The Prime Numbers Hidden Symmetric Structure and its Relation to the Twin Prime Infinitude and an Improved Prime Number Theorem.

Transcription

1 The Prime Numbers Hidde Symmetric Structure ad its Relati t the Twi Prime Ifiitude ad a Imprved Prime Number Therem. Imre Mikss Uiversidad Simó Blívar, Dep. de Frmació Itegral y Ciecias Básicas, Valle de Sarteejas, Baruta, Caracas, Veezuela. mikss@htmail.cm Due t the sievig prcess represeted by a Secdary Sievig Map; durig the geerati f the prime umbers, gemetric structures with defiite symmetries are frmed which becme evidet thrugh their gemetrical represetatis. The study f these structures allws the develpmet f a cstructive prime geeratig frmula. This defies a mea prime desity yieldig a secd rder recursive ad discrete prime prducig frmula ad a secd rder differetial equati whse slutis prduce a imprved Prime Number Therem. Applyig these results t twi prime pairs is pssible t geerate a Twi Prime Number Therem ad imprtat cclusis abut the ifiitude f the twi primes. Itrducti Mst f the kwledge abut the sequece f umbers amed primes is a set f uprved therems ad cjectures 1. The reas f this fact remais as elusive as the very prfs. The apprach f this wrk prpses a ew ad heuristic way f treatig these prblems. As it is well kw, sievig algrithms are the ly efficiet way t prduce primes. This fact shuld be take as a idicati that sievig is the atural way f prducig primes. A myth has bee geerated abut the sequece f primes, ad may attempts have bee udertake t fid sme prperties which shuld be itrisic t the sequece itself, despite its geeratig prcedure 2. Mst f them (perhaps all f them) are based the famus Euler frmula which relates the sequece f primes with the Zeta fucti 3. This frmula is thig else tha a aalytic represetati f the sievig prcess 4. Thrugh the cstructi f this frmula a limit is take, which elimiates the very heart f the prcess. Due t this limit, the relevace f the erased part has remaied hidde frm the scietific cmmuity thrugh ceturies. I the preset wrk the structure f this hidde part is made evidet thrugh the iterati f a Secdary Sievig Map. Thrugh the structure created by the cstructi f the primes, the reas why the twi primes shuld be ifiite becmes clear. This last is kw as the Twi Prime Cjecture 1, which is e f the uprved ics i the mder umber thery. Usig a mea prime desity derived frm the gemetric cstructi, a discrete secd rder equati is btaied as well as a ctiuus versi f it, which is a secd rder liear differetial equati. This is a first apprximati fr a prime differetial equati ad is used t demstrate cstructively ad t imprve the best kw apprximati t the primes, kw as the Prime Number Therem. Sievig as a Recursive Map Usually recursive maps act subsets f the real umbers, althugh sme f them are defied gemetrical bjects 5. A recursive map which acts ifiite ad discrete sequeces f umbers is prpsed here. This map called Secdary Sievig Map (SSM) is deted with the Greek letter b. Give:

2 2 { α β χ δ } η= η, η, η, η,... A ifiite ad discrete sequece f atural umbers which satisfies: ηα < ηβ < ηχ < ηδ <... < way: The b act this sequece h i the fllwig ( ) * β η = η η = η η η η* = { η η, η η, η η, η η... } β α β β β γ β δ β The mius sig meas elemet extracti. The temptati t factrize h shuld be avided because the resultig equati is t the rigial; h b is a umber whereas h is a set. I rder t make the last defiiti peratial, h ad h * shuld fulfill h û h *. The secd elemet f the rigial sequece h b, is called the pivt umber. The utcmes f applyig b is amed geerati. The atural umbers set (withut the cer) * = cmplies with all the features required t be a argumet f b. Applyig b recursively is similar, but t equal, t the Eratsthees sieve, because the last e is t applied a ifiite sequece ad it was t cceived as a iterative mappig. The SSM applied t, culd be writte i Mathematica as Nest[Cmplemet[#,#[[2]]*#]&,Rage[m1],m2], where m1 is the size f the atural umbers subset which it will be applied, ad m2 is the desired iteratis umber (it is impssible t act a ifiite set with a cmputer). Actig ce whit b prduces the first geerati, the set f dd umbers: β 1 ( ) = ( ) { } 2 = 1,3,5,7,... =2.+1 Usig this last tati, 0 = (geerati cer). Observe that the first pivt umber (the umber 2 which appears as a sub idex i paretheses) used t geerate de dd umbers, is t preset i them. Hwever the umber e des appear. This is a persistig characteristic i b s successive applicatis : the umber e always survives ad the pivt umber used t geerate the last iterati bviusly disappears frm it, because it was multiplied by e ad extracted. I the secd geerati, the mai features f these sequeces start t emerge: ( ) ( ) β ( ) = β ( β ( ) ) = β ( 2) = 2,3 = { 1, 5, 7,11,13,17,19, 23, 25, 29,... } = { 1, 7,13,19, 25,... } { 5,11,17, 23, 29,... } = ( ) ( ) Where U meas as usual the ui f tw sets ad the last equati is a cveiet way t write the existece f tw verlapped liear behavirs. Observe that the set f pivt umbers (which recrd is kept i s sub idex) starts t frm the set f prime umbers. Nte als that the sequeces (2, 3) 2 ad (2) 1 are qualitatively differet: a splittig has ccurred i the first e. Istead f e liear fucti f, there are tw verlapped ad simultaeus. I rder t uderstad this segregati, the symmetric features f the SSM shuld be examied thrugh a heuristic gemetrical represetati f it, shw i the ext secti. As a fial setece fr this secti, it shuld be ted that i the last decades sme isight has bee gaied abut a fractal structure f the primes 6, ad it is well kw i dyamical systems that fractal structures are prduced by iterative mappigs 5. The SSM actig culd be the basis fr the fractal structure f the primes. Mirrr Symmetry ad Peridicity i Geeratis, as a Gemetrical Image f a Multiple Liear Represetati The SSM ca be represeted gemetrically usig a ifiite chai f curves (jumps) which cects the sequece f umbers deted with h * (see equatis 1 ad 2). The -tuched umbers crrespds t the =

3 3 Figure 1 Geeratis ad Discrete Scale Ivariace Graphical Represetatis. I Part A the first three geeratis gemetric structure is shw thrugh a aalgy betwee extracted ad tuched umbers. The peridicity i each geerati ad their mirrr symmetries are evidet i the structures f the tuchig curves. I Part B the DSI f the Secd Geerati is shw. first geerati. Further iterative applicatis f b ca be made the same graph simply verlappig the gemetrical represetatis f the crrespdig h *. Fr example i Figure 1, the represetatis f three geeratis are see i separate lies. The tuched umbers uder b s acti, are just tuched ce whereas i the graphical represetati multiple tuchig is allwed. I this way the gemetrical features are better bserved. Frm these drawigs ca be iferred that due t the successive applicatis f the SSM, peridic structures frm sptaeusly. The perid f a particular geerati is give by the multiplicati f all the previus pivt umbers (the frmal demstrati will be published elsewhere). Actually these structures are peridic bth i the geeratis but als i the superpsiti f the h * prduced ad extracted durig each geerati. I fact this last superpsiti is the mst trius i the gemetrical represetatis frm Figure 1, Part A. As it ca be bserved i the same figure, the splittig metied i the past secti is caused by the icmmesurability f the first pivt umber (2) ad the secd (3) which sets the first utuched umbers (1 ad 5) t lie symmetrically arud the umber three. The, due t the six-fld peridicity, the tw liear behaviurs represeted i equati (3) are prduced. Nte that mirrr symmetry arud the umbers 3, 9, 15 as well as arud the umbers 6, 12, 18 starts t emerge. This symmetry will becme mre evidet i the third geerati ad afterwards: ( ) ( ) β 3 = 3 = = 2,3,5 {1, 7,11,13,17,19, 23, 29, 31...} =

4 4 This last equati is agai a cdesed ad cveiet way t represet the third geerati. I Figure 1 the crrespdig represetati shws a ew perid (30 = 2.3.5) ad a ew mirrr symmetry arud the multiples f 30 ad their halves. I rder t advace i a descripti f the subject, sme defiitis are eeded. The umbers i the eclum matrix (1,5 fr the secd geerati ad 1,7,11,13,17,19,23,29 fr the third geerati) are called seeds. Each seed frm a brach thrugh the prper peridicity f its geerati. A brach represets a liear mappig f with the crrespdet perid betwee their elemets ad its seed as startig phase. I the big matrix f the third geerati, their 8 ifiite braches are writte startig with their crrespdig 8 seeds. The mirrr symmetry metied earlier ca be easily idetified i the seeds structure. Fr example, the third geerati culd be writte as: thrugh its geerati will certaily frecast all the primes betwee itself ad its square, this iterval is called PCI. The PCI is depleted frm the pivt umber multiples ad the ther umbers ctaied withi have their squares, cubes, etc certaily i the utside f this iterval. I this sese, the geeratis are successively better apprximatis t the sequece f primes. The pristie atural umbers, are the geerati cer. They already predict the umber 3 as the secd prime. I the fllwig table, the begiig frm the first three geeratis, are see with the pivt umbers ad their squares i italic, ad the predicted primes i bld: 1,2,3,4 1,3,5,7,9 1, 5, 7,11,13,17,19,23, ( 2,3,5) = This seeds separati is quite iterestig because each has a six-fld peridicity which is bviusly a previus geerati remat. Each fur member sequece is defied as a stem. The mirrrig betwee these tw stems is quite easily expressed mathematically: the sum f the first ad the secd stem iverted, results i the geerati s perid, = = = = 30. Stems f higher geeratis becme mre cmplicated. Fr example the stems f the furth geerati are t aymre s rdered as f the third es. This has t d with a imprtat geerati s feature; they perate tw characteristics legths: e legth is the perid ad the ther is the prime cfidece iterval (PCI). The PCI is defied thrugh the mst imprtat characteristic f the geeratis: their ability t frecast ew primes. It was already metied that the set f pivt umbers ted t frm the set f primes. But each pivt umber After the third geerati each brach has its w PCI, ad the the umber 29 is als predicted. The PCI grws with the square f the pivt umbers ad the perid grws as a factrial (the prduct f all previus pivt umbers, this prime factrial is usually called primrial i the literature). The PCI is bigger that the perid (super-peridic regi) at the first geeratis ad the perid utgrw the PCI after the furth geerati (sub-peridic regi). I the superperidic regi the stems are cmpsed f seeds which are primes, i the sub-peridic regi this is t the case aymre, the seeds ca be primes r t. This has imprtat csequeces the structure f higher geeratis due t what is called iteral sievig. A last setece (disclaimer) fr this secti: prbably may f the features metied here are already kw i mdular algebra, but it has bee preferred t t lik these results with ay established mathematical thery i rder t avid cfusis.

5 5 Thse liks, if eeded, surely will be cstructed i future wrks. Relati with the Euler Idetity After the expsiti f the mai ideas develped util w, a questi arises: des this apprach shed ew light the prime umbers? The aswer is yes. The symmetric structures prduced by the SSM actually are als preset i the Euler equati: ζ ( s) = = s s s s s s s s s ( ) ( ) ( ) ( ) The fucti i the left is the Riema Zeta Fucti. Durig the deducti f this equality a limit is take i which the third step is shw here: s s s ζ ( s ) = s s s s s s s s The limit csists i repeatig ifiitely the prcedure which prduced the last equality 4. Whe the first iverse prime after the umber e, i the right side, get extremely big, ifiity i fact; the Euler idetity is prve. But te that i the exemplified step the sequece f demiatrs are exactly the same as the sequece f umbers crrespdig t the third geerati. This is t surprisig because as it has bee metied previusly; the SSM as well as the Euler idetity are thig else tha prime sieve represetatis. But i Euler s idetity deducti, the structure f these sequeces has bee eglected due t the limit ad fllwig a aalytical prime represetati gal. The kerel f the prblem is the successively brke symmetry f the prime sequece, which is trius i the ed result but gives hit a priri f its rigi, if the partially brke symmetries (the geeratis i the preset wrk meclature) are eglected. The it is imprtat t kw exactly hw each geerati is prduced frm the previus thrugh the SSM. Due t the geerati s primrial peridicity, it is pssible t restrict the elemet extracti t the ze crrespdig t the ext primrial durig the cstructi (applicati f SSM) f the ext geerati. Fr istace i the cstructi f the furth geerati frm the third geerati: ( 2, 3, 5, 7 ) =

6 6 The eclsed umbers crrespd t the sequece h * f the furth geerati. Nte that the braches have a cut-ff just befre the first perid ( = 210) because further elemets are uiterestig; they fllw a repetitive patter due t the peridicity. Nte als that the ew pivt umber is 7, ad there are 7 clums i the matrix. But e elemet will be extracted i each rw fr the ext geerati, leavig 6. This gives the hit that the seed umber i each geerati is the prduct f each pivt umber mius e: 1, 2, 8, 48, etc. (the frmal demstrati will be published elsewhere). Frm the previus matrix, ce the eclsed elemets have bee extracted ad the tw dimesial matrix reduced t e dimesi, e btais the large multi-liear represetati i the previus page. All what was writte befre suggest that is pssible t recast the SSM i a frm which ivlves ifiite sequece. This wuld mea a equati which is clser tha ever t a prime geeratig frmula. I fact this gal ca be achieved ad it is de i the ext secti. Discrete Scale Ivariace ad Iteral Sievig Leadig t a Fiite ad Cstructive Frmula fr Prime Geerati The SSM is based elemet extracti. This is de thrugh h* i the defiig equatis f β. But h* ad h are bth ifiite sequeces. I precedig sectis strg argumets have bee give idicatig that all features f the SSM ca be reduced t the first perid, due precisely t the itrisic peridicity f this mappig s results. Ca a ew mappig based i the symmetries frm the SSM cstructed with fiite sets r sequeces as argumet? The affirmative aswer t this questi was partially give i the last secti whe the furth geerati was btaied frm the third i just the ew geerati iterval (210). But hw shuld be reduced the elemet extracti t e perid? The sluti t this questi lies i a ew symmetry f the SSM. This symmetry culd be expressed as fllws: The patters prduced by the segregati (with primrial peridicity ad mirrr symmetries as already metied) betwee extracted ad -extracted elemets f, after b s successive applicatis, are the same see frm the perspective f a 1-peridic ifiite sequece (the rigial ) as well as frm the perspective f ay r-peridic ifiite sequece if r is relative prime with the set f all the pivt umbers used durig the mappigs f b. I particular all the future r still t used pivt umbers (which i fact are all the rest f the prime umbers) make sequeces which shw the same patters r symmetric structures. I Figure 1, Part B, tw examples f this symmetry are shw i the last rw. This last defiiti have all the features f a Discrete Scale Ivariace (DSI) 7 but e: the kw DSI has a preferred scale ad the rest f the scales are pwers f the fudametal e. I the preset case there is preferred scale ad ivariat discrete scale is a pwer f ay ther because all are relative primes. The prf f the DSI fr the sequeces studied i this wrk will be published elsewhere. This pwerful symmetry has dramatic csequeces what is w defied as iteral sievig. The SSM ctais a rmal sieve which is perfrmed with, ad, ifiite sequeces. The DSI symmetry allws t fid the extracted elemets restricted t the first perid. As the already extracted elemets, by all the previus mappigs f β, leaved the same patters all the still uused pivt umbers; the patter f the set f umbers which will be extracted by the ext pivt umber is kw. It is simply the patter embedded i the sequece f seeds multiplied by the pivt umber. Fr example; i the previus cstructi f the furth geerati, the extracted umbers fr the ext geerati are: (7, 49, 77, 91, 119, 133, 161, 203) = 7. (1, 7, 11, 13, 17, 19, 23, 29). The fiite sequece i the last parethesis is the set f the seeds frm the third geerati. The the Iteral Sievig (IS) is defied as the sievig restricted t the first perid. The set f the elemets which will be extracted is cstructed with the prduct betwee the seeds frm the previus geerati ad the ext pivt umber. With these

7 7 itra-perid rules the W mappig will be defied, actig fiite sequeces ad prducig primes i the frm f used pivt umbers. Omega acts iitially tw umbers {{S}, T} where T is the iitial perid T=1, ad S is the iitial seed S=1 (the cer geerati). Because 2 is the start ad the ed f the first brach (2 is the first pivt umber), Omega adds t the ly seed 1 the perid 1 just ce, frmig the sequece: (S, S+T) = (1, 2). The the utput frm the first mappig f mega is (the first geerati): ({{ 1 },1}) {{ 1, 2} { 2 },1.2} {{ 1 }, 2} Ω = = Repeatig the prcedure the first geerati: ({{ 1 }, 2} ) {{ 1, 3, 5} { 3 }, 2.3} {{ 1, 5 }, 6} Ω = = Here agai t the uique seed 1, the perid 2 is successively added util the last umber (5) is still smaller tha the perid (6 = 2.3); the 3 times the set f the seeds is extracted. The ly elemet frm this last set which is ctaied i the uique brach is 3. The the umber 3 is extracted leavig the already kw biliear superpsiti. Observe that it is redudat t maitai a recrd f the perid (the primrial right part) because it ca be btaied summig the tw extremes f the seeds. Hwever it is kept fr the sake f clarity. After the first geerati, the acti f W becmes systematic: ({{ 1,5 },6} ) {{ 1,7,13,19,25,5,11,17,23,29 } { 5,25 },30} {{ 1, 7,11,13,17,19, 23, 29 },30} Ω = = Ad i geeral: ( ) 0, 1, 2( 1 ),..., ( 1)( 1) Ω σ = σ + σ+ σ 1+ σ+ σ 1+ σ+ σ2 σ 1+ σ2 σ Where s 2 ad s -1, are s s secd ad last elemets, ad s is used as clum vectr frmig the matrix betwee the brackets. Here agai the mius sig after the clsig bracket meas elemet extracti. A vectrial versi fr W is give as fllws: ( ) ( ) i ( 1, 1) Ω σ = σ ν σ + σ σ σ Here s is the vectr f seeds, is the matrix exteral geeralized prduct (i which the last perati betwee idividual elemets is a tw dimesial vectr istead the usual prduct). The vectr is cmpsed frm the atural umbers startig at cer util the secd seed s2 mius e. The umber s -1 is the last seed ad the bld pit is a scalar prduct. The lie ver the etire perati meas that the resultig matrix is prjected i e dimesi ( vectrized ) ad rdered. The mius sig is a set perati ad meas elemet extracti. Fr example peratig with Omega (1, 5): (( )) ( ) ( ) ( ) i( ) ( ) Ω 1,5 = 1,5 0,1, 2,3, 4 1,6 5 1,5 = ( 1,0 ),( 1,1 ),( 1,2 ),( 1,3 ),( 1,4 ) ( 5,0 ),( 5,1 ),( 5,2 ),( 5,3 ),( 5,4) i ( 1,6 ) ( 5,25) = 1, 7,13,19, 25 ( 5,11,17, 23, 29) ( 5, 25) = ( 1, 7,11,13,17,19, 23, 29) These peratis ca be resumed i e lie f Mathematica cde: [[ ]] { [[ ]] } [[ ]] { } Nest Cmplemet Flatte Outer List,#, Rage # ,# 1 + 1,# 2 *# &, 1,5,3 This cmmad has already the iterative rder Nest embedded as the last perati. Observe that i this example 3 iteratis are prduced. Nte als that the startig geerati is the secd e; {1, 5} istead the cer geerati {1}. This is due t the W s sigular behaviur whe it is started at the cer geerati. After the secd geerati the brach prducti becmes systematic. A fial reflecti W; reducig the acti f the SSM t the first perid f each geerati, made evidet that the primes are the remat f a decimatig machie. This machie whe

8 8 Figure 2 Secd Order Discrete Prime Prducig Equati Gdess. The clseess f the results is quatized i stripes belw ad ver the real prime values. The differece values lk like etire discrete quatities, but i fact they are fractial values very clse t the itegers. applied t the atural umbers elimiates all f them but the umber e. The differece betwee the primes ad the -primes is the dr thrugh they are leavig the set f the umbers mdified by W; thse which leaves thrugh the last dr (activated by the multiplicati by 1, which is always i the set f seeds) are primes, thse leavig thrugh the rest f the iteral sievig, are t. This prime geeratig decimati machie is self-regulated: If the desity f the pssible primes is high (lw) i ay regi this will mea that i the ext geerati the decimati will be high (lw), lwerig (elevatig) i this way the desity. This prperty is trius at the very first geeratis where the desity f pssible primes is the highest ever, leadig t a fierce decimati ad t a rapid stabilizati f the desity f primes. A Mea Field Thery fr Primes leads t a Imprved Prime Number Therem. Usig sme prperties frm Omega, it is easy t btai the mea behavir f the prime umbers. Each time W is applied; the perid is icreased by the pivt umber p used at that iterati ad the umber f seeds by (p-1). I the PCI all the seeds are primes ad e ca make the suppsiti that their desity is the same there ad i the ze utside the PCI. The applyig Omega chages the mea desity f primes frm D -1 t D thrugh the fllwig factrs: ( p ) 1 1 = 1 = 11 p p But D ad D -1 ca be expressed as the iverse f the differece f tw csecutive prime umbers: 1 1 = ; = ( p p ) ( p p ) As i a lattice f atms, betwee tw sites e ca cut the distace as crrespdig t e atm, e prime i this case. Substitutig these equatis i the previus e ad slvig fr p +1, e btais a secd rder recursive equati fr the ext prime: p +1 = ( 2 1) ( ) 1 p 1 p p p The accuracy f the frmer equati is variable, but i a fracti f the cases it gives the ext prime almst exactly. I Figure 2 the differeces betwee the

9 9 true primes ad the umbers prduced thrugh the frmer frmula with the tw previus true primes as argumets, are shw fr the first 1000 primes. This is a lgarithmic plt i the abscissa ad shws that the errrs i the prime predicti are apprximately symmetrically distributed arud the cer. I a fracti f the trials, quasi-exact (the differece t the true prime is a fracti less tha e) slutis are btaied. Besides the first few cases, all the predicted primes are almst etire umbers ad separated frm the real es by a eve almst iteger differece (prducig the quatized stripes frm the graph). The cversi f the frmer discrete equati i a ctiuus e with derivatives; has the gal f reprducig the results f the Prime Number Therem ad eve t imprve them. If the fucti g() is able t geerate sequetially the prime umbers with atural umbers as argumets, the g() = p where p is the th prime. This meas that the miimal measurable differece betwee tw argumets has t be 1. The the best apprximati fr a derivative is: ( ) γ = γ γ ( ) ( ) The geeral sluti t this equati ca be fud i terms f expetial itegrals: ( ) γ = e ( ( )) ( 1) A A+ Ei e + B Where A ad B, are the itegrati cstats ad Ei (-1) is the iverse fucti f the expetial itegral give by: ( ) Ei x = x t e dt t If the cstats are set t cer; A = B = 0 the the simplest versi is btaied: ( ) γ = e Ei ( 1 ) ( ) ( + 1) γ ( ) γ ( ) = = p p 1 γ + 1 Substitutig these results i the discrete equati f prime desities, the fllwig equati is btaied: 1 γ ( -1) = γ ( ) 1 γ ( ) Calculatig the secd derivative f g i a similar way: ( ) = ( ) ( -1) γ γ γ Substitutig this last result i the frmer equati, the fllwig simple, -liear differetial equati is btaied (these are apprximatis t γ ad are called γ ) : The fucti p () is defied as the umber f primes which exist i the iterval (0, ) 8. If p is the th prime, the there are primes i the iterval (0, p ). But p = g(), applyig p t bth sides: ( ) ( ) π γ ( ) π p = = The p ad g, are the iverse f each ther. Kwig the simplest apprximate sluti fr g, e ca btai the crrespdet p ivertig it: t e = Ei l ( γ ) = dt l ( γ ) t Substitutig t = - l[z] the fllwig equati is btaied: γ 1 = d z = L i 0 l ( z ) ( γ )

10 10 Figure 3 Cmparis f the Old ad New Apprximatis t π(). The values f π() are shw as islated pits, the ew apprximati as a ctiuus lie ad Li(x) as a dted lie. I the Part B; Li(x) remais clse t π(), whereas the ew apprximati diverges slwly. But is the umber f primes less r equal tha g = p, the is p (p ), which cstitutes the first cstructive demstrati f the Prime Number Therem (the right side defies the Lgarithm Itegral fucti). Eve mre, it is the first cstructi f a apprximate differetial equati fr the prime umbers. This is the simplest versi f the sluti. If the cstats A ad B are left free, the a imprved Prime Number Therem is btaied: π Ω 1 e ( x) A l( x) A ( ) Ei l x + A = B = A e t e dt t B This equati has a curius behavir depedig the values f A ad B. With apprpriate values give t A ad B (A =.52; B = -0.71, btaied thrugh the best fit fr the 100 first primes) the curve the graph f Figure 3, Part A, is remarkably clser t p tha the Lgarithm Itegral. But this clseess is limited; if e lks deeper the fucti crsses p () ad the remais far away frm it. This meas that certai values f A, prduces almst exact results fr buded regis f the argumet. This cfirms the fact that the previus differetial equati is a apprximati t the real differetial equati f γ(). Fr a 100 times bigger dmai widw the apprximati remais belw f the real p () after certai pit, as it shw i Figure 3, Part B. The apprximate differetial equati fr g ca be trasfrmed a apprximate differetial equati fr p just applyig the chai rule t the iverse fucti: d 2 π ( y ) 2 d y ( y ) 2 d π d y + = 0 y This last equati has the same sluti as the previus. O the Ifiitude f the Twi Primes As it was metied, the successive geeratis are fier ad fier apprximatis t the prime umbers. I this sese the first geerati states that all the primes are dd umbers. It als states that all the rest f the primes after the umber tw shuld be sieved ut frm a ifiite sequece f twi primes. The first geerati is the cradle f all twi primes. But i the first geerati, versi Omega, the seed 1 des t express the presece f twi primes. Just i the secd geerati with the seeds 1 ad 5 the twi primes are explicitly preset i betwee the seeds (i a srt f

11 11 peridic budaries the braches crrespdig t 5 first ad the t 1 have all their elemets separated by tw uits). The, all twi primes bigger tha the pair (3, 5), are ctaied i the sequece f pairs (6. + 5, ). Oly i the third geerati the existece f twi primes is evidet i the seed structure: 1, 7, 11, 13, 17, 19, 23, 29. The first twi pair is shw i bld ad the secd i italic. As i the previus geerati the limitig braches crrespdig t the seeds 1 ad 29 als cstitutes a pair f twi prime prducig braches. If the iteral sievig is eglected, these umber f ptetial twi primes wuld grw at a similar rate as the braches, with (p-1)#. The twi prime structure frm the third geerati esures that all the frthcmig twi primes will belg t e f the three sets: ( , ), ( , ) r ( , ). Durig the sythesis f a ew geerati each brach lses e member due t the iteral sievig. The a simple arithmetic prves that the umber f twi primes is ifiity; startig at the third geerati, the umber f pssible twi primes will grw i each geerati with a factr p-2, where p is the geerati s pivt umber. This is because each brach grws like the pivt umber mius e (due t the iteral sievig). But each member f a twi prime pair belgs t differet braches, the the iteral sievig elimiates 2 twis istead f e (repetiti f the iteral sievig i the same twi pair is impssible because this wuld mea that tw itegers are far apart i lees tha e uit). This last setece, expressed i equatis, meas that the mea desity f twi primes will chage as a fucti f the geeratis i the fllwig way: + 1 p 2 τ = τ p Where τ is the mea desity f twi primes ad the superscript idicates its geerati, beig p the geerati s pivt umber. This last equati shws that the twi primes ever fade ut: By cstructi f either β r Ω the umber f pssible twi primes i each geerati is ever cer. If the pivt umbers limits t ifiity, the mea desity f twi primes i each geerati teds t be the same cer quatity. As well as i the case f the plai prime umbers, this discrete equati ca be cverted i a differetial equati: τ τ ( ) '' 2 = ( ) ' γ ( ) Where τ() represets a hypthetical fucti which gives the th prime umber belgig t a twi Figure 4 Gdess f the τ() Apprximati t the th Twi Prime. The real psitis f the Twi Primes are shw with discrete pits ad the apprximati with a ctiuus lie.

12 12 prime ad γ () is already kw. Usig the simplest sluti fr: γ () = Ei (-1) (); the fllwig sluti fr τ is btaied: τ ( ) ( 1) Ei ( ) ( 1) ( ) A B e Ei ( ) = Where A ad B are the itegrati cstats. Settig A = 4.5 ad B = 0.61 the curve shw i Figure 4 is prduced. I the same plt, the true rdered values f the prime umbers belgig t twi primes are shw as pits. If the geeral sluti fr γ () is used, the 4 itegratis cstats are preset: ( ( ( ))) ( 1) D C+ Ei ( e ( + C) ) ( 1) D B e 1+ Ei e + C A+ C ( 1) D Ei ( e ( + C) ) Fially it shuld be ted that it is impssible t ivert the sluti fr τ() (lkig fr a equivalet π fucti f the twi primes) due t the multiplicity f the fucti Ei (-1) () i it. Cclusis I e f the classical papers abut the prime umbers, it is stated frm where cmes the isight t hypthesize ad the t prve that the fucti Li(x) is a gd apprximati fr p(x). There were reass beyd the experimetal fact ad its plttig i a lgarithmic scale 8. A direct csequece f this was the extreme difficulty t prve aythig abut this fact ad abut the prime umbers i geeral. A lack f a wrkig mdel fr prime umbers has built the geeral sese that they were beyd the reach f the huma itellectual pwers 2. Withut such a mdel the mathematicias relied feeble hits hidde i the apparetly t uderstadable structure f the prime umbers. This made the few available prfs it huge effrts very difficult t elabrate ad practically just the level f sme specialists. I this wrk a simple ad beautiful mdel f the prime umbers is cstructed startig frm the gemetrical symmetries frmed durig the sievig prcess. These symmetries are related with the relative psiti f the prime umbers ad t with the umbers itself r with their restricted divisibility. The study f this restricted divisibility, as a fudametal fact abut the Primes, was ather misleadig path. The restricted divisibility is mre a csequece tha a cause i the set f features which characterize the prime umbers. Fllwig the right path f this mdel t its lgical csequeces, it was pssible t elabrate a thery fr the mea quatities f the primes. This thery leads t a secd rder discrete ad apprximate equati ad t a secd rder ctiuus differetial equati fr the primes. This is the first time that such equatis are preseted. The simplest sluti f this equati prduces the Li(x) apprximati as well as a geeral sluti with tw cstats assciated with the secd rder differetial equati. The behavirs f the slutis uder the adjustmet frm these cstats demstrate that the thery elabrated i this wrk is icmplete, ad that eve deeper symmetries are still lurkig i the structure f the prime umbers. Perhaps the biggest advace f this wrk is the elabrati f a recursive ad fiite frmula fr the prducti f prime umbers. I this frmula the ifiitude f the twi primes is explicitly shw thrugh its structure. This leads, i cmbiati with the frmerly metied differetial equati fr the primes; t a apprximati f a differetial equati fr the twi primes which is aalytically slvable. All these successes shw that the mdel fr the prime umbers preseted i this wrk is the crrect e. Furthermre, sme ew results, which were reprted as breakthrughs, are simply atural csequeces f the preset mdel. Fr example the existece f arithmetic prgressis f primes is bligatry due t the structure f the braches 9. This mdel is straightfrwardly cected with the Riema Hypthesis thrugh the Euler idetity. Hwever, as was shw i this wrk, the presece f the uique argumet s i the Riema fucti seems t be irrelevat t the structure f the primes, at last i

13 13 relati with their mea prperties. I this reasig, the zers related t this argumet, are als irrelevat. I fact the symmetric structures frmed i β ad Ω are the same which frm the Euler idetity with s = 1 r with ay ther value. It is the authr s belief that this wrk has just scratched the surface f the Primes real structure. Fllwig the lead set by it, it will be pssible t fially fid a exact sluti fr the prime prblem. Sme f the deeper symmetries are already evisied by the authr ad will be published s elsewhere. Fially a pssible bus: the already discvered symmetries suggest that uveilig the cmplete prime umber structure wuld be the same as btaiig the uique atural cutig system, t biary, t decimal, but prime. 5 Peitge H., Jürges H., Saupe D. Chas ad Fractals: New Frtiers f Sciece. 1st ed. (Spriger- Verlag, New Yrk, 1992) Chapter 1. 6 Wlf M., Multifractality f Prime Numbers. Physica A 160, (1989). 7 Srette D., Discrete scale ivariace ad cmplex dimesis. Phys. Rep. 297, (1998) p Lagarias J. C., Odlyzk A. M., Cmputig pi(x): A Aalytic Methd. J. Algrithms, 8 (1987) p Gre B., Ta T., The Primes Ctai Arbitrarily Lg Arithmetic Prgressis. Ackwledgemets The authr wishes t thak Dr. Eduard Greaves fr the revisi f this dcumet ad als the creatrs f the wderful tl Mathematica. Withut this amazig piece f sftware it wuld be impssible t d this wrk. 1 Guy, R.K. Uslved Prblems i Number Thery. 2d ed. (Spriger-Verlag, New Yrk, 1994). 2 Watkis, M. R. Number Thery ad Physics Archive. cs.htm. 3 Riema, B. Über die Azahl der Primazahle uter eier Gegebe Gröβe, Gesammelte Matematische Werke VII, 2d ed. (Teuber, Leipzig, 1982). 4 Derbyshire, J. Prime Obsessi: Berhard Riema ad the Greatest Uslved Prblem i Mathematic.,1st ed. (Jseph Hery Press,Washigt D.C., 2003) p. 102.