Proving the Computer Science Theory P = NP? With the General Term of the Riemann Zeta Function

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1 Research Joural of Mahemacs ad Sascs 3(2): 72-76, 20 ISSN: Maxwell Scefc Orgazao, 20 Receved: Jauary 08, 20 Acceped: February 03, 20 Publshed: May 25, 20 Provg he ompuer Scece Theory P NP? Wh he Geeral Term of he Rema Zea Fuco M.V. Aovgba Deparme of urrculum ad Teachg, Beue Sae Uversy, P.M.B. 029 Makurd, Ngera Absrac: The sudy ams a showg ha he geeral erm or sequece of he Rema zea fuco s a polyomal me algorhm or Turg mache M whch s used o resolve he compuer scece heory: P NP? >(s)depeds o he se of aalyc zeros s F+ as raw maerals whle s depeds o, where ad Fare real umbers. The work shows ha polyomal me S( ) ad for all srgs of eger values $, M s closed ad bouded of real values: [0, ]. The algorhm M sasfes ook s heorem whch s a NP-omplee problem. Hece, M s NP-Hard ad hece NP-omplee ad hus resolves he P NP? problem a polyomal me s( ). Key words: Algorhm, closed bouded real values, complex varable, real-valued fuco, polyomal me, rema zea fuco, urg mache INTRODUTION The compuer scece heory: P NP? s a compuao complexy problem amed a locag wheher here exss a polyomal me algorhm whch a well cofgured compuer ca uderake boh a deermsc ask P ad oher odeermsc processes NP a he polyomal me (Wkpeda, 200a). I has already bee observed ha all algorhms P are also NP (Tordable, 200); bu he ope problem s wheher P NP or o. The Wkpeda (200b) oes ha he P versus NP problem s a major usolved problem compuer scece whch asks wheher every problem whose soluo ca be effcely checked by a compuer ca also be effcely solved by a compuer. A problem s sad o be he NP class f s solvable polyomal me by a odeermsc Turg mache, ad a P problem whose me s bouded by a polyomal s always NP, whle a problem s cosdered NP-hard f a algorhm for solvg ca be raslaed o oe for solvg ay oher NP-problem, ad a problem whch s boh NP ad NP-hard s called a NP-complee problem (Wesse, 200). The proof of he P NP? Theory wll have esmaed cosequeces (Wkpeda, 200b): A proof ha P NP could have sug praccal cosequeces, f he proof leads o effce mehods for solvg some of he mpora problems NP. I s also possble ha a proof would o lead o effce mehods, perhaps f he proof s ocosrucve, or he sze of he boudg polyomal s oo bg o be effce pracce. The cosequeces, boh posve ad egave, arse sce varous NP-complee problems are fudameal may felds. There are eormous posve cosequeces ha would follow from rederg racable may currely mahemacally racable problems. For sace, may problems operaos research are NP-complee, such as some ypes of eger programmg, ad he ravellg salesma problem, o ame wo of he mos famous examples. Effce soluos o hese problems would have eormous mplcaos for logscs. May oher mpora problems, such as some problems proe srucure predco, are also NP-complee; f hese problems were effcely solvable could spur cosderable advaces bology. Ths research, however, ams a resolvg he P NP? problem usg he geeral erm (or sequece) of he Rema zea fuco: where ζ ( s) s + s () (2) where s magary ad F, are real ad specfcallyf s a cosa whle s varable. Ths s show usg absolue covergece of he geeral erm of (). Thus f for some sequece u (x): 72

2 Res. J. Mah. Sa., 3(2): 72-76, 20 + Lm x ( x) ( x) < (Spegel, 2005) Now, sce (2)Y s() ad ()Y H(s)H(F+) wh F a cosa (predeermed) real umber, he geeral erm for () s: ( ) ( ) (3) The research wll focus o showg ha (3) s a polyomal me algorhm M whch s NP, NP-Hard, ad hece NP-omplee ad hus aswers he P NP? complexy problem. MATERIALS AND METHODS Ths research was sared sce Sepember 5, 200 alog wh hs researcher s work owards provg he Rema Hypohess. Ths was because hs researcher srogly beleved he Rema zea fuco of raw maerals sf+, o be a polyomal me algorhm M, NP-Hard ad NP-omplee ad ca be used o prove he compuer scece heory: P NP? The maerals used hs work clude he Rema zea fuco: ς( s) s () where s F+, for real ad magary, whle, 2,. The mehod used he work s provg for fe seres absolue covergece, esablshg exsece of a leas oe srg whch geeraes he value of uy () for he Turg mache M some defed me rage L, esablshg ha M uquely produces 0 for all srg ousde L, ad showg ha M ca be appled o solve ay exsg NP-omplee ask. Proof of he P NP heory: To prove he compuer scece heory eals defyg a polyomal me algorhm M such ha a polyomal me S, P NP (Wkpeda, 200a) whch requres provg ha: a) M ca, as a verfer, perform a deermsc process P ad also perform oher odeermsc processes NP, b) M ca perform boh processes a he same polyomal me, ad c) M ca be used o solve ay exsg NP-omplee problem. To prove (a) requres frsly o esablsh ha here exss a polyomal me algorhm or Turg mache M as a verfer wh a laguage L beg NP sasfyg he codo: f ad oly f here exs some polyomals p ad q, ad a deermsc mache M, such ha: rero : For all x ad y, M rus me pu (x, y). p( x) rero : For all x L, here exss a srg y of legh q x such ha M(x, y). rero : For all x o L ad all srgs y of legh, M(x, y) 0 (Wkpeda, 200a). q( x) To sasfy hese crera, le x ad y ad suppose ha: M s (4) whch s he geeral erm (3) of he sequece geeraed from he Rema zea fuco () such ha for polyomals p ad q: ad p x s + or p + ( + ) s p q x or q p rero holds sce M rus me (, ). rero also holds: a srg of legh + 2 L: L < ( + ) q s (5) o o pu, here exss such ha M sce rased o ay umber reurs. rero also holds sce for all srgs of legh ( + ) q L, ad, M(x,y) 0. Ths s because L geeraes M 0 as he sequece absoluely coverges o 0, ha s: By he rao es for absolue covergece of he geeral erm of he sequece: 73

3 Res. J. Mah. Sa., 3(2): 72-76, 20 ( + ) or ().. whch coverges absoluely o 0 sce: + Lm ( ) ( ) < mplyg covergece of he sequece. Ths s doe akg ad + 2 So ha: p + adq p (7) (8) () Wh he rage -4#L<4, for ay L, f, M. Ousde of hs rage, p q M 0 Thus wh p ad q polyomals makg M as a verfer o have closed bouded real values [0,], ad sce M mples he exsece of polyomals p ad q, he M s a verfer wh a laguage L beg NP. To prove (b), wheher M ca perform boh processes of solvg a problem ad verfyg he soluo a he same polyomal (fas) me, so ha P NP a he sad polyomal me: We are o prove ha M s NP-Hard ad hece NP-omplee. By NP-Hard requres ha M s used o solve a leas oe of he NP-omplee problems (Wesse, 200). I s oed ha addae M solves a NP-omplee problem, amely, ook s heorem, whch goes by: + () Lm () Lm ( + ) ( + ) ( 0) 2 Lm < {} (9) Thus M 0 for o L, whch sasfes rero. The facor (7) dmshes slower ha small o:t() o( k ) o( - ) where 4 as o( k ) -4 Hece, M s bouded wh small o:t() o( k ) where k -4 as o( k ) -4 Ths makes he umeraor o be zero ad hece he ere sequece hals o zero. Hece M s a polyomal me algorhm. Reversg he process (o sasfy he f ad f oly codo), Le: ( + ) p M p q (0) The (0) mples he exsece of polyomals p ad q: Gve a srg S ad a No-Deermsc Turg mache M, creae a NF expresso E(S, M) whch s sasfable f ad oly f M acceps S polyomal me. Demosrae a algorhm for geerag E(S, M) for ay par S, M. Prove ha he algorhm for geerag E(S, M) rus polyomal me. By polyomal me algorhm s mea f he polyomal s rug me s upper bouded by a polyomal he sze of he pu for he algorhm so ha some T() O( k ) for some cosa k (Wkpeda, 200d). Now, rewrg ook s heorem so ha: ( ) (, ): +, + ESM s M (2) he algorhm M rus polyomal me so ha for all, + here exss of legh q ad ( + ) M, ad L, ad all srg ( + ) > of legh q( x ) q, M 0 because L geeraes M 0 as he sequece coverges absoluely o 0, ha s: 74

4 Res. J. Mah. Sa., 3(2): 72-76, 20 M s covergece o 0 also wh absolue covergece of he geeral erm of he sequece whch coverges absoluely o 0 as demosraed (8) ad (9). Thus E s sasfable sce M acceps S polyomal me producg a whe L: L < ad 0 a 4 Reversg he process o sasfy he f ad oly f codo, le M + ad le M, he: Hece; Also, ( + ) ( ) ad ( ) M 0 0 whch s oly possble f k o ( ) 0 Thus M acceps S polyomal me, gve he small 0 osequely, M sasfes ook s heorem whch s NP-omplee. Hece M s NP-Hard ad s a polyomal me algorhm ha resolves he P NP? problem. orollary: Suppose ha (2) s a complex-valued fuco of a complex varable he M always for all srg a ad M 0 a. Proof: makes s F+ 0, hece for all > M 0. Hece for srg of solvg P ad srg + k (k, 2, ) of verfyg he soluo NP, M mplyg ha P NP always a Remark o he corollary: To have s F+ as a complex-valued fuco of a complex varable s smply o ake me o be magary as used by quaum physcss ad cosmologss (Wkpeda, 200c) as complex me s geomercally posed o be o he vercal axs ad perpedcularly crosses he real me horzoal axs. Thus a compuer ca be cosruced o ake ay suable real F value ad a accompayg complex values ad execue he programme M + so ha wh <, he compuer s ape akes for all srg > ad reurs. Else he compuer akes ad reurs 0 wheever >. The summary of he algorhm M + wll he be: If, he M f >, he : <, M RESULTS AND DISUSSION The research has foud ou ha here exss a polyomal me algorhm M + whch s he geeral erm of he Rema zea fuco, whch s verfed o be NP-Hard, NP-omplee ad hece resolves he P NP? problem. Hece he compuer mgh be sruced o execue: Programme P NP Ipu : eger,2, ad : real : ; < Oupu M: Boolea: >, M M + sasfes ook s heorem ad ca hece be used o solve may oher NP-omplee problems. The fdgs show ha he algorhm explas he very foudaos of Boolea algebra ha he values of he Rema zea fuco oscllae bewee ad 0 gve wo dsc regos, respecvely L : < ad L 2 :. If complex values are ake L, he he value of he algorhm for all srg s whe, ad f be L 2 he value of he algorhm s 0 as moves o fy. Thus he Rema zea fuco, for all posve eger values, has fe closed bouded lower ad upper real-value bouds [0, ] ad s defed o he ere complex couum:. For he compuer o work Boolea order wh he oly wo dgs 0 ad, he compuer could he be programmed so ha wll obey he sruco: pu : eger oupu M: Boolea verfed wh: M : <, 75

5 Res. J. Mah. Sa., 3(2): 72-76, 20 I fac wll be suffce o cofgure some compuer o effcely work or perform ay sruco me s():, for all srg. Hece P s performed a he same me as NP. The compuer wll fuco f programmed: Gve a verfer M + ad srg, 2, 3, for ay operao, pu (, ) oupu M (, ) : > The Turg mache M(, ) performs boh P ad NP operaos a s F + polyomal me where for, P NP for all >, ad for 4, P NP (0, ): 0 f ad f. The fdgs have educaoal value of expadg he rage of mahemacal expereces wh whch o broade he currculum ad whch ca afford he learer a hgher level of solvg complexy ad oher compuaoal problems. The sudy provdes added expereces from whch mahemacs ad compuer scece educao currculum could selec supplemeary ems. ONLUSION The work se ou o ad has esablshed ha M (, ) + he geeral erm of he Rema zea fuco s a polyomal me algorhm ha resolves he P NP? problem. A corollary o he proof exss by exedg he algorhm s raw maerals s F + from he z-plae rasformed o he w-plae by leg o be complex hus producg complex values whch makes P NP a (F, 4) ad he algorhm M geeraes closed bouded real bary values [, 0] o he se of frs complex eres (-4, 4). REFERENES Spegel, M.R., Advaced Mahemacs for Egeers ad Scess. Taa McGraw-Hll, New Delh. Tordable, J., 200. P versus NP. Rereved from: hp:// (Accessed o: December 02, 200.) Wesse, E.W., 200. NP-Problem. From MahWorld - A Wolfram web resource. Rereved from: hp://mahworld.wolfram.com/np-problem.hml, (Accessed o: December 02, 200). Wkpeda, 200a. NP complexy. Rereved from: hp://e.wkpeda.org/wk/np_(complexy), (Accessed o: November, 200). Wkpeda, 200b. P versus NP Problem. Rereved from: hp://e.wkpeda.org/wk/p_versus_np_problem, (Accessed o: November, 200). Wkpeda, 200c. Imagary me. Rereved from: hp://e.wkpeda.org/wk/magary_me, (Accessed o: November 5, 200). Wkpeda, 200d. Polyomal me. Rereved from: hp://e.wkpeda.org/wk/polyomal_me#poly omql_me, (Accessed o: November 5, 200). 76