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Drushinin Victor Vladimirovich, Lazarev Alexey Alexandrovich, National research nuclear University "MEPHI”, Sarov physical-technical Institute Sarov, E-mail: vvdr@newmail.ru
The theory of arbitrary pairs of primes
Abstract: For generating pairs ofprimes with an arbitrary distance between them we created the sieve-type sieve of Eratosthenes.We identified buffer zones and showed an infinite amount such pairs on the real axis.
Keywords: sieve of Eratosthenes, the Euler function, probability.
If we consider the various associations of primes {pi}
, analysis of the properties of prime numbers p e P are simplified: twins {p;pi+1 = p + 2}; a pair of four {; Pi+i = Pi + 4}; a pair of sixes {; p+i = p, + 6}; quartets ofprimes — two pairs of twins separated by a composite number, and other formations. A recently published article I. Chan [1] about a pairs of primes, separated by a set of 70000000 composite numbers, is called a breakthrough in the solution of several problems in the theory of numbers. We have created a sieve to generate pairs of twins, pair of fours, pairs of sixes and quartets, proved endless number their on the real axis [2, 3, 4, 5]. In this paper, we carried out a generalization these results and received the sieve-type sieve of Eratosthenes to generate a pair-2N (P2N) of primes and proved an existence of an infinite number such pairs on the real axis.
P2N mean the formation of (N +1) consecutive odd numbers, the extreme ofthem are primes {p ; Pi+i = Pi + 2 N}, and (N -1) numbers standing between them are composite. For example, P10 — {241;243;245;247;249;251}, where primes are shown in bold. In the theory of primes it is showed that the distance between two successive primes can be arbitrarily large. For example, if we take the set of odd numbers {n!+ (2k +l), 1 < k < [(n -1) ) 2], where [x] denotes the largest integer not exceeding x, we get a set of consecutive composite numbers
[( -1) / 2]. If we expand this set from left and right side, we will encounter the primes and thus we formed P2N. It turns out that one observed P2N generates on a real axis arithmetic sequence (AS) centers such numerical units. Some of them are P2N. We distinguish at least two reasons for the study P2N. Firstly, we can get search arbitrarily large primes by reproduction of such pairs. Secondly, P2N abruptly increases the size of the first buffer zone. If we remove all the composite numbers of multiples (p = 3;p2 = 5;p3 = 7;...;pn) ,only primes remain on segment of the real axis [3; Dn ]. If the last prime pn includes the left side in P2N, then the next prime number is pn+1 = pn + 2N. Where Dn = ( + 4Npn + 4N2 - 2). For example, we took pn = p5l = 239, then the next p52 = 241. These two primes form a pair of twins, according to our classification, N = 1. D51 = 58079. If we cleared by the sieve of Eratosthenes to pn = p52 = 241, we find that the following prime p53 = 251 , i. e., N = 5. The last two primes form P10, so D52 = 62999. After increasing the last tested prime on to “2”, we have expanded the zone of guaranteed prime numbers by almost 10%.
In order to reproduce P2N, it is necessary to find one such couple and distinguish its center
MN = p, +2[N/2] + {(-fT +1}/2. (1)
In this case, if N is even, MN — the number is odd and vice versa, p = MN - N, pt+1 = MN + N. For example, a
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The theory of arbitrary pairs of primes
pair of eights P8 {359; 361;363;365;367} has M4 = 363, and a couple-dozen P10 {181; 183; 185; 187; 189; 191} hascentre M 5 = 186. Prepare AS in the next form
MNi (k) = MN + TNi ■ k, (2)
where (k +1) e N, and the difference TNi is the multiplication of the primes that are included in the composite numbers of internal P2N, except or include the center, multiplied by “2”. Moreover, we take one at a prime from composite numbers without repeating. There are a few differences. In the example P8: 361 = 192,365 = 5 • 73, there are two possible ctlifE'erte nce (41 = 19 • 2 • 5 = 190, T42 = 19 • 2 • 73 = = 2774. Take one TNi and fix a set of primes. All other primes in ascending order remove bad k in (2). They are bad because of MN(k)=MN + TNi -к ±N one or both are a composite numbers, i. e. there is no P2N. Within the buffer zone, the pair not removed k is a good index and they create a new P2N. This type of sieve of Eratosthenes is illustrated by the example of P8. We write AP {363 +190 • k ± 4}. In “190” include {;19} therefore, we will check all remaining primes. Solving Diophantine equations:
1. 359 + 190k i3,k = 1 + 3(t -1), t e N.
2. 367 + 190k i3,k = 2 + 3(t -1).
3. 359 + 190k : 7,k = 5 + 7 (t -1).
4. 367 + 190k ':7,k = 4 + 7 (t -1).
5. 359 + 190k': 11,k = 1 + 5(t -1).
6. 367 + 190k': 11,k = 6 + 3(t -1).
And so on. Removing bad k on the interval [1,15], we find only two not deleted k = {3;15}. They give two new pair of eights primes {929; 937} and {3209; 3217}.
We made the program for calculating pairs of arbitrary order, which fully confirmed the correctness of the proposed algorithm. Here are some results. For P8 by the formula (93 + 210k ± 4) good k are (7;9; 12; 16;20;30;31;33; 34;41;45;48;50;58;59;60;61;...).
For P10 by the formula (186 + 330k ± 5) good indexes k are (3;17;18;20;22;28;37;21;51;52;56;79;88;93;95;100;...; ...; 1998;...).
For P12 by the formula (205 + 2310k ± 6) good indexes k are (7;Г;11;16;19;2Т;26;2Г;31;45;52;55;6Г;7Г;Г5;9Т; 94;...; 19Г2;...).
Discuss the question about the buffer zone in the calculation of index pairs. If we work according to the formula (MN + TNi ■ k ± N) andwant to know whether the number A is the right side of P2N, we need to check on Diophantine equations all primes up to kA <s[ä . Thus all indices not deleted by this set on the interval [1, kA ] is a good index k . The number of pairs of primes for arbitrary value on the number line is infinite. We proved in the previous article [2 - 5] that the method is true for arbitrary pairs.
The authors thank to Professor, doctor physical-mathematical sciences Shevyahov N. S. for interest in the work and valuable comments.
References:
1. Yitang Zhang. Annals of Mathematics, 1014, v.179, 3, P. 1121-1174.
2. Druzhinin V. V. NTVP, 2014, № 1, P. 22-25.
3. Druzhinin V. V., Lazarev A. A., Sirotkina A. G. Life Science Journal, 2014, v. 11 (10s), P. 346-348.
4. Druzhinin V. V., Lazarev A. A. NTVP, 2014, № 4, P. 19-21.
5. Druzhinin V. V., Lazarev A. A. NTVP, 2015, № 1, P. 17-19.
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