2 – 1 = 89 ∈ PN− Thus, the number 362 is decomposable into sums of 2 primes: 11 + 353, 113 + 251. In Examples 7 and 8 we have the same type of terms θ1, and θ2. And since inequality (17) is always feasible on Λ1 and Λ2, the elements of the sets ПTw, are always present in . Therefore, the validity of the Goldbach – Euler binary problem as a whole remains valid for all even numbers and It is proved that any even numbers decomposes into a sum of 2 primes, & since the expansion of even numbers ζ ≤ 8 is verified directly, the binary Goldbach – Euler problem for even numbers is proved. Conclusion A comprehensive study of the problem of finding and distributing primes and composite numbers, twins of primes, twins of composite numbers, including a theoretical study by means of computer software made it possible to obtain great results, to solve the secrets of the structures of subsets of the set of natural numbers and to give the definitions, to obtain a new algorithm for finding and distributing primes P (≥5), to fulfil the calculation of the exact number of primes and to obtain the method of distribution primes by their ordinal numbers n and vice versa in the set of primes. The method of their distribution law in interval [ ) has been found. Any even number has the form , where ν = (ζ − m) / 6 is an even parameter. Any even number can be represented by the sums of 2 elements respectively, over the remainders m . For any even numbers on the segment there always exists a pair of numbers and such that both ∈ ΠTw ∪ (FN + ∆FN−).
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