CALCULATING THE NUMBER OF TWIN PRIMES WITH SPECIFIED DISTANCE BETWEEN THEM BASED ON THE SIMPLEST PROBABILISTIC MODEL

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1 U.P.B. Sci. Bull. Seies A, Vol. 80, Iss.3, 018 ISSN CALCULATING THE NUMBER OF TWIN PRIMES WITH SPECIFIED DISTANCE BETWEEN THEM BASED ON THE SIMPLEST PROBABILISTIC MODEL Sasengali ABDYMANAPOV 1, Seik ALTYNBEK, Alua TURGINBAYEVA 3 Building moden cyptosystems equies vey big pimes. This eseach studies the chaacteization of pimes, the poblem of detecting and assessing the numbe of pimes in a numeical seies and povides othe esults egading the distibution of pimes in a natual seies. It should be noted that the esults wee obtained within the famewok of the simplest pobabilistic model. This eseach veifies the pospects of obtaining an estimate fo the numbe of -pais based on some statistical models of thei fomation. Keywods: composite numbes; divisibility citeion, pimes; sequence of numbes; twin pimes, cyptogaphy. 1. Intoduction Pimes ae involved in moden cyptogaphy the science of secet message coding and decoding. Since many cyptogaphic systems ely on the eseach of pimes and elated theoetical and numeical poblems, technological and algoithmic advances in this field have become paamount. Ou ability to find big pimes and pove that they ae pime pedetemined ou ability to influence the situation that makes cyptogaphes comfotable. Based on estimate made fo the numbe of twin pimes on a paticula inteval, one can build a pseudoandom numbe geneato o use distibution of pimes in cyptogaphy. The latte will be the subject of ou eseach calculating the numbe of pimes on a specified inteval and an attempt to solve the poblem of pobabilistic distibution of twin pimes. Numeous publications have been devoted to methods fo synthesizing the PRBS geneatos and analyzing thei chaacteistics [1-]. In [3], thee ae outlined pinciples of system appoach to cetificating PR numbe geneatos in infomation secuity systems. Appaently, PRBS geneatos built based on shiftes with linea feedbacks ae the most widespead geneatos. Such methods ae widely studied and calculated. Thus, we shall have an altenative method fo inceasing cyptogaphic stability. 1 Kazakh Univesity of Economics, Finance and Intenational Tade, Kazakhstan, ecto@kuef.kz Kazakh Univesity of Economics, Finance and Intenational Tade, Kazakhstan, seik.altynb@gmail.com, Seik_AA@bk.u 3 Euasian National Univesity by named L.N. Gumileva, Kazakhstan, tasheat@mail.u

2 184 Sasengali Abdymanapov, Seik Altynbek, Alua Tuginbayeva Specific instances of the pime numbe theoem wee fomulated and poved by Cal Fiedich Gauss and Benhad Riemann [4]. Existing methods equie n steps O(n e ) fo abitay whole numbes, whee e is a cetain faction, such as 1/3 o 1/4. Thus, expeimental esults show that each of these methods faces cetain untestable numbes at about n = It is easy to show that if the matte at hand is the distibution of pimes in aithmetic pogessions, thee exist simila appoaches to specific cases, such as the zeta function in the simila genealized Riemann hypothesis [5]. The local density of pimes in a natual seies appeas to be essentially andom. Howeve, the following theoem establishes the asymptotic density of pimes. The known pime numbe theoem gives an asymptotic assessment of the density of pimes that ae less than o equal to a positive eal numbe. Let s conside a case of twin pimes: two pimes that diffe by. One could find such pais easily: 11, 13 o 197, 199. Relatively bigge-pais ae not that easy to find, but they can still be found. The biggest ones ae as follows: * ± 1, pai * 3900 ± 1, pais of double numbes * ± 1, and * 8005 ± 1, and * ± 1. Is it possible to pedict asymptotically how many such pais exist within set boundaies? Let us ty to think heuistically. The pobability of a andom numbe in the neighbohood of x being a pime numbe was about 1/ln x. Thus, the x dt hypothesis emeged that π (x)~c. ln(t) Using the expession fo the pobability density of pimes in the natual seies, Davenpot (1967), suggested that the numbe π (x) of twin pimes smalle than x is asymptotically equal to x dt ( x) C, C (1) ln( t) and, theefoe, infinite. Recently eceived a fundamentally new esult in this diection (see Yitang, 014), which assets the existence of an infinite numbe of Pimes such that the distance between each pai does not exceed a cetain numbe (70 million). This obviously means that at least one set of pais with distance less than 70 million is infinite. Subsequently, Zh. Yitang s method [6] and close methods have shown that infinite, at least one set of pais with distance, is less than 46 [7]. The poposed wok involves veification of the possibility to obtain an estimate of the numbe of -pais on the basis of some statistical models of thei fomation, the simplest pobabilistic models of the empiical fomula of the numbe of -pais, including the case of = is given and numeical compaison with eal data povided.

3 The estimate of the numbe of twin pimes with a given distance [ ] pobabilistic model 185. Basic definitions Place the numbe of positive integes in a matix B. The fist d seies membes wite down - fist column of the matix B. Second d seies membes wite similaly, as the second column, etc. the esult is a matix of size (d, ). In each ow of this matix S a, thee is an aithmetic pogession with diffeence d and with the appopiate initial membe of the a-element of the fist column of the matix (1 a (close to [8]). If the numbes a and d ae elatively Pime, we will call such sting a given sting (the given sequence), since this line coesponds to a given deduction. Conside the submatix B(d,x) of matix B containing only the fist x+1 columns. It fully contains the segment of the natual numbe fom one to xd. We denote, as is customay, the numbe of pimes in this inteval using π(x. Accoding to the theoem of Diichlet, in each given ow of the matix B contains infinitely many pimes 1. Moeove, fo any x in each given ow of the matix B(d,x) and the Diichlet theoem, the numbe π(d,x) of Pimes is the same and popotional to the numbe of such lines: ( d, x) ( x / ( () whee, φ( the Eule function [9]. If the diffeence between two numbes is equal to < d, they ae in diffeent ows of matix B. Let a 1 + nd, a + md two numbes (a 1 a, n m) and ( a a ) + ( n m) d =. (3) 1 If a 1 a > 0, then (3) n-m=0 numbes with a diffeence < d lie in two lines defined by the equation a 1 a = and in the same column. If a 1 a < 0, then (3) n-m=1 and d (a 1 a ) =. In this case, the numbe of the specified distance lie in the lines a 1 and a, but in adjacent columns. Let s place the numbes fom the matix B columns on a cicle and define its diection fom smalle values to bigge ones. Then any two numbes a 1 and a ae the two distances: fom a 1 to a and fom a to a 1. A necessay condition fo the existence of infinitely many twin pimes with a distance within a pai is the existence of any matix B fo < d thee ae at least two given lines with the distance between them. Indeed, fo any value of d in the matix B contains the whole ow of natual numbes. If thee is a matix B whee all ows of c pimes ae at a distance diffeent fom, then the distance between a pai of Pimes is not implemented eve. Howeve, obtained esult shows that matix B does not exist as well [10] any specified matix B ows ae at even distances smalle than d. Thus, necessay condition fo the existence of twin pimes with a distance smalle than d fo any d is met.

4 186 Sasengali Abdymanapov, Seik Altynbek, Alua Tuginbayeva 3. Pobability model In the famewok of pesenting natual seies as matix B, we can conside the emegence of twin pimes with the given distance as a andom event. Namely, let s conside the matix B=B( and let the numbe of pimes n 1 and n be on a cetain segment of a sequence fom υ to υ in two specified ows with the distance between them. Let s conside the simplest pobabilistic model, which is that the pobability of value being a pime numbe does not depend on the numbes in the sequence and is the same fo all positions occupied by odd numbes in this sequence. Based on this assumption (see Appendix), the expected numbe of cases when the Pimes of both sequences (ows of matix B) ae clealy elated to each othe positions in these sequences is equal to ~ n1n (( ), = (4) n whee n an odd numbe of membes of the pogession fom numbe υ to numbe υ. If d is an even numbe, then n = (υ υ )/d if d is odd, then n = (υ υ )/d. Thus, the numbe of -twin pimes can be estimated by the atio ( i, = 1 1 ( ) = ( j, = 1 nin j (5) n dist( i, j) = whee d - the diffeence between pogession, n i - the numbe of Pimes in ow numbe i in the coesponding segment of the sequence, the summation is fo all ows i, j of matix B, with numbes co-pime with d ((i,=1, (j,=1) and with distance between them equal to (dist(i,j)=). In paticula, to estimate the numbe of pais with diffeence not exceeding a given 0 you can use deived fom (5) atio of ( i, = 1 1 ( ) = ( j, = 1 nin j (6) 0 0 n dist( i, j) = Estimations of the numbe of pais by the fomulas (5, 6) ae essentially compaable to a diect calculation of this quantity. In the wok of these assessments ae used to analyze the adequacy of the accepted model. Calculating the numbe of Pimes n 1 xd and n xd by the fomula () and using the known atio of π(x)~ x we obtain the estimate Π ln(x) (x, be the numbe of -pais in the inteval of the natual numbes fom 1 to xd: d( )( ( x) x ( x, = ( ) ( x = C (7) n( ( ) ln( x

5 The estimate of the numbe of twin pimes with a given distance [ ] pobabilistic model 187 d ( ) Whee: C = ( ( ) at even d and d( ) C = at odd d. (8) ( ( ) Whee: μ() the numbe of pais given stings with a distance in the matix B. The elation (7) is essentially Bun's theoem [8] howeve, the coefficient C (7) can be selected in a unique way, depending on the diffeence d pogession. Ratio (8) can be simplified as follows. In computed the quantity μ() equal distances fo all possible pais of given stings [10]. p ( ) = ( (9) d p 1 Whee: φ( Eule's function, p is the Pime numbe "p " means that p is not a diviso of, and "p d" means that p is a diviso of d. If we apply μ() into p p d d d p 1 d p 1 (8), we will get: C = at even d and C = at odd d. ( ( (10) Thus, fomulas (7, 10) give efficiently computable estimate of the numbe of -pais in the inteval of the natual numbes, stating with unit 4. This assessment has a paamete diffeence between d, which detemines the coefficient C. In d( d ) paticula, if d is a pime numbe, then (10) we will be as follows C = ( d 1), and this atio does not depend on. If d is the poduct of m consecutive pimes p 1, p,, p m, stating with two, then (10) will be as follows at = (the case of twin pi d im m pi 1 i= pi ( pi ) pimes): C = = (11) m ( i= ( pi 1) This facto asymptotically m is close to the value C in the fomula (1) (Twin Pime Constant). As with any value of d the set of simple pais of twins is the same, thee is a possibility to impove the assessment of the choice of d paamete. Anothe numbe of pais with distance smalle than 0 easily follows fom x the pevious one: (, ) = ( ) 0 x d C 0 (1) ln( x At d=p fom (10) we obtain fo the case 0 < p: p x (, ) ( 0 ) 0 x d = p ( p 1) ln( x (13) ) 4 Obviously, a simila fomula can be obtained fo any segment of the natual numbes, if we use the coesponding estimate of the numbe of Pimes.

6 188 Sasengali Abdymanapov, Seik Altynbek, Alua Tuginbayeva 4. Numeical expeiments Let s use the fomula (6) to compute the total numbe of pais fo the fist 5 even values of distances =,4,...,50 and even values of d anging fom 100 to 400. We have calculated this value twice (a) fo membes in aithmetic pogessions, stating with the fist (υ = 1, υ = 10000) and (b) fo 100,000 membes of the pogession, stating with 10 5 ( υ = 10 5, υ = 10 6 ). In addition, let us compute the same value by the fomula (1) fo case (a). Figue 1 shows the elationship of the eal numbe of -pais with the calculated fomula fo descibed cases. The aveage values of elationships computed diectly by (6) is close to the unit element. In all cases, the diffeence is close to 5%. The elationship magnitude depends on the d value. Cuve calculated accoding to the fomula (1) also deviates fom unity by no moe than 5%, although it has a diffeent shape. Fig. 1. Relationship between the actual numbe of -pais to the numbe calculated accoding to the fomulas (6, 1). In Y-diection, thee ae elationship values. The points maked with an asteisk and a zeo ae calculated by (6) fo pogession fagments in the ange fom 1 to membes and fom to membes, espectively. Points maked with plus ae calculated by (1) fo pogession fagments in the ange fom 1 to membes, as well as the points maked with an asteisk. Calculations with the same paametes, but just fo twin pimes, ae shown in Figue. The value of elationships between the actual numbes calculated by (5) fo two fagments of the sequence is lagely the same. It is seen that fo the fomula (5) (denote the cicle and asteisk), the elations in the anges , sometimes eaching Thee ae two goups of values fo calculating by fomula (7): fom 1 to 1.05 and fom 0.8 to 0.85.

7 The estimate of the numbe of twin pimes with a given distance [ ] pobabilistic model 189 Fig.. Relationship between the actual numbe of -pais (twin pimes) and numbes calculated accoding to equations (5,7,). In X-diection, value d vaies fom 100 to 400 with a step of. In Y-diection, thee ae the aveage values of elations. The points maked with stas and cicles calculated by the fomula (5) fo fagments of the pogessions fom 1 to 10,000 membes and fom 100,000 to 100,000 membes, espectively. Points maked with a plus sign - the same piece ate, which is used to calculate the stas, but ae calculated accoding to the fomula (7). Diffeent values of pogession d diffeence povide diffeent estimates fo the numbe of twin pimes. We have tested d = i seies as a candidate fo this optimal value, whee i vaies fom 1 to 66. Figue 3 shows the esults of calculations made by (5, 7). Calculations made by the model fomula (5) have % -3% eo while those made by the fomula (7) up to 10% eo. Fig. 3. Relationship between the actual numbe of twin pimes and those calculated accoding to equations (5,7) when the value is equal to d = i, whee i vaies fom 1 to 66. In X-diection, thee is the d value; in Y-diection elationship values. Squaes fo fomula (5), cicles fo (7). We now compae the numeical standad of a hypothetical estimate of (1) a vaiety of simple pais of twins π (x) with the estimate (7) on the set of matices B=(d,x) with values of d in the sequence of numbes of the fom 10i, whee i uns though the inteval fom 1 to 5000, x= The atio of the numbe of pais of twins, calculated by the fomula (7) to the numbe calculated by the fomula (1) shown in Figue 4. Although this atio is not highe than 0.95, it is essential that the schedule clealy has a hoizontal asymptote, i.e., misalignment estimates could be coected by multiplying ou estimate of the empiical constant.

8 190 Sasengali Abdymanapov, Seik Altynbek, Alua Tuginbayeva Fig. 4. The atio of the numbe of twin pimes, calculated by the fomula (7) to the numbe calculated by the fomula (1) when the value of d = i, whee i vaies fom 1 to In X-diection, thee is the d value; in Y-diection elationship values. 5. Conclusion The esults of numeical expeiments show that the estimate of the numbe of numbes-twins in the simplest pobabilistic model coelates with the actual numbe, howeve, the absolute eo can be quite lage. It is shown that fo diffeent values of d, namely fo diffeent ways to split the natual numbes in aithmetic pogession. We have eceived effectively computable estimate of the numbe of - pais on the inteval of natual numbes. At cetain paamete values, it asymptotically coincides with the known. Acknowledgment The pesent wok was caied out on the basis of the scientific and technical pogam No. BR of the Science Committee of the Ministy of Education and Science of the Republic of Kazakhstan. R E F E R E N C E S [1] Y.S. Khain, V.I. Benik, G.V. Matveev, S.V. Agievich, Mathematical and Compute Foundations of Cyptology, in Minsk: New Knowledge, 003, pp. 38. [] B. Schneie, Applied cyptology: Potocols, algoithms, souce texts in SI language, in Moscow: Tiumph, 003, pp. 816 [3] A.V. Potiy, A.K. Pesteev, Pinciples of the system appoach to cetificating PR numbe geneatos in infomation secuity systems, in Radio engineeing, vol. 104, 1997, pp [4] C. F. Gauss, Disquisitiones aithmeticae, in Yale Univesity Pess, [5] K. Soundaaajan, The distibution of pimes. Equidistibution in numbe theoy, an Intoduction, in Spinge Nethelands, 007, pp [6] Zh. Yitang, Bounded gaps between pimes, in Annals of Mathematics, vol. 179, 014, pp. 3. [7] A. Ganville, About the cove: a new mathematical celebity, in Bulletin of the Ameican Mathematical Society, vol. 5, no., 015, pp [8] S. Baibekov, A. Dumagambetov, An infinite numbe Twin pime, in Geneal Mathematics, vol. 19, 016. [9] H. Davenpot, Multiplicative Numbe Theoy, in Spinge-Velag, Belin, [10] M. Oazov, The lowe bound fo a numbe of epesentations in additive numbe theoy poblems, in Moden Tends in Engineeing Sciences: Poceedings of Intenational Scientific Confeence, Ufa, Leto, 011, pp [11] V. Bun, On Goldbach's law and the numbe of pime pais, in Achiv fo Mathematik og Natuvidenskab, vol. 34, no. 8, 1915, pp