A proof of Goldbach's hypothesis that all even numbers greater than four are the sum of two primes.

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1 A roof of Goldbch's hyothesis tht ll eve umbers greter th four re the sum of two rimes By Ket G Sliker Abstrct I this er I itroduce model which llows oe to rove Goldbchs hyothesis The model is roduced by studyig Goldbch rtitios s dislyed by iverted mirror imge of ll the rimes u to some eve umber equl to the lst rime lus three The bottom hlf of the model is the moved to the right i stes of two which exhibit the Goldbch rtitios for the ext eve umber As log s the model cotis ll the rimes u to the resultig eve umber mius three, the Goldbch s hyothesis c be rove if it c be show tht ech move must roduce Goldbch rtitio util oe reches the ext rime lus oe I show tht this must be the cse k m Nottio Ay eve umber A secific eve umber, lwys three more th some rime A rime umber (2, the oly eve rime, is ot cosidered i this er) Ay rime i the list which mkes u Goldbch block The lst rime (lso the gretest) i Goldbch block + 1 The ext rime fter The mirror imge couter-rt of i Goldbch block r The g betwee d i Goldbch block r A eve umber dded to m to obti k t Itroductio Christi Goldbch ( ) roosed, i ow fmous letter to Euler i 1742, tht every eve umber greter th four c be exressed s the sum of two rimes This ws lso rt of Hilbert s 8 th cojecture t the 2 d Itertiol Cogress i Pris i 1900 Hrdy d Littlewood stted i er which ered i Act Mthemtic i 1922, There is o resoble doubt tht the theory is correct, d tht the umber of reresettios is lrge whe m is lrge; but ll ttemts to obti roof hve bee comletely usuccessful Severl comuttiol results hve show tht Goldbch's hyothesis is true for lrge umber of eve umbers Jvier Echevrri verified the hyothesis u to , d Mtti Siislo to 4x 10 (RG) Sheldo clculted tht the robbility of 1

2 14 some eve umber m > 4x10 ot hvig Goldbch rtitio to be 150,000,000,000 roximtely 10 (NS) Imressive s this my be, it is ot roof, d the ossibility remis tht there does exist some eve umber which is ot the sum of two rimes The seemig rdom distributio of rimes hve mde the cojecture quite difficult to rove, log with the lck of techiques relted to the dditive roerties of rimes It is certily ossible tht, eve though very rre, there re ifiite umber of excetios to Goldbch's cojecture The questio is oe s to why o roof of Goldbch's hyothesis hs ered I suggest tht the fult does ot rest i the existig methods of lysis, but rther i the lck of model to guide tht lysis If I m correct, d the followig model reseted i this er does rovide roof of Goldbch's cojecture, the this will be o smll victory for model theory d should reemhsize the eed for the costructio of models s id to existig techiques of lysis The Model To rove Goldbch's cojecture I will emloy model bsed o the followig simle equtio: Where is some odd rime, k eve umber, d 3 (k - 3): ( k ) + = k This equtio sulies oe of the ecessry rimes; if it c be demostrted tht ( k ) c lwys be equl to other rime for every eve umber k, the the hyothesis is rove Exmle: Tke for istce the umber 26 It hs three Goldbch rtitios: = = = 26 For ech rime i the rtitio: {3, 7, 13, 19}, the differeces: {(26-3), (26-7), (26-13), (26-19} re lso rime We c ow costruct model tht exresses these rtitios We first write dow ech rime s block u to (26-3), sice 23 is rime it forms the to rime row The to row cotis the itegers u to 26 for referece 2

3 Figure 1 If we ow mirror d ivert the rime rows of this model, we c see the Goldbch rtitios s listed bove Figure 2 Let us ow move the right side of the mirror imge 2 moves to the right This will yield the Goldbch rtitio for (26 + 2): Figure 3 The resultig cofigurtio will show ll the Goldbch rtitios for 28, which hs oly two: 5+ 23, d Figure 3 shows these rtitios log with their commuttive equivlets: , d

4 We could move the Goldbch block two more moves to the right to obti ll the Goldbch rtitios for 30, however, s soo s we move the block we lose oe Goldbch rtitio for 32, mely This is becuse our model does ot coti ll the rimes u to 32 3 If we restrict our moves to the right such tht from the iitil ositio t m every move to the right cotis ll the rimes u to the resultig eve umber mius three, we c rove Goldbch s hyothesis by showig tht every such move must roduce Goldbch rtitio Goldbch Blocks Sice this etire lysis rests o wht I hve termed Goldbch block, let us ow defie them Ech Goldbch block will strt t some iitil stte rereseted by some eve umber m, d cosist of two mirror imges of the list of ll rimes { 0, 1, 2 }, with the first rime beig 0 = 3 d the lst rime, lwys equl to m 3 (see Figures 1 d 2), hece: [Def 10] The first rime 0 0 = 3 [Def 11] The lst rime i the Goldbch block = m 0 [Def 12] The vlue of m t the Goldbch block s iitil ositio m = 0 + For exmle, i Figure 4 below, the rime list goes from {,, } , so tht = 7 which hs vlue of 23 The vlue of 3 is 11, d the vlue of 7 3= 4 is 13 The vlue of m is 0 +, or = 26 We will ow desigte the "g" betwee ech rime,, d its mirror imge couter rt, -, s r t the Goldbch block's iitil cofigurtio t m Figure 4 shows the differet r s lied to Figure 2: 4

5 Figure 4 From the fct tht m = + 0, r 0 is lwys equl to zero The vlue of ech idividul r c be see to be: [Eq 10] The vlue of ech g i the Goldbch block ( ) r = m +, or solvig for m: [Eq 11] m = r + + Exmle: I the bove figure m = 26, 2 = 7, d =, hece ( ) r 2 = = 2 By defiitio, Goldbch blocks t the iitil cofigurtio t m hve t lest oe Goldbch rtitio, sice m = 0 + Let us exmie Goldbch block with the right hd mirror imge shifted to the right 4 sces to obti m + 4 Figure 5 shows this cofigurtio: Figure 5 5

6 Sice r 0 = 0 by defiitio, we desigte r t s the umber of moves to the right from the iitil ositio t m = + 0, hece r t t m is zero We will defie the ew sum, rereseted by m+ r s k, hece: t [Eq 20] The vlue of Goldbch block shifted to the right k = m+ r t Where m+ rt Solvig for r t i terms of k d m: [Eq 21] k m r t = Where ( ) r + m t Defiitio of Goldbch Prtitio Let us ow defie Goldbch rtitio for these Goldbch blocks Sice rtitio occurs whe rime i the bottom mirror imge mtches rime i the to imge, the if some rime ( ) i the rime list lus the origil g ( r ) lus the move to the right ( r t ) equls other rime higher u o the list, the the resultig Goldbch block exhibits Goldbch rtitio Let m be other rime i the rime list { 0, 1, 2 7} such tht m > the: [Theorem 10] Defiitio of Goldbch Prtitio If + r + rt = m, the k hs Goldbch rtitio equl to m + ( k m) A esy Corollry to [Theorem 10] is: [Corollry 11] k + r + r =, the k hs Goldbch rtitio equl to + ( k ) If ( ) Proof: t q From [Theorem 10], + r + rt = mwhich llows us to re-write [Corollry 11] s k m = q, which leds immeditely to k = q + m q q Exmle: Secificlly, i the bove exmle for 30 (Figure 5), oe Goldbch rtitio is Hece from Theorem 10: 6

7 + r + rt = m Isttitig + r 1+ r1 1+ r1+ rt = m Relcig ech vrible with the vlues give i Figure 5: = 11 Hece, from [Theorem 10], the Goldbch rtitio for 30 is give by: 11 + (30 11) = 30, or: = 30 Proof of Goldbch s Hyothesis We c ow rove tht ech k must hve Goldbch rtitio Proof: From [Eq 20] k = m+ r t If r t = 0, the k = m, d hece hs Goldbch rtitio from [Def 12] Suose rt 0, the from [Eq 21] k m = r t But from [Eq 11] m = r + + So we c write: ( ) k r + + = r t Rerrgig terms: ( ) k + r + r = t 7

8 But from [Corollry 11], if ( ) k + r + r =, the k hs Goldbch rtitio t q equl to + ( k ), hece k hs Goldbch rtitio t + ( k ) q q Ket Sliker Pim Commuity College ksliker@imedu 8

9 Refereces [NS] NEIL SHELDON A Sttistici's Aroch to Goldbch's Cojecture Techig Sttistics Vol 25, No 1 Srig 2003 [RG] RICHARD K GUY Usolved Problems i Number Theory Sriger-Verlg New York, 1994 [WY] WANG YUAN (editor) Goldbch Cojecture World Scietific Publishig Co Pte Ltd Sigore,

A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of

A. Description: A simple queueing system is shown in Fig. 16-1. Customers arrive randomly at an average rate of Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig