From Hardy-Lttlewood(9) To 04 All rme apers Are ot Eve Wrog (0Abel rze Is ot Eve Wrog,0Abel rze Is ot Eve Wrog) Chu-Xua Jag The Hardy-Lttlewood prme k-tuples cojecture[8,9,4] ad Erdos-Tura cojecture(every set of tegers of postve upper desty cotas arbtrarly log arthmetc progressos )[4,5,6,7,0,5] are ot eve wrog. Usg the crcle method ad the seve method oe do ot prove smplest tw prme cojecture( there exst ftely may pars of tw prmes) ad the smplest Goldbach cojecture (every eve umber >4 s the sum of of two prmes).therefore from Hardy-Lttlewood(9) to 04 all prme papers are ot eve wrog[-50].they do ot prove ay prme problems. 006Felds medal(gree-tao theorem[0]),007wolf prze(fursteberg theorem[5]) ad 0Abel prze(szemered theorem[4]) s ot eve wrog, they do ot uderstad arthmetc progressos. The correct arthmetc progressos s Example 8[6,p68-74].Isttute for Advaced study(math) has log bee recogzed as the leadg teratoal ceter of research pure mathematcs. A.of Math.publshed may wrog prme papers, for example:gree-tao[0,4],goldsto-tz-yldrm[8],wles-taylor[48,49], Zhag[8] ad other. Ther papers are related to the Hardy-Lttlewood wrog prme k-tuples cojecture[8,9,4,50].therefore ther papers are ot eve wrog. All Rema hypothess s ot eve wrog[5].all zeros of all zeta fuctos are ot eve wrog. 0Abel prze s ot eve wrog. But A.of Math reject Jag papers. Edtors of A.of Math do ot uderstad the prme theory ad wat to publsh wrog prme papers. Tw prmes theorem[6,p4]. = + We have Jag fucto to see example J ( ) ( ) 0 ω =Π We prove that there exst ftely may prmes such that + s prme.therefore we prove tw prmes theorem.
We have (,) = { : + = prme} ~ Π ( ) log Goldbach theorem[6,p4]. = + We have Jag fucto to see example J( ω) = Π( ) Π 0 We prove that every eve umber 6 s the sum of two prmes.therefore we prove Goldbach theorem. We have (,) = { : = prme} ~ Π ( ) Π log Usg above method we prove about 000 prme theorems[].ths paper s oly correct prme theory, other prme theores are wrog, because they do ot prove the smplest tw prmes theorem ad the smplest Goldbach theorem.the prme papers of ICM006,ICM00 ad ICM04 are wrog.if ICM do ot recogze ths paper,the the prme papers of ICM08 ad ICM0 also are wrog.tao does ot prove that every odd umber s the sum of fve prmes,hs proof s wrog[].i [6,p70-00] we establsh the theory of prme table.we prove that prmes tuples there exst ftely may prme solutos ad ftely may prme solutos. Let k be a gve prme,jp+k-j(j=,...,k-),there exst ftely may prme p such that each of jp+k-j s a prme[6].let k be a gve prme,jp^+k-j(j=,...,k-),we prove t has ftely may prme solutos ad ftely may prme solutos[7]. ********************************************************************* rme dstrbuto s regularty J ( ) + ω rather tha probablty /log to see formula(8)
************************************* Jag s fucto J ( ) + ω prme dstrbuto Chu-Xua Jag 中国航天科工集团理论部. O. Box 94, Bejg 00854,. R. Cha jcxua@sa.com Abstract We defe that prme equatos f (,, ),, f (, ) (5) k are polyomals (wth teger coeffcets) rreducble over tegers, where,, are all the prme. If Jag s fucto J ( ) 0 + ω = the (5)has fte prme solutos. If J ( ) 0 + ω the there are ftely may prmes,, such that f, fk are prmes. We obta a ute prme formula prme dstrbuto k + (, + ) = {,, : f,, f k are k prmes} k k J+ ( ωω ) = (deg f ) ( + o()). (8) k k+ k+ (!) φ ( ω) log = Jag s fucto s accurate seve fucto. Usg Jag s fucto we prove about 600 prme theorems [6]. Jag s fucto provdes proofs of the prme theorems whch are smple eough to uderstad ad accurate eough to be useful. Mathematcas have tred va to dscover some order the sequece of prme umbers but we have every reaso to beleve that there are some mysteres whch the huma md wll ever peetrate. Leohard Euler
It wll be aother mllo years, at least, before we uderstad the prmes. aul Erdös Suppose that Euler totet fucto φω ( ) = Π( ) = as ω, () where ω = Π s called prmoral. Suppose that ( ω, h ) =, where =,, φω ( ). We have prme equatos = ω+,, = ω+ h () φω ( ) φω ( ) where = 0,,,. ()s called ftely may prme equatos (IME). Every equato has ftely may prme solutos. We have ( ) h = ( o()). = +, () φω ( ) h(mod ω) where h deotes the umber of prmes = ω+ h = 0,,,, ( ) the umber of prmes less tha or equal to. We replace sets of prme umbers by IME. () s the fudametal tool for provg the prme theorems prme dstrbuto. Let ω = 0 ad φ (0) = 8. From () we have eght prme equatos = 0+, = 0+ 7, = 0+, 4 = 0+, 5 = 0+ 7, 6 = 0+ 9, 7 = 0+, 8 = 0+ 9, = 0,,, (4) Every equato has ftely may prme solutos. THEOREM. We defe that prme equatos f (,, ),, f (,, ) (5) k are polyomals (wth teger coeffcets) rreducble over tegers, where,, are prmes. If Jag s fucto J + ( ω) = 0 the (5) has fte prme solutos. If J + ( ω) 0 the there exst ftely may prmes,, such that each f k s a prme. ROOF. Frstly, we have Jag s fucto [-] J ( ω) = Π[( ) χ( )], (6) + where χ ( ) s called seve costat ad deotes the umber of solutos for the followg specal cogruece k Π f ( q,, q ) 0 (mod ), (7) = where q =,,,, q =,,. J ( ) + ω deotes the umber of sets of,, prme equatos such that 4
f(,, ),, fk(,, ) are prme equatos. If J ( ) 0 + ω = the (5) has fte prme solutos. If J ( ) 0 + ω usg χ ( ) we sft out from () prme equatos whch ca ot be represeted,, the resdual prme equatos of () are,,, prme equatos such that f (,, ),, fk(,, ) are prme equatos. Therefore we prove that there exst ftely may prmes,, such that f(,, ),, fk(,, ) are prmes. Secodly, we have the best asymptotc formula [,,4,6] k + (, + ) = {,, : f,, f k are k prmes} k k J+ ( ωω ) = (deg f ) ( + o()). (8) k k+ k+ (!) φ ( ω) log = ( 8 ) s called a ute prme formula prme dstrbuto. Let =, k = 0, J ( ω) φω ( ) =. From (8) we have prme umber theorem (, ) = { : s prme } = ( + o()).. (9) log umber theorsts beleve that there are ftely may tw prmes, but they do ot have rgorous proof of ths old cojecture by ay method. All the prme theorems are cojectures except the prme umber theorem, because they do ot prove that prme equatos have ftely may prme solutos. We prove the followg cojectures by ths theorem. Example. Tw prmes +, (00BC). From (6) ad (7) we have Jag s fucto J ( ω) = Π( ) 0. (0) Sce J ( ) 0 ω () exst ftely may prme equatos such that + s a prme equato. Therefore we prove that there are ftely may prmes such that + s a prme. Let ω = 0 ad J (0) =. From (4) we have three prme equatos = 0+, = 0+ 7, = 0+ 9. 5 8 From (8) we have the best asymptotc formula J( ωω ) (, ) = { : + prme } = ( o()) φ ( ω) log + = Π ( o()). + ( ) log () 5
I 996 we proved tw prmes cojecture [] Remark. J ( ω ) deotes the umber of prme equatos, ω ( + o()) φ ( ω) log the umber of solutos of prmes for every prme equato. Example. Eve Goldbach s cojecture = +. Every eve umber 6 s the sum of two prmes. From (6) ad (7) we have Jag s fucto J( ω) = Π( ) Π 0. () Sce J ( ω) 0 as () exst ftely may prme equatos such that s a prme equato. Therefore we prove that every eve umber 6 s the sum of two prmes. From (8) we have the best asymptotc formula J( ωω ) (, ) = {, prme } = ( o()). φ ( ω) log + = Π ( o()) Π +. () ( ) log I 996 we proved eve Goldbach s cojecture [] Example. rme equatos, +, + 6. From (6) ad (7) we have Jag s fucto J ( ω) = Π( ) 0, 5 J ( ) ω s deotes the umber of prme equatos such that + ad + 6 are prme equatos. Sce J ( ) 0 ω () exst ftely may prme equatos such that + ad + 6 are prme equatos. Therefore we prove that there are ftely may prmes such that + ad + 6 are prmes. Let ω = 0, J(0) =. From (4) we have two prme equatos 5 = 0+, = 0+ 7. From (8) we have the best asymptotc formula J ( ω) ω (,) = { : +, + 6are prmes} = ( + o()). (4) φ ( ω) log Example 4. Odd Goldbach s cojecture = + +. Every odd umber 9 s the sum of three prmes. From (6) ad (7) we have Jag s fucto 6
J( ω) = Π ( + ) ) Π 0. (5) + Sce J ( ω) 0 as () exst ftely may pars of ad prme equatos such that s a prme equato. Therefore we prove that every odd umber 9 s the sum of three prmes. From (8) we have the best asymptotc formula J( ωω ) (, ) = {, : prme } = ( o()) φ ( ω) log +. = Π + ( o()) Π +. (6) ( ) + log Usg very complex crcle method Helfgott deduces the Hardy-Lttlewood formula of three prme problem[0,],but Hardy-Lttlewood-Vogradov-Helfgott do ot prove that every odd umber >7 s the sum of three prme umbers.therefore ther proofs are wrog. Example 5. rme equato = +. From (6) ad (7) we have Jag s fucto ( ) J ( ω) = Π + 0 (7) J ( ω ) deotes the umber of pars of ad prme equatos such that s a prme equato. Sce J ( ω) 0 () exst ftely may pars of ad prme equatos such that s a prme equato. Therefore we prove that there are ftely may pars of prmes ad such that s a prme. From (8) we have the best asymptotc formula J( ωω ) (,) = {, : + prme } = ( o()). 4 φ ( ω) log + (8) ote. deg ( ) =. Example 6 []. rme equato = +. From (6) ad (7) we have Jag s fucto J, (9) ( ω) = Π ( ) χ( ) 0 where χ ( ) = ( ) f χ ( ) = otherwse. (mod ) ; χ ( ) = 0 f / (mod ) ; Sce J ( ω) 0 () there are ftely may pars of ad prme equatos such that s a prme equato. Therefore we prove that there are ftely may pars of prmes ad such that s a prme. 7
From (8) we have the best asymptotc formula ( J ( ω) ω,) = {, : + prme} = ( + o()). (0) 6φ ( ω) log 4 Example 7 []. rme equato = + ( + ). From (6) ad (7) we have Jag s fucto J( ω) = Π ( ) χ( ) 0 () where χ ( ) = ( ) f (mod 4) ; χ ( ) = ( ) f (mod 8) ; χ ( ) = 0 otherwse. Sce J ( ω) 0 () there are ftely may pars of ad prme equatos such that s a prme equato. Therefore we prove that there are ftely may pars of prmes ad such that s a prme. From (8) we have the best asymptotc formula J( ωω ) (,) = {, : prme } = ( o()). 8 φ ( ω) log + () Example 8 [4-0]. Arthmetc progressos cosstg oly of prmes. We defe the arthmetc progressos of legth k., = + d, = + d,, = + ( k ) d, ( d, ) =. () k From (8) we have the best asymptotc formula (,) = { :, + d,, + ( k ) d are prmes} ( ) k J ωω = ( + o()).. (4) k k φ ( ω) log If J ( ω ) = 0 the () has fte prme solutos. If J ( ) 0 ω the there are ftely may prmes such that,, k are prmes. To elmate d from () we have j =, = ( j ) ( j ), j k. (5) From (6) ad (7) we have Jag s fucto J ( ω) = Π ( ) Π( )( k+ ) 0 (6) < k k Sce J ( ω) 0 there are ftely may pars of ad prme equatos such that,, k are prme equatos. Therefore we prove that there are ftely may pars of prmes ad such that,, k are prmes. From (8) we have the best asymptotc formula { } k (,) =, : ( j ) ( j ) prme, j k 8
k J( ωω ) = ( + o()) k k k φ ( ω) log k k ( k+ ) = Π Π ( + o()). (7) k k k k < k ( ) k ( ) log Example 9. It s a well-kow cojecture that oe of,, + + s always dvsble by. To geeralze above to the k prmes, we prove the followg cojectures. Let be a square-free eve umber..,, + +, where ( + ). From (6) ad (7) we have J () = 0, hece oe of,, + + s always dvsble by. 4., +, +,, +, where 5 ( + b), b=,. 4 From (6) ad (7) we have J (5) = 0, hece oe of, +, +,, + s always dvsble by 5. 6., +, +,, +, where 7 ( + b), b=, 4. 6 From (6) ad (7) we have J (7) = 0, hece oe of, +, +,, + s always dvsble by 7. 0 4., +, +,, +, where ( + b), b=,4,5,9. 0 From (6) ad (7) we have J () = 0, hece oe of, +, +,, + s always dvsble by. 5., +, +,, +, where ( + b), b=,6,7,. From (6) ad (7) we have J () = 0, hece oe of, +, +,, + s always dvsble by. 6 6., +, +,, +, where 7 ( + b), b=,5,6,7,0,,,4,5. 6 From (6) ad (7) we have J (7 ) = 0, hece oe of, +, +,, + s always dvsble by 7. 8 7., +, +,, +, where 9 ( + b), b= 4,5,6,9,6.7. 8 From (6) ad (7) we have J (9) = 0, hece oe of, +, +,, + s always dvsble by 9. Example 0. Let be a eve umber.., +, =,,5,,k+, 9
From (6) ad (7) we have J ( ) 0 ω. Therefore we prove that there exst ftely may prmes such that, + are prmes for ay k.., +, =, 4,6,, k. From (6) ad (7) we have J ( ) 0 ω. Therefore we prove that there exst ftely may prmes such that, + are prmes for ay k. Example. rme equato = + From (6) ad (7) we have Jag s fucto J ( ω) = Π( + ) 0. (8) Sce J ( ω) 0 () there are ftely may pars of ad prme equatos such that s prme equatos. Therefore we prove that there are ftely may pars of prmes ad such that s a prme. From (8) we have the best asymptotc formula J( ωω ) (,) = {, : prme } = ( o()). φ ( ω) log + (9) I the same way we ca prove = + whch has the same Jag s fucto. Jag s fucto s accurate seve fucto. Usg t we ca prove ay rreducble prme equatos prme dstrbuto. There are ftely may tw prmes but we do ot have rgorous proof of ths old cojecture by ay method []. As strog as the umercal evdece may be, we stll do ot eve kow whether there are ftely may pars of tw prmes []. All the prme theorems are cojectures except the prme umber theorem, because they do ot prove the smplest tw prmes. They cojecture that the prme dstrbuto s probablty[-8,-5,8-47]. Refereces [] Chu-Xua Jag, O the Yu-Goldbach prme theorem, Guagx Sceces (Chese) (996), 9-. [] Chu-Xua Jag, Foudatos of Satll s soumber theory, art I, Algebras Groups ad Geometres, 5(998), 5-9. [] ChuXua Jag, Foudatos of Satll s soumber theory, art II, Algebras Groups ad Geometres, 5(998), 509-544. [4] Chu-Xua Jag, Foudatos Satll s soumber theory, I: Fudametal ope problems sceces at the ed of the mlleum, T. Gll, K. Lu ad E. Trell (Eds) Hadroc ress, USA, (999), 05-9. [5] Chu-Xua Jag, roof of Schzel s hypothess, Algebras Groups ad Geometres, 8(00), 4-40. [6] Chu-Xua Jag, Foudatos of Satll s somuber theory wth applcatos to ew cryptograms, Fermat s theorem ad Goldbach s cojecture, Iter. Acad. ress, 00, MR004c: 00, http://www.-b-r.org/docs/jag.pdf [7] Chu-Xua Jag, rme theorem Satll s soumber theory,algebras Groups ad Geometres, 9(00), 475-494. 0
[8] Chu-Xua Jag, rme theorem Satll s soumber theory (II), Algebras Groups ad Geometres, 0(00), 49-70. [9] Chu-Xua Jag, Dsproof s of Rema s hypothess, Algebras Groups ad Geometres, (005), -6. http://www.-b-r.org/docs/jag Rema.pdf [0] Chu-Xua Jag, Fftee cosecutve tegers wth exactly k prme factors, Algebras Groups ad Geometres, (006), 9-4. [] Chu-Xua Jag, The smplest proofs of both arbtrarly log arthmetc progressos of prmes, preprt, 006. [] D. R. Heath-Brow, rmes represeted by -84. x + y, Acta Math., 86 (00), 4 [] J. Fredlader ad H. Iwaec, The polyomal x + y captures ts prmes, A. Of Math., 48(998), 945-040. [4] E. Szemeréd, O sets of tegers cotag o k elemets arthmetc progressos, Acta Arth., 7(975), 99-45. [5] H. Fursteberg, Ergodc behavor of dagoal measures ad a theorem of Szemeréd o arthmetc progressos, J. Aalyse Math., (997), 04-56. [6] T. Gowers,Hypergraph regularty ad the multdmesoal Szemered theorem,a. of Math.,66(007),897-946. [7] T.Gowers,A ew proof of Szemered theorem,gafa,(997),465-588. [8] A.Odlyzko,M.Rubste ad M.Wolf,Jumpg Champos,Expermet Math.8,(999),07-8. [9] B. Kra, The Gree-Tao theorem o arthmetc progressos the prmes: A ergodc pot of vew, Bull. Amer. Math. Soc., 4(006), -. [0] B. Gree ad T. Tao, The prmes cota arbtrarly log arthmetc progressos, A.of Math., 67(08), 48-547. [] T. Tao, The dchotomy betwee structure ad radomess, arthmetc progressos, ad the prmes, I: roceedgs of the teratoal cogress of mathematcas (Madrd. 006), Europ. Math. Soc. Vol. 58-608, 007. [] B. Gree, Log arthmetc progressos of prmes, Clay Mathematcs roceedgs Vol. 7, 007,49-59. [] H. Iwace ad E. Kowalsk, Aalytc umber theory, Amer. Math. Soc., rovdece, RI, 004 [4] R. Cradall ad C. omerace, rme umbers a computatoal perspectve, Sprg-Verlag, ew York, 005. [5] B. Gree, Geeralsg the Hardy-Lttlewood method for prmes, I: roceedgs of the teratoal cogress of mathematcas (Madrd. 006), Europ. Math. Soc., Vol. II, 7-99, 007. [6] K. Soudararaja, Small gaps betwee prme umbers: The work of Goldsto-tz-Yldrm, Bull. Amer. Math. Soc., 44(007), -8. [7] A. Gravlle, Harald Cramér ad dstrbuto of prme umbers, Scad. Actuar. J, 995() (995), -8. [8] Ytag Zhag,Bouded gaps betwee prmes, A.of Math., 79(04)-74.
[9] Chu-Xua Jag,The Hardy-Lttlewood prme k-tuple cojecture s false. http://vxra.org/pdf/00.04v.pdf. [0] H.A.Helfgott,Major arcs for Goldbach problem, http://arxv.org/pdf/05.897v.pdf [] H.A.Helfgott,Mor arcs for Goldbach problem, http://arxv.org/pdf/05.55v.pdf [] http://vxra.org/author/chu-xua_jag http://vxra.org/pdf/0.0050v.pdf [] T.Tao,Every odd umber greater tha s the sum of at most fve prmes, http://arxv.org/pdf/0.6656v.pdf,to Math.Comp 8(04),997-08. [4] G.H.Hardy ad J.E.Lttlewood,Some problems of artto umerorum ;III:O the expresso of a umber as a sum of prmes,acta Math.,44(9),-70. [5].Erdos ad.tura,o some sequeces of tegers,j.lodo Math.Soc.,(96),6-64. [6] Chu-Xua Jag,The ew prme theorem (5), http://vxra.org/pdf/004.00v.pdf [7] Chu-Xua Jag,The ew prme theorem (4), http://vxra.org/pdf/004.0v.pdf [8]D.Goldsto,J.tz ad C.Yldrm, rmes tuples I, A. of Math.,70(009),89-86. [9] D.Goldsto,Y.Motohash,J.tz,ad C.Yldrm,Small gaps betwee prmes exst,roc.japa Acad.Ser.A Math.Sc,8(006),6-65. [40] D.Goldsto,S.Graham,J.tz,ad Y.Yldrm,Small gaps betwee products of two prmes,roc Lodo Math.Soc.()98(009),74-774.. [4] B.Gree ad T.Tao,Lear equatos prmes,a.of Math.,7(00),75-850. [4] J.Bourga,A.Gamburd ad.sarak,affe lear seve,expaders,ad sum-product,ivet Math,79(00),559-644.Bourga 获 00 年邵逸夫数学奖,Sarak 获 04 年邵逸夫数学奖. 这都是丘成桐推荐的 [4] M.I.Vogradov,Represetatos of a odd umber as a sum of three prmes,dokl.akad.auk SSSR 5(97),9-94. [44] T.Tao ad V.Vu,Addtve combatorcs, Cambrdge Uversty ress.cambrdge(006). [45] B.L.va der Waerde,Bewes eer Baudetsche Vermutug, euw Arch.Wsk.,5(97),-6. [46] B.Host ad B.Kra,Covergece of polyomal ergodc averages,israel J.Math,49(005),-9. [47]B.Host ad B.Kra,ocovetoal ergodc averages ad lmafolds,a of Math,6(005),97-488. [48] A.Wles,Modular ellptc curves ad Fermat last theorem, A.of Math.,4(995),44-55. [49] R.Taylor ad A.Wles,Rg-theoretc propertes of certa Hecke algebras,a of Math,4(995),55-57. [50]D.H.J.olymath,ew equdstrbuto estmates of Zhag type,ad bouded gaps
betwee prmes,http://arxv.org/pdf/40.08.pdf. [5] Chu-Xua Jag,Dsproofs of Rema hypothess,algebras Groups ad Geometres,-6(005).http://www.-b-r.org/docs/JagRema.pdf http://www.mrelatvty.et/apers/45/jagrema.pdf; http://www.vxra.org/pdf/004.008v.pdf; http://www.doc.com/p-4759967.html 995 年我们用新方法证明了 tw prmes theorem ad Goldbach theorem[].995 年 0 月 8-0 日参加首届全国 [ 余新河数学题 ] 研讨会 我论文排在第一位, 中科院组织会议不允许我发言, 以后文集没我论文,996 年在 [ 广西科学 ] 上发表, 中科院去信不允许发表, 但文章巳印好, 最后在 [ 证明 ] 贴上 [ 探讨 ] 发表. 中国一篇划时代论文在中国这样悲惨遭遇. 以后在美国多次发表, 至今无人反驳和否定 从 (7) 我们使用一种特殊同余式 f(q) 三 0 (mod p) q=,...,p-; 不使用 q=,...,p-,p 共有 p 个元素, 这是过去所有数论中没有的, 这样我们创立新素数理论 这是 Euler fucto 推广, 因为 p 的 Euler 函数互素只有 p- 个,Jag fucto 作建立 ISO 数学, 中科院吓坏了, 用保存在 Euler fucto 中与研究素数方程有关的数,Euler fucto ad Jag fucto 都是研究素数的工具, 这一点是统一的 00 年 0 月 5 日科技日报头版报道蒋春暄证明哥德巴赫猜想, 证明费马大定理, 否定黎曼假设和改组科技日报, 下令不允许再报道蒋春暄工作. 蒋春暄母校北京航空航天大学校长沈士团于 00-- 和 0-0-8 召开两次会议邀请蒋春暄去北航成立数学小组, 展开蒋春暄开创工作研究工作, 新校长李未上台, 坚决反对蒋春暄去北航工作, 从北大中科院调干部去北航工作, 死死控制北航, 在北航成立华罗庚学习班. 这样完成整个中国对蒋春暄全面封杀. 中国只能宣传陈景润 +, 出书 [ 从哥德巴赫到陈景润 From Goldbach to Chejgru], 中国不承认蒋春暄正确素数理论, 外国也不承认蒋春暄正确素数理论 但他们都在读蒋春暄的书和论文 目前国内外数学杂志没有素数论文, 无人证明 tw prmes ad Goldbach cojecture. 这种不死不活场面还要继续下去, 我们继续宣传本文 从 Hardy(9) 到 94 年 90 年发表的素数论文都是错的 GY do ot prove that rmes tuples are admssble ad admssble.gy papers are 00% wrog, Ytag Zhag paper also s wrog[8]. O the sgular seres the Jag prme k-tuples theorem Chu-Xua Jag. O. Box 94, Bejg 00854,. R. Cha jcxua@sa.com Abstract
Usg Jag fucto we prove Jag prme k -tuples theorem.we fd true sgular seres. Usg the examples we prove the Hardy-Lttlewood prme k -tuples cojecture wth wrog sgular seres.. Jag prme k -tuples theorem wll replace the Hardy-Lttlewood prme k -tuples cojecture. (A) Jag prme k -tuples theorem wth true sgular seres[, ]. We defe the prme k -tuples equato p p+, (), where, =, k. we have Jag fucto [, ] J ( ) ( ( )) ω =Π χ, () where ω = Π, χ ( ) s the umber of solutos of the followg specal cogruece whch s true. k Π ( q+ ) 0 (mo d), q =,, p. () = If χ ( ) < the J ( ) 0 ω. There exst ftely may prmes such that each of + s prme. If χ ( ) = the J ( ) 0 ω =. There exst ftely may prmes such that each of + s prme. J ( ) ω s a subset of Euler fucto φω ( )[]. If J ( ) 0 ω, the we have the best asymptotc formula of the umber of prme [, ] (,) { : } ~ ( ) k J ωω k prme C( k) k k k = + = = (4) φ ( ω) log log φω ( ) =Π( ) + χ( ) Ck ( ) =Π s Jag true sgular seres. Example. Let k =,, +, tw prmes theorem. From () we have k (5) χ() = 0, χ( ) = f >, (6) Substtutg (6) to () we have 4
J ( ω) = Π( ) 0 (7) There exst ftely may prmes such that + s prme. Substtutg (7) to (4) we have the best asymptotc formula k (, ) = { : + = prme} ~ Π( ). ( ) log (8) Example. Let k =,, +, + 4. From () we have From () we have χ() = 0, χ() = (9) J ( ω ) = 0. (0) It has oly a soluto =, + = 5, + 4= 7. Oe of, +, + 4 s always dvsble by. Example. Let k= 4,, +, where =,6,8. From () we have Substtutg () to () we have χ( = )0, χ() =, χ( ) = f >. () J ( ω) = Π( 4) 0, () 5 There exst ftely may prmes such that each of + s prme. Substtutg () to (4) we have the best asymptotc formula 7 ( 4) 4(,) = { : + = prme} ~ Π 5 ( ) 4 log 4 () Example 4. Let k = 5,, +, where =,6,8,. From () we have Substtutg (4) to () we have χ( = )0, χ() =, χ(5) =, χ( ) = 4 f > 5 (4) J ( ω) = Π( 5) 0 (5) 7 There exst ftely may prmes such that each of + s prme. Substtutg (5) to (4) we have the best asymptotc formula 4 4 5 ( 5) 5(,) = { : + = prme} ~ Π 7 ( ) 5 log 5 (6) 5
Example 5. Let k = 6,, +, where =,6,8,,4. From () ad () we have χ() = 0, χ() =, χ(5) = 4, J (5) = 0 (7) It has oly a soluto = 5, + = 7, + 6 =, + 8 =, + = 7, + 4 = 9. Oe of + s always dvsble by 5. (B)The Hardy-Lttlewood prme k -tuples cojecture wth wrog sgular seres[-6]. Ths cojecture s geerally beleved to be true, but has ot bee proved(odlyzko et al.jumpg champo,expermet math,8(999),07-8). We defe the prme k -tuples equato where, =,, k., + (8) I 9 Hardy ad Lttlewood cojectured the asymptotc formula where k(,) = { : + = prme} ~ H ( k), (9) log k ν ( ) Hk ( ) =Π s Hardy-Lttlewood wrog sgular seres, k (0) ν ( ) s the umber of solutos of cogruece whch s wrog. k Π ( q+ ) 0 (mo d), q=,,. () = From () we have ν ( ) < ad Hk ( ) 0. For ay prme k -tuples equato there exst ftely may prmes such that each of + s prme, whch s false. Cojecture. Let k =,, +, tw prmes theorem From () we have ν ( ) = () Substtutg () to (0) we have H () = Π () 6
Substtutg () to (9) we have the asymptotc formula (,) = { : + = prme} ~ Π log (4) whch s wrog see example. Cojecture. Let k =,, +, + 4. From () we have Substtutg (5) to (0) we have ν() =, ν( ) = f > (5) H () = 4 Π ( ) ( ) (6) Substtutg (6) to (9) we have asymptotc formula ( ) ( ) log (,) = { : + = prme, + 4 = prm} ~ 4 Π whch s wrog see example. Cojecture. Let k = 4,, +, where =,6,8. (7) From () we have ν( = ), ν() =, ν( ) = f > (8) Substtutg (8) to (0) we have H (4) = Π 7 ( ) > 4 ( ) (9) Substtutg (9) to (9) we have asymptotc formula 7 ( ) 4 = { + = prme} Π > 4 4 (,) : ~ ( ) log Whch s wrog see example. Cojecture 4. Let k = 5,, +, where =,6,8, (0) From () we have ν( = ), ν() =, ν(5) =, ν( ) = 4 f > 5 () Substtutg () to (0) we have 4 4 5 ( 4) H (5) = Π 4 ( ) 5 > 5 5 () 7
Substtutg () to (9) we have asymptotc formula 4 4 5 ( 4) 5 = { + = prme} Π 5 > 5 5 5 (,) : ~ 4 ( ) log Whch s wrog see example 4. Cojecture 5. Let k = 6,, +, where =,6,8,,4. () From () we have ν() =, ν() =, ν(5) = 4, ν( ) = 5 f > 5 (4) Substtutg (4) to (0) we have 5 ( 5) H (6) = Π 5 6 > ( ) 5 5 (5) Substtutg (5) to (9) we have asymptotc formula 5 5 5 ( 5) 6(,) = { : + = prme} ~ Π > 5 ( ) 6 log 6 (6) whch s wrog see example 5. Cocluso.From Hardy-Lttlewood(9) to 04 all prme papers are wrog. The Jag prme k-tuples theorem has true sgular seres.the Hardy-Lttlewood prme k -tuples cojecture has wrog sgular seres. The tool of addtve prme umber theory s bascally the Hardy-Lttlewood wrog prme k-tuples cojecture [-5]. Usg Jag true sgular seres we prove almost all prme theorems. Jag prme k -tuples theorem wll replace Hardy-Lttlewood prme k -tuples cojecture. There caot be really moder prme theory wthout Jag fucto. Refereces [] Chu-Xua Jag, Foudatos of Satll s soumber theory wth applcatos to ew cryptograms, Fermat s theorem ad Goldbach s cojecture. Iter. Acad. ress, 00,MR004c:00,(http://www.-b-r.org/docs/jag.pdf) (http://www.wbab.et/math/xua. pdf). [] Chu-Xua Jag, Jag s fucto J ( ) + ω prme dstrbuto. (http:// www. wbab. et/math/ xua. pdf) (http://vxra.org/pdf/08.0004v.pdf) 8
[] G. H. Hardy ad J. E. Lttlewood, Some problems of artto umerorum, III: O the expresso of a umber as a sum of prmes, Acta Math, 44(9), -70. [4] B. Gree ad T. Tao, The prmes cota arbtrarly log arthmetc progressos, A. Math., 67(008), 48-547. [5] D. A. Goldsto, S. W. Graham, J. tz ad C. Y. Yldrm, Small gaps betwee products of two prmes, roc. Lodo Math. Soc., () 98 (009) 74-774. [6] D. A. Goldsto, S. W. Graham, J. tz ad C. Y. Yldrm, Small gaps betwee prmes or almost prmes, Tras. Amer. Math. Soc., 6(009) 585-50. [7] D. A. Goldsto, J. tz ad C. Y. Yldrm, rmes tulpes I, A.of Math., 70(009) 89-86. [8]. Rbebom, The ew book of prme umber records, rd edto, Sprger-Verlag, ew York, Y, 995. 409-4. [9] H.Halberstam ad H.-E.Rchert,Seve methods, Academc ress,974. [0] A.Schzel ad W.Serpsk, Sur certaes hypotheses cocerat les ombres premers,acta Arth.,4(958)85-08. [].T.Batema ad R.A.Hor,A heurstc asymptotc formula cocerg the dstrbuto of prme umbers,math.comp.,6(96)6-67 [] W.arkewcz,The developmet of prme umber theory,from Eucld to Hardy ad Lttlewood,Sprger-Verlag,ew York,Y,000,-5. [] B.Gree ad T.Tao,Lear equatos prmes, A.of Math.7(00)75-850. [4] T.Tao,Recet progress addtve prme umber theory, http://terrytao.fles.wordpress.com/009/08/prme-umber-theory.pdf [5]Ytag Zhag,Bouded gaps betwee prmes,a.of Math.,.79(04)-74. [6] D.H.J.olymath,ew equdstrbuto estmates of Zhag type,ad bouded gaps betwee prmes.http://arxv.org/pdf/40.08.pdf. The ew rme theorem(5) 9
, j + k j( j =,, k ) Chu-Xua Jag. O. Box 94, Bejg 00854,. R. Cha jagchuxua@vp.sohu.com Abstract Usg Jag fucto we prove that there exst ftely may prmes such that each j + k j s a prme. Theorem. Let k be a gve prme., j + k j( j =,, k ) () There exst ftely may prmes such that each of j + k j s a prme. roof. We have Jag fucto[] where ω = Π, χ ( ) s the umber of solutos of cogruece q =,,. k j= J ( ) [ ( )] ω =Π χ, () Π ( jq + k j) 0 (mo d), () From () we have χ () = 0, f < k the χ ( ) =, χ ( k) =, f k < the χ ( ) = k. From () ad () we have J ( ) ( ) ( ) 0 ω = k Π k. (4) We prove that there exst ftely may prmes such that each of j + k j s a prme We have the asymptotc formula [] (,) { : } ~ ( ) k J ωω k = j + k j = prme, (5) k k φ ( ω) log k< where φω ( ) =Π ( ). Referece [] Chu-Xua Jag, Jag s fucto J ( ) + ω prme dstrbuto. http://www. wbab.et/math /xua. pdf. 0
All zeros of all zeta fuctos are wrog.0abel prze s wrog Rema s Hypothess ad Cojecture of Brch ad Swerto-Dyer are False Chu-Xua Jag. O. Box 94, Bejg 00854 Cha jcxua@sa.com Abstract All eyes are o the Rema s hypothess, zeta ad L-fuctos, whch are false, read ths paper. The Euler product coverges absolutely over the whole complex plae. Usg factorzato method we ca prove that Remam s hypothess ad cojecture of Brch ad Swerto-Dyer are false. All zero computatos are false, accurate to sx decmal places. Rema s zeta fuctos ad L fuctos are useless ad false mathematcal tools. Usg t oe caot prove ay problems umber theory. Euler totet fucto φ ( ) ad Jag s fucto J ( ) + ω wll replace zeta ad L fuctos.all Rema hypothess cludg Wel Rema hypothess are false.. Itroducto The fucto ζ () s defed by the absolute coverget seres ζ () s = s () complex half-plae Re () s > s called the Rema s zeta fucto. The Rema s zeta fucto has a smple pole wth the resdue at s = ad the fucto =
ζ () s s aalytcally cotued to whole complex plae. We the defe the ζ () s by the Euler product s, () ζ ( s) = ( ) where the product s take all prmes, s = σ + t, =, σ ad t are real. The Rema s zeta fucto ζ () s has o zeros Re () s >. The zeros of ζ () s 0 < Re () s < are called the otrval zeros. I 859 G. Rema cojectured that every zero of ζ () s would le o the le Re ( s ) = /. It s called the Reme s hypothess. [] We have We defe the ellptc curve [] where D s the cogruet umber. ζ( s = σ + t, σ ) 0 () ED : y x Dx =, (4) Assume that D s square-free. Let be a prme umber whch does ot dvde D. Let deote the umbers of pars ( xy, ) where x ad y ru over the tegers modulo, whch satsfy the cogruece ut y x Dx mod. (5) a = (6) We the defe the L fucto of E D by the Euler product s s (7) (, D) = LE (, s) = ( a + ) D where the product s take over all prmes whch do ot dvde D. The Euler product coverges absolutely over the half plae Re ( s ) > /, but t ca be aalytcally cotued over the whole complex plae. For ths fucto, t s the vertcal le Re () s = whch plays the aalogue of the le Re ( s ) = / for the Rema zeta fucto ad the Drchlet L fuctos. Of course, we beleve that every zero of LE ( D, s ) Re () s > 0 should le o the le Re () s =. It s called a cojecture of Brch ad Swerto-Dyer (BSD). We have. Rema s Hypothess s false L( E, s = σ + t, σ / ) 0 (8) D Theorem. Euler product coverges absolutely Re () s >. Let s 0 = /+ t, usg
factorzato method we have ζ ( s = / + t) 0 (9) 0 roof. Let s= s0,. s0,.8 s0, s0, 4 s0, 5 s0, s 0 0 We have the followg Euler product equatos s0 ζ( s0) = ζ( s0) ( + ) 0, (0).s0.s 0 ζ(. s0) = ζ( s0) ( + ) 0, () s0.8s0.8s 0 ζ(.8 s0) = ζ( s0) ( + ) 0, () s0 ζ ζ ζ ζ, () 0 0 ( 0) ( 0) ( s s s = ζ s + + ) 0, (4) 0 0 (4 0) ( 0) ( s s s = ζ s + ) ( + ) 0, (5) 0 0 0 4 0 (5 0) ( 0) ( s s s s ζ s s = + + + + ) 0 ( s 0 0) = ζ( s0) ( 0 ) s0 ( + + ) 0, (6) Sce the Euler product coverges absolutely Re () s >, the equato (0)-(6) are true. From (0)-(6) we obta ζ ( s ) 0 (9) 0 All zero computatos are false ad approxmate, accurate to sx decmal places. Usg three methods we proved the RH s false []. Usg the same Method we are able to prove that all Rema s hypotheses also are false. All L fuctos are false ad useless for umber theory.. The Cojecture of Brch ad Swerto-Dyer s false. Theorem. Euler product coverges absolutely Re ( s ) > /. Let s = + t. Usg factorzato method we have roof. Let s = s, s,4 s, we have the followg Euler product equatos. L( E, s = + t) 0 (7) D s s s ( a + ) + a LE ( D, s) = LE ( D, s) + 0 s s (, D) = a + (8)
s 4s 5s 4s a + a LE ( D, s) = LE ( D, s) + 0 s s (, D) = a + (9) 4s 6s 7s 6s a + a LE ( D,4 s) = LE ( D, s) + 0 s s (, D) = a + (0) Sce the Euler product coverges absolutely Re ( s ) > /, equatos (8)-(0) are true. From (8)-(0) we obta LE (, s) 0 (7) D All zero computatos are false ad approxmate. Usg the same method we are able to prove all LEs (, ) 0 whole complex plae. The ellptc curves are ot related wth the Dophate equatos ad umber theory [4]. Frey ad Rbet dd ot prove the lk betwee the ellptc curve ad Fermat s equato [4,5]. Wles proved Tayama-Shmura cojecture based o the works of Frey, Serre, Rbet, Mazuer ad Taylor, whch have othg to do wth Fermat s last theorem [6]. Tayama-Shmura cojecture was obscurty for about 0 years tll people serously started thkg about ellptc curves. Mathematcal proof does ot proceed by persoal abuse, but by show careful logcal argumet. Wles proof of Fermat s last theorem s false [7-9]. I 99 Jag proved drectly Fermat s last heorem [0,]. 4. Cocluso. The zero computatos of zeta fuctos ad L fuctos are false. Rema s zeta fuctos ad L fuctos are useless ad false mathematcal tools. Usg t oe caot prove ay problems umber theory []. The heart of Laglads program(l) s the L fuctos []. Therefore L s false. Wles proof of Fermat last theorem s the frst step L. Usg L oe caot prove ay problems umber theory, for example Fermat s last theorem [6]. Euler totet fucto φ ( ) ad Jag s fucto J ( ) + ω wll replace Rema s zeta fuctos ad L fuctos [-5]. Refereces [] B. Rema, Uber de Azahl der rmzahle uder eer gegebeer GrÖsse, Moatsber. Akad. Berl. 67-680 (859) [] Joh Coates, umber theory, Acet ad Moder, I: ICCM 007. vol. I, -. [] Chu-Xua Jag, Dsproofs of Rema s hypothess. Algebras, Groups ad Geometres,, -6(005) http://www.-b-r.org/docs/jagrema.pdf; http://vxra.org/abs/004.008; http://www.wbab.et/math/xua9.pdf [4] G. Frey, Lks betwee stable ellptc curves ad certa dophate equatos, Aales Uverstats Saravess (986), -40. [5] K. A. Rbet, O modular represetatos of Gal ( Q/ Q ) arsg from modular forms, Ivet. 4
Math. 00 (990), 4-476. [6] A. Wles, Modular ellptc curves ad Fermat s last theorem, A. of Math. 4(995), 44-55. [7] G. erelma, erelma dsproves Wles proof of Fermat s theorem. http://jouralsm.cedar-falls.k.a.us/pctures/4./etzartcle.pdf. http://documetsearch.org/pdf/perelma-dsproves-wles-proof-of [8] Y. G. Zhvotov, Fermat s last theorem ad Keeth Rbet s mstakes, http://www.wbab.et/fles/448_zhvotov.pdf; http://ewrotor.arod.ru/eglsh. html. [9] Y. G. Zhvotov, Fermat s last theorem ad mstakes of Adrew Wles, http://www.ewrotor.arod.ru/fermat.html http://www.readfree.et/bbs/read.php?td=44659 [0] Chu-Xua Jag, Automorphc fucto ad Fermat s last theorem () http://www.wbab.et/tham/xua67.pdf http://www.vxra.org/pdf/0.0007v.pdf [] Chu-Xua Jag, Automorphc fucto ad Fermat s last theorem ()(Fermat s proof of FLT), http://www.wbab.et/tham/xua69.pdf http://www.vxra.org/pdf/0.0009v.pdf [] Arthur, et al., edtors, O certa L fuctos. AMS, CMI. 0. volume. [] S. Gelbart, A elemetary troducto to the Laglads program, Bull, of AMS. 0(984) 77-9. [4] Chu-Xua Jag, The Hardy-Lttlewood prme k tuple cojecture s false. (http://www.wbab.et/math/xua77.pdf); (http://vxra.org/pdf/00.04v.pdf) [5] Chu-Xua Jag, Jag s fucto J ( ) + ω prme dstrbuto, (http://www.wbab.et/math/xua.pdf) (http://www.vxra.org/pdf/08.0004v.pdf) [6] Chu-Xua Jag, Foudatos of Satll s soumber theory wth applcatos to ew cryptograms, Fermat s theorem ad Goldbach s cojecture, Iter. Acad. ress, 00. MR004c:/00. (http://www.-b-r.org/docs/jag.pdf); (http://www.wbab.et/math/xua.pdf) 最后说明 Euler product 是绝对收敛的 方程 (0) (6) 和 (8) (0) 是绝对正确的 0 世纪数论上没有重大突破, 主要大家集中力量研究黎曼假设 (RH) 利用 RH 来研究数论问题 RH 所有零点计算都是错误的, 因为无穷 ` 级数无法计算精确值, 这些错误零点计算使 00 年来所有数学家都相信 RH 是正确的, 顶尖数学家都集中力量证明 RH, 并指出 RH 是 世纪最大要解决问题 所有 RH 专家都集中在 IAS Isttute of advaced study was the udsputed Mecca of the Rema hypothess. 0 世纪所有伟大数学家都研究 RH 例如 Hbert. Hardy, Wel, If I were to awake after havg stept for fve hudred years, my frst questo would be: Has the Rema hypothess bee prove? (Davd Hbert) 如果没有蒋春暄否定 RH, 那末再过 500 年 RH 仍不能解决.RH 是数论的基础 利用 RH 许多数学家证明上千个定理, 这些定理都是错的 RH 零点是错的, 那末 L fuctos 零点也是错的 利用 zeta 函数和 5
L fuctos 不能证明数论中任何问题 从 RH 出发利用 L 函数和椭圆曲线 R. L. Laglads 提出 Laglads program(l) 把许多没有关系数学问题统一联系起来 利用中国成语 张冠李戴 办法解决所有数学问题, 怀尔斯证明费马大定理就是 L 第一次应用最大成果, 怀尔斯失败, 身败名裂, 也是 L 失败 蒋春暄否定 RH 在 AIM, CLAYMA, IAS, THES, MIM, MSRI 已无人研究 RH 和数论 但是他们仍抓住 L 函数和椭圆曲线不放继续研究, 所以他们集中力量研究 BSD 猜想 目前国内外数论专家下岗或改行 但他们对蒋春暄开创数论新时代, 他们都不讲话, 因为中国对蒋春暄数论成果不承认, 定为伪科学 以 RH 开创数论时代已结束 一个新的数论时代将开始 0 年由卢昌海著王元序 [ 黎曼猜想漫谈 The Rema hypothess(rh)] 到 04 年由清华大学出版社巳印了三次, 大力宣传 RH, 不承认蒋春暄否定 RH. 王元又在中国宣传陈氏定理 +, 不承认蒋春暄 996 年证明 +. 中华民族就这么落后. 又如何成为世界科学强国. reprt (Jauary 994). After Wles was about to aouce hs proof of FLT to the world o Jue, 99. Jag wrote ths paper. Tepper Gll, Kex Lu, ad Erc Trell, Edtors Fudametal Ope roblems Scece at the Ed of the Mleum roceedgs of the Bejg Workshop, August 997 Hadroc ress, alm Harbor, FL 468-577, U. S. A ISB -57485-09-6, pp. 555-558. Fermat Last Theorem was roved 99 Jag, Chu-xua. O. Box 94, Bejg,. R. Cha We foud out a ew method for provg Fermat last theorem (FLT) o the afteroo of 6
October 5, 99. We proved FLT at oe stroke for all prme expoets p >, It led to the dscovery to calculate = 5,,5,05,. To ths date, o oe dsprove ths proof. Ayoe ca ot dey t, because t s a smple ad marvelous proof. It ca ft the marg of Fermat book. I 974 we foud out Euler formula of the cyclotomc real umbers the cyclotomc felds []. exp tj = SJ, () = = where J deotes a - th root of uty, J =, s a odd umber, t are the real umbers. S s called the complex hyperbolc fuctos of order wth varables, where A ( ) j B j θ j j = ( ) j S = [ e + ( ) e cos( + ( ) )], () α j α j A= t, B = t ( ) cos, = ( ) t ( ) s α j j+ α j α j α θ j α α= α= α=, A+ B = 0. () = Usg () the cyclostomes theory may exted to totally real umber felds. It s called the hypercomplex varable theory []. () may be wrtte the matrx form 0 0 A S ( ) e cos s s B S e cosθ ( ) B S s cos s s e θ =, S exp( B )s( θ ) ( ) ( ) ( ) cos s s (4) where ( ) / s a eve umber. From (4) we may obta ts verse trasformato 7
A e ( ) cos cos cos S B e cosθ S B e sθ ( ) = 0 s s s S exp( B )s( θ ) S 0 s s s ( ) ( ) ( ) From (5) we have A j j e = S, e cos = S + S ( ) cos e B j θ j + = =, j s ( ) ( ) s. (6) B j j+ θ j = S + j = I () ad (6) t ad S have the same formulas such that every factor of has a Fermat equato. Assume S 0, S 0, S = 0 where =, 4,, S=. 0 are determate equatos wth varables. From (6) we have A B j j j e = S+ S, e = S + S + SS ( ) cos. (7) From () ad (7) we may obta the Fermat equato (5). exp j j A+ B j = ( S+ S) ( S + S + SS ( ) cos ) = S + S =. j= j= (8) Theorem. Fermat last theorem has o ratoal solutos wth SS 0 for all odd expoets. roof. The proof of FLT s dffcult whe s a odd prme. We cosder s a composte umber. Let = Π, where rages over all odd umber. From () we have f f f f (9) j j= f α = exp( A+ B ) = [exp( t α )] From (7) we have f f f (0) j j= f exp( A+ B ) = S + S where f s a factor of. From (9) ad (0) we may obta Fermat equato 8
f f f f f f j j= f α = () exp( A+ B ) = S + S = [exp( t α )] Every factor of has a Fermat equato. From () we have f =, B = B = 0, e = S + S = exp( t α ) () A 0 α = 0 j = f= t, = t = 0, exp( A+ B) = S + S = () α = f =,exp( A+ B ) = S + S = [exp( t α )] (4) If S =, S = 0 ad S = 0, S =, the A= B j = 0. Euler proved (), therefore () has o ratoal solutos wth SS 0 (ad so o teger solutos wth SS 0 ) for all odd expoets f. () ad () ca ft the marg of Fermat book. Let = p where p s a odd prme. From () ad (7) we may derve Fermat equtatos p p p p p j (5) = exp( A+ B ) = S + S = ( S ) + ( S ) = p α = exp( A+ Bp ) = S + S = [exp t α )] (6) p p p j p p p (7) = exp( A+ B ) = S + S = [exp( t + t )] Euler proved (5) ad (6), therefore (7) have o ratoal solutos wth SS 0 (ad so o teger solutos wth SS 0 ) for ay odd prme p >. (5)-(7) ca ft the marg Let = 5 p where p s a odd prme. From () ad (7) we may derve Fermat eqatos 5p 5p 5p j (8) j= exp( A+ B ) = S + S = p 5 5 5 p 5 α = exp( A+ Bp + B ) = S + S = [exp t α )] (9) p 4 p p p 5j p j= α = (0) exp( A+ B ) = S + S = [exp( t α )] 9
(8)-(0) ca ft the marg. Let = 7 p where p s a odd prme. From () ad (7) we may derve Fermat equatos 7p 7p 7p j () = exp( A+ B ) = S + S = p 7 7 7 p p p 7 α = exp( A+ B + B + B ) = S + S = [exp t α )] () p 6 p p p 7 j p = α = () exp( A+ B ) = S + S = [exp t α )] ()-() ca also ft the marg. Usg ths method we proved FLT 99 [-5]. a Refereces. Jag, Chu-xua. Hypercomplex varable theory, reprts, 989.. Jag, Chu-xua. Fermat last theorem has bee proved (Chese, Eglsh summary) Qa Kexue, (99)7-0. reprts (Eglsh), December, 99. (It s suffcet to prove S S + = for FLT of odd expoets).. Jag, Chu-xua. More tha 00 years ago Fermat last theorem was proved (Chese, Eglsh summary). Qa Kexue, 6(99) 8-0. (It s suffcet to prove S 4 4 S = for FLT.) 4. Jag, Chu-xua. Fermat proof for FLT. reprts (Eglsh), March, 99. 5. Jag, Chu-xua. Factorzato theorem for Fermat equato. reprts (Eglsh), May, 99. ote. Let oe kew the mportat results, we gave out about 600 preprts 99-99. There were my preprts rceto, Harvard, Berkeley, MIT, Uchcago, Columba, Marylad, Oho, Wscos, Yale,, Eglad, Caada, Japa, olad, Germay, Frace, Flad,, A. Math., Mathematka, J. umber Theory, Glasgow Math. J., Lodo Math. Soc., I. J. Math. Math. Sc., Acta Arth., Ca. Math. Bull. (They refused the publcatos of my papers). Both papers were publshed Chese. FLT s as smple as ythagorea theorem. Ths proof ca ft the marg of Fermat book. We thk the game s up. We set dept of math (rceto Uversty) a preprt o Ja. 5, 99. Wles clams the secod proof of FLT Eglad (ot U. S. A.) after two years. We wsh Wles ad hs supporters dsprove my proof, otherwse Wles work s oly the secod ad complex proof of FLT. We beleve that the rceto s the farest Uversty ad hstory wll pass the farest judgmet o proofs of FLT ad other problems. We are watg for word from the experts who are studyg ths paper. 0