From Hardy-Littlewood(1923) To 2013 All Prime Papers Are Wrong

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1 From Hardy-Lttlewood(19) To 01 All rme apers Are Wrong The Hardy-Lttlewood prme -tuples conjecture[18,9,4] and Erdos-Turan conjecture(every set of ntegers of postve upper densty contans arbtrarly long arthmetc progressons )[14,15,16,17,0,5] are wrong.usng the crcle method and the seve method one do not prove smplest twn prme conjecture( there exst nfntely many pars of twn prmes) and the smplest Goldbach conjecture (every even number >4 s the sum of of two prmes).therefore from Hardy-Lttlewood(19) to 01 all prme papers are wrong.they do not prove any prme problems. 006Felds medal(green-tao theorem[0]),007wolf prze(furstenberg theorem[15]) and 01Abel prze(szemered theorem[14]) are wrong,they do not understand arthmetc progressons.the correct arthmetc progressons s Example 8[6,p68-74].Insttute for Advanced study(math) has long been recognzed as the leadng nternatonal center of research n pure mathematcs. Ann.of Math.publshed many wrong prme papers, for example:green-tao[0,41],goldston-ntz-yldrm[8],wles-taylor[48,49] and other. Ther papers are related to the Hardy-Lttlewood wrong prme -tuples conjecture[18,9,4].therefore ther papers are wrong.but Ann.of Math reject Jang papers. Edtors of Ann.of Math do not understand the prme theory and want to publsh wrong prme papers. Twn prmes theorem[6,p41]. 1 We have Jang functon to see example 1 J ( ) ( ) 0 We prove that there exst nfntely many prmes 1 such that 1 s prme.therefore we prove twn prmes theorem. We have 1 (,) 1 : 1 prme ~ 1 ( 1) log Goldbach theorem[6,p41]. We have Jang functon to see example 1 1

2 1 J( ) ( ) 0 We prove that every even number 6 s the sum of two prmes.therefore we prove Goldbach theorem. We have 1 1 (,) 1 : prme ~ 1 ( 1) log Usng above method we prove about 000 prme theorems[].ths paper s only correct prme theory, other prme theores are wrong, because they do not prove the smplest twn prmes theorem and the smplest Goldbach theorem.the prme papers of ICM006,ICM010 and ICM014 are wrong.if ICM do not recognze ths paper,then the prme papers of ICM018 and ICM0 also are wrong.tao does not prove that every odd number s the sum of fve prmes,hs proof s wrong[].in [6,p170-00] we establsh the theory of prme table.we prove that n prmes n tuples there exst nfntely many prme solutons and fntely many prme solutons. Let be a gven prme,jp+-j(j=1,...,-1),there exst nfntely many prme p such that each of jp+-j s a prme[6].let be a gven prme,jp^+-j(j=1,...,-1),we prove t has nfntely many prme solutons and fntely many prme solutons[7]. ********************************************************************* rme dstrbuton s regularty J ( ) n1 rather than probablty 1/log x *************************************

3 Jang s functon J ( ) 1 n dstrbuton Chun-Xuan Jang. O. Box 94, Bejng ,. R. Chna n prme Abstract We defne that prme equatons f (,, ),, f (, ) (5) 1 1 n 1 n are polynomals (wth nteger coeffcents) rreducble over ntegers, where,, 1 n are all the prme. If Jang s functon J ( ) 0 n1 then (5)has fnte prme solutons. If J ( ) 0 n1 then there are nfntely many prmes,, 1 n such that f, 1 f are prmes. We obtan a unte prme formula n prme dstrbuton 1(, n 1) { 1,, n : f1,, f are prmes} n 1 Jn 1( ) (deg f ) (1 o(1)). (8) n n ( n!) ( ) log 1 Jang s functon s accurate seve functon. Usng Jang s functon we prove about 600 prme theorems [6]. Jang s functon provdes proofs of the prme theorems whch are smple enough to understand and accurate enough to be useful. Mathematcans have tred n van to dscover some order n the sequence of prme numbers but we have every reason to beleve that there are some mysteres whch the human mnd wll never penetrate. Leonhard Euler It wll be another mllon years, at least, before we understand the prmes. aul Erdös

4 Suppose that Euler totent functon ( ) ( 1) as, (1) where s called prmoral. Suppose that (, h ) 1,where 1,, ( ). We have prme equatons n1,, nh () 1 ( ) ( ) where n 0,1,,. ()s called nfntely many prme equatons (IME). Every equaton has nfntely many prme solutons. We have ( ) h 1 (1 o(1))., () ( ) h(mod ) where h denotes the number of prmes n n h n 0,1,,, ( ) the number of prmes less than or equal to. We replace sets of prme numbers by IME. () s the fundamental tool for provng the prme theorems n prme dstrbuton. Let 0 and (0) 8. From () we have eght prme equatons 1 0n 1, 0n 7, 0n 11, 4 0n 1, 5 0n 17, 6 0n 19, 7 0n, 8 0n 9, n 0,1,, (4) Every equaton has nfntely many prme solutons. THEOREM. We defne that prme equatons f (,, ),, f (,, ) (5) 1 1 n 1 n are polynomals (wth nteger coeffcents) rreducble over ntegers, where,, 1 n are prmes. If Jang s functon Jn 1( ) 0 then (5) has fnte prme solutons. If J n1( ) 0 then there exst nfntely many prmes,, 1 n such that each f s aprme. ROOF. Frstly, we have Jang s functon [1-11] n J ( ) [( 1) ( )], (6) n1 where ( ) s called seve constant and denotes the number of solutons for the followng specal congruence f ( q,, q ) 0 (mod ), (7) 1 where q1 1,, 1,, q 1,, 1. 1 n n J ( ) n1 denotes the number of sets of,, 1 n prme equatons such that f1( 1,, n),, f( 1,, n) are prme equatons. If J ( ) 0 n1 then (5) has fnte prme solutons. If J ( ) 0 n1 usng ( ) we sft out from () prme equatons 4

5 whch can not be represented,, 1 n, then resdual prme equatons of () are,, 1 n prme equatons such that f1( 1,, n ),, f( 1,, n) are prme equatons. Therefore we prove that there exst nfntely many prmes,, 1 n such that f1( 1,, n ),, f( 1,, n) are prmes. Secondly, we have the best asymptotc formula [,,4,6] 1(, n 1) { 1,, n : f1,, f are prmes} n 1 Jn 1( ) (deg f ) (1 o(1)). (8) n n ( n!) ( ) log 1 ( 8 ) s called a unte prme formula n prme dstrbuton. Let n1, 0, J ( ) ( ). From (8) we have prme number theorem 1(, ) 1 : 1s prme (1 o(1)).. (9) log umber theorsts beleve that there are nfntely many twn prmes, but they do not have rgorous proof of ths old conjecture by any method. All the prme theorems are conjectures except the prme number theorem, because they do not prove that prme equatons have nfntely many prme solutons. We prove the followng conjectures by ths theorem. Example 1. Twn prmes, (00BC). From (6) and (7) we have Jang s functon J ( ) ( ) 0. (10) Snce J ( ) 0 n () exst nfntely many prme equatons such that s a prme equaton. Therefore we prove that there are nfntely many prmes such that s a prme. Let 0 and J (0). From (4) we have three prme equatons 0n11, 0n17, 0n From (8) we have the best asymptotc formula J ( ) (,) : prme (1 o(1)) ( ) log 1 1 (1 o(1)). ( 1) log In 1996 we proved twn prmes conjecture [1] (11) 5

6 Remar. J ( ) denotes the number of prme equatons, (1 o(1)) ( ) log the number of solutons of prmes for every prme equaton. Example. Even Goldbach s conjecture 1. Every even number 6 s the sum of two prmes. From (6) and (7) we have Jang s functon 1 J( ) ( ) 0. (1) Snce ( 0 1 prme equatons such that 1 s a prme equaton. Therefore we prove that every even number 6 s the sum of two prmes. From (8) we have the best asymptotc formula J( ) (, ) 1, 1 prme (1 o(1)). ( ) log (1 o(1)). (1) ( 1) log In 1996 we proved even Goldbach s conjecture [1] Example. rme equatons,, 6. From (6) and (7) we have Jang s functon J ( ) ( ) 0, 5 J ( ) s denotes the number of prme equatons such that and 6 are prme equatons. Snce J ( ) 0 n () exst nfntely many prme equatons such that and 6 are prme equatons. Therefore we prove that there are nfntely many prmes such that and 6 are prmes. Let 0, J(0). From (4) we have two prme equatons 5 0n11, 0n 17. From (8) we have the best asymptotc formula J ( ) (,) { :, 6are prmes} (1 o(1)). ( ) log (14) Example 4. Odd Goldbach s conjecture 1. Every odd number 9 s the sum of three prmes. From (6) and (7) we have Jang s functon 1 J( ) ) 1 0. (15) 6

7 Snce J ( ) 0 as n () exst nfntely many pars of 1 and prme equatons such that 1 s a prme equaton. Therefore we prove that every odd number 9 s the sum of three prmes. From (8) we have the best asymptotc formula J( ) (,) 1, : 1 prme (1 o(1)) ( ) log (1 (1)) o. (16) ( 1) log Usng very complex crcle method Helfgott deduces the Hardy-Lttlewood formula of three prme problem[0,1],but Hardy-Lttlewood-Vnogradov-Helfgott do not prove that every odd number >7 s the sum of three prme numbers.therefore ther proofs are wrong. Example 5. rme equaton 1. From (6) and (7) we have Jang s functon ( ) 0 J (17) J ( ) denotes the number of pars of 1 and prme equatons such that s a prme equaton. Snce J ( ) 0 n () exst nfntely many pars of 1 and prme equatons such that s a prme equaton. Therefore we prove that there are nfntely many pars of prmes 1 and such that s a prme. From (8) we have the best asymptotc formula J( ) (, ) 1, : 1 prme (1 o(1)). 4 ( ) log (18) ote. deg ( 1 ). Example 6 [1]. rme equaton 1. From (6) and (7) we have Jang s functon J, (19) ( ) ( 1) ( ) 0 where ( ) ( 1) f ( ) 1 otherwse. 1 1(mod ) ; ( ) 0 f 1 1(mod ) ; Snce J ( ) 0 n () there are nfntely many pars of 1 and prme equatons such that s a prme equaton. Therefore we prove that there are nfntely many pars of prmes 1 and such that s a prme. From (8) we have the best asymptotc formula ( J ( ),) { 1, : 1 prme} (1 o(1)). (0) 6 ( ) log 7

8 4 Example 7 [1]. rme equaton 1 ( 1). From (6) and (7) we have Jang s functon J (1) ( ) ( 1) ( ) 0 where ( ) ( 1) f 1(mod 4) ; ( ) ( ) f 1(mod 8) ; ( ) 0 otherwse. Snce J ( ) 0 n () there are nfntely many pars of 1 and prme equatons such that s a prme equaton. Therefore we prove that there are nfntely many pars of prmes 1 and such that s a prme. From (8) we have the best asymptotc formula J( ) (,) 1, : prme (1 o(1)). 8 ( ) log () Example 8 [14-0]. Arthmetc progressons consstng only of prmes. We defne the arthmetc progressons of length., d, d,, ( 1) d,(, d) 1. () From (8) we have the best asymptotc formula (,) { 1 : 1, 1 d,, 1 ( 1) d are prmes} ( ) 1 J (1 o(1)).. (4) ( ) log If J ( ) 0 then () has fnte prme solutons. If J ( ) 0 then there are nfntely many prmes 1 such that,, are prmes. To elmnate d from () we have 1 j 1, ( j1) ( j), j. (5) From (6) and (7) we have Jang s functon J ( ) ( 1) ( 1)( 1) 0 (6) Snce J ( ) 0 there are nfntely many pars of 1 and prme equatons such that,, are prme equatons. Therefore we prove that there are nfntely many pars of prmes 1 and such that,, are prmes. From (8) we have the best asymptotc formula 1(,) 1, : ( j 1) ( j ) 1 prme, j J ( ) ( ) log (1 o(1)) 8

9 1 ( 1) (1 o(1)). (7) 1 1 ( 1) ( 1) log Example 9. It s a well-nown conjecture that one of,, s always dvsble by. To generalze above to the prmes, we prove the followng conjectures. Let n be a square-free even number. 1., n, n, where ( n 1). From (6) and (7) we have J () 0, hence one of, n, n s always dvsble by. 4., n, n,, n, where 5( nb), b,. From (6) and (7) we have J (5) 0, hence one of 4, n, n,, n s always dvsble by 5. 6., n, n,, n, where 7( nb), b,4. From (6) and (7) we have J (7) 0, hence one of 6, n, n,, n s always dvsble by , n, n,, n, where 11 ( nb), b,4,5,9. From (6) and (7) we have J (11) 0, hence one of 10, n, n,, n s always dvsble by , n, n,, n, where 1 ( nb), b,6,7,11. From (6) and (7) we have J (1) 0, hence one of 1, n, n,, n s always dvsble by , n, n,, n, where 17 ( nb), b,5,6,7,10,11,1,14,15. From (6) and (7) we have J (17) 0, hence one of 16, n, n,, n s always dvsble by , n, n,, n, where 19 ( nb), b4,5,6,9, From (6) and (7) we have J (19) 0, hence one of 18, n, n,, n s always dvsble by 19. Example 10.Let n be an even number. 1., n, 1,,5,,1, From (6) and (7) we have J ( ) 0. Therefore we prove that there exst nfntely many prmes such that, n are prmes for any.., n,, 4,6,,. 9

10 From (6) and (7) we have J ( ) 0. Therefore we prove that there exst nfntely many prmes such that, n are prmes for any. Example 11. rme equaton 1 From (6) and (7) we have Jang s functon J. (8) ( ) ( ) 0 Snce J ( ) 0 n () there are nfntely many pars of 1 and prme equatons such that s prme equatons. Therefore we prove that there are nfntely many pars of prmes 1 and such that s a prme. From (8) we have the best asymptotc formula J( ) (,) 1, : prme (1 o(1)). ( ) log (9) In the same way we can prove 1 whch has the same Jang s functon. Jang s functon s accurate seve functon. Usng t we can prove any rreducble prme equatons n prme dstrbuton. There are nfntely many twn prmes but we do not have rgorous proof of ths old conjecture by any method []. As strong as the numercal evdence may be, we stll do not even now whether there are nfntely many pars of twn prmes []. All the prme theorems are conjectures except the prme number theorem, because they do not prove the smplest twn prmes. They conjecture that the prme dstrbuton s probablty[1-8,-5,8-47]. References [1] Chun-Xuan Jang, On the Yu-Goldbach prme theorem, Guangx Scences (Chnese) (1996), 91-. [] Chun-Xuan Jang, Foundatons of Santll s sonumber theory, art I, Algebras Groups and Geometres, 15(1998), [] ChunXuan Jang, Foundatons of Santll s sonumber theory, art II, Algebras Groups and Geometres, 15(1998), [4] Chun-Xuan Jang, Foundatons Santll s sonumber theory, In: Fundamental open problems n scences at the end of the mllennum, T. Gll, K. Lu and E. Trell (Eds) Hadronc ress, USA, (1999), [5] Chun-Xuan Jang, roof of Schnzel s hypothess, Algebras Groups and Geometres, 18(001), [6] Chun-Xuan Jang, Foundatons of Santll s sonmuber theory wth applcatons to new cryptograms, Fermat s theorem and Goldbach s conjecture, Inter. Acad. ress, 00, MR004c: 11001, [7] Chun-Xuan Jang, rme theorem n Santll s sonumber theory,algebras Groups and Geometres, 19(00), [8] Chun-Xuan Jang, rme theorem n Santll s sonumber theory (II), Algebras Groups and Geometres, 0(00), [9] Chun-Xuan Jang, Dsproof s of Remann s hypothess, Algebras Groups and 10

11 Geometres, (005), Remann.pdf [10] Chun-Xuan Jang, Ffteen consecutve ntegers wth exactly prme factors, Algebras Groups and Geometres, (006), 9-4. [11] Chun-Xuan Jang, The smplest proofs of both arbtrarly long arthmetc progressons of prmes, preprnt, 006. [1] D. R. Heath-Brown, rmes represented by x y, Acta Math., 186 (001), [1] J. Fredlander and H. Iwanec, The polynomal 4 x y captures ts prmes, Ann. Of Math., 148(1998), [14] E. Szemeréd, On sets of ntegers contanng no elements n arthmetc progressons, Acta Arth., 7(1975), [15] H. Furstenberg, Ergodc behavor of dagonal measures and a theorem of Szemeréd on arthmetc progressons, J. Analyse Math., 1(1997), [16] T. Gowers,Hypergraph regularty and the multdmensonal Szemered theorem,ann. of Math.,166(007), [17] T.Gowers,A new proof of Szemered theorem,gafa,11(1997), [18] A.Odlyzo,M.Rubnsten and M.Wolf,Jumpng Champons,Experment Math.8,(1999), [19] B. Kra, The Green-Tao theorem on arthmetc progressons n the prmes: An ergodc pont of vew, Bull. Amer. Math. Soc., 4(006), -. [0] B. Green and T. Tao, The prmes contan arbtrarly long arthmetc progressons, Ann.of Math., 167(08), [1] T. Tao, The dchotomy between structure and randomness, arthmetc progressons, and the prmes, In: roceedngs of the nternatonal congress of mathematcans (Madrd. 006), Europ. Math. Soc. Vol , 007. [] B. Green, Long arthmetc progressons of prmes, Clay Mathematcs roceedngs Vol. 7, 007, [] H. Iwance and E. Kowals, Analytc number theory, Amer. Math. Soc., rovdence, RI, 004 [4] R. Crandall and C. omerance, rme numbers a computatonal perspectve, Sprng-Verlag, ew Yor, 005. [5] B. Green, Generalsng the Hardy-Lttlewood method for prmes, In: roceedngs of the nternatonal congress of mathematcans (Madrd. 006), Europ. Math. Soc., Vol. II, 7-99, 007. [6] K. Soundararajan, Small gaps between prme numbers: The wor of Goldston-ntz-Yldrm, Bull. Amer. Math. Soc., 44(007), [7] A. Granvlle, Harald Cramér and dstrbuton of prme numbers, Scand. Actuar. J, 1995(1) (1995), 1-8. [8] Ytang Zhang,Bounded gaps between prmes,to submt Ann.of Math. [9] Chun-Xuan Jang,The Hardy-Lttlewood prme -tuple conjecture s false. [0] H.A.Helfgott,Major arcs for Goldbach problem, 11

12 [1] H.A.Helfgott,Mnor arcs for Goldbach problem, [] [] T.Tao,Every odd number greater than 1 s the sum of at most fve prmes, appear n Math.Comp. [4] G.H.Hardy and J.E.Lttlewood,Some problems of artto umerorum ;III:On the expresson of a number as a sum of prmes,acta Math.,44(19),1-70. [5].Erdos and.turan,on some sequences of ntegers,j.london Math.Soc.,11(196), [6] Chun-Xuan Jang,The new prme theorem (5), [7] Chun-Xuan Jang,The new prme theorem (4), [8]D.Goldston,J.ntz and C.Yldrm, rmes n tuples I, Ann. of Math.,170(009), [9] D.Goldston,Y.Motohash,J.ntz,and C.Yldrm,Small gaps between prmes exst,roc.japan Acad.Ser.A Math.Sc,8(006), [40] D.Goldston,S.Graham,J.ntz,and Y.Yldrm,Small gaps between products of two prmes,roc London Math.Soc.()98(009), [41] B.Green and T.Tao,Lnear equatons n prmes,ann.of Math.,171(010), [4] J.Bourgan,A.Gamburd and.sarna,affne lnear seve,expanders,and sum-product,invent Math,179(010), [4] M.I.Vnogradov,Representatons of an odd number as a sum of three prmes,dol.aad.au SSSR 15(197), [44] T.Tao and V.Vu,Addtve combnatorcs, Cambrdge Unversty ress.cambrdge(006). [45] B.L.van der Waerden,Bewes ener Baudetschen Vermutung, euw Arch.Ws.,15(197),1-16. [46] B.Host and B.Kra,Convergence of polynomal ergodc averages,israel J.Math,149(005),1-19. [47]B.Host and B.Kra,onconventonal ergodc averages and nlmanfolds,ann of Math,161(005), [48] A.Wles,Modular ellptc curves and Fermat last theorem, Ann.of Math.,141(1995), [49] R.Taylor and A.Wles,Rng-theoretc propertes of certan Hece algebras,ann of Math,141(1995), 年我们用新方法证明了 twn prmes theorem and Goldbach theorem[1].1995 年 10 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,1996 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 1

13 定 从 (7) 我们使用一种特殊同余式 f(q) 三 0 (mod p) q=1,...,p-1; 不使用 q=1,...,p-1,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 functon 推广, 因为 p 的 Euler 函数互素只有 p-1 个,Jang functon 作建立 ISO 数学, 中科院吓坏了, 用保存在 Euler functon 中与研究素数方程有关的数,Euler functon and Jang functon 都是研究素数的工具, 这一点是统一的 001 年 10 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 和 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 1+, 出书 [ 从哥德巴赫到陈景润 From Goldbach to Chenjngrun], 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 twn prmes and Goldbach conjecture. 这种不死不活场面还要继续下去, 我们继续宣传本文 从 Hardy(19) 到 91 年 90 年发表的素数论文都是错的 GY do not prove that rmes n tuples are admssble and nadmssble.gy papers are 100% wrong. 最近张益唐根据 GY 错误文章继续工作, 在国内外大作舆论, 张益唐文章也 100% 错的, 国内王元一批人为张益唐文章起哄. On the sngular seres n the Jang prme -tuples theorem Chun-Xuan Jang. O. Box 94, Bejng ,. R. Chna Abstract Usng Jang functon we prove Jang prme -tuples theorem.we fnd true sngular seres. Usng the examples we prove the Hardy-Lttlewood prme -tuples conjecture wth wrong sngular seres.. Jang prme -tuples theorem wll replace the Hardy-Lttlewood prme -tuples conjecture. 1

14 (A) Jang prme -tuples theorem wth true sngular seres[1, ]. We defne the prme -tuples equaton p p n, (1), where n, 1, 1. we have Jang functon [1, ] J ( ) ( 1 ( )), () where, ( ) s the number of solutons of the followng specal congruence whch s true. 1 ( qn ) 0 (mod ), q 1,, p1. () 1 If ( ) 1 then J ( ) 0. There exst nfntely many prmes such that each of n s prme. If ( ) 1 then J ( ) 0. There exst fntely many prmes such that each of n s prme. J ( ) s a subset of Euler functon ( )[]. If J ( ) 0, then we have the best asymptotc formula of the number of prme [1, ] 1 (,) : ~ ( ) J n prme C( ) (4) ( ) log log ( ) ( 1) 1 ( ) 1 C ( ) 1 1 s Jang true sngular seres. Example 1.Let,,, twn prmes theorem. From () we have (5) () 0, ( ) 1 f, (6) Substtutng (6) nto () we have J ( ) ( ) 0 (7) There exst nfntely many prmes such that (4) we have the best asymptotc formula s prme. Substtutng (7) nto 14

15 1 (,) : prme ~ (1 ). ( 1) log (8) Example.Let,,, 4. From () we have From () we have () 0, () (9) J ( ) 0. (10) It has only a soluton, 5, 4 7. One of,, 4 s always dvsble by. Example.Let 4,, n,where n,6,8. From () we have Substtutng (11) nto () we have () 0, () 1, ( ) f. (11) J ( ) ( 4) 0, (1) 5 There exst nfntely many prmes such that each of n Substtutng (1) nto (4) we have the best asymptotc formula s prme. 7 ( 4) 4(,) : n prme ~ 5 ( 1) 4 log 4 (1) Example 4. Let 5,, n,where n,6,8,1. From () we have Substtutng (14) nto () we have () 0, () 1, (5), ( ) 4 f 5 (14) J ( ) ( 5) 0 (15) 7 There exst nfntely many prmes such that each of n (15) nto (4) we have the best asymptotc formula s prme. Substtutng ( 5) 5(,) : n prme ~ 11 7 ( 1) 5 log 5 (16) Example 5.Let 6,, n,where n,6,8,1,14. From () and () we have () 0, () 1, (5) 4, J (5) 0 (17) 15

16 It has only a soluton 5, 7, 6 11, 8 1, 1 17, One of n s always dvsble by 5. ( B ) The Hardy-Lttlewood prme -tuples conjecture wth wrong sngular seres[-14]. Ths conjecture s generally beleved to be true, but has not been proved(odlyzo et al.jumpng champon,experment math,8(1999), ). We defne the prme -tuples equaton where n, 1,, 1. n (18), In 19 Hardy and Lttlewood conjectured the asymptotc formula where (,) : n prme ~ H( ), (19) log ( ) 1 H ( ) 1 1 s Hardy-Lttlewood wrong sngular seres, (0) ( ) s the number of solutons of congruence whch s wrong. 1 ( qn ) 0 (mod ), q 1,,. (1) 1 From (1) we have ( ) and H ( ) 0.Foranyprme -tuples equaton there exst nfntely many prmes such that each of n s prme, whch s false. Conjecture 1.Let,,, twn prmes theorem From (1) we have ( ) 1 () Substtutng () nto (0) we have H () () 1 Substtutng () nto (19) we have the asymptotc formula (,) : prme ~ 1log (4) 16

17 whch s wrong see example 1. Conjecture.Let,,, 4. From (1) we have Substtutng (5) nto (0) we have () 1, ( ) f (5) H () 4 ( ) ( 1) (6) Substtutng (6) nto (19) we have asymptotc formula ( ) (,) : prme, 4 prm ~ 4 whch s wrong see example. Conjecture.Let 4,, n,where n,6,8. ( 1) log (7) From (1) we have () 1, (), ( ) f (8) Substtutng (8) nto (0) we have H (4) 7 ( ) 4 ( 1) (9) Substtutng (9) nto (19) we have asymptotc formula 7 ( ) 4(,) : n prme ~ ( 1) 4 log 4 Whch s wrong see example. Conjecture 4.Let 5,, n,where n,6,8,1 (0) From (1) we have () 1, (), (5), ( ) 4 f 5 (1) Substtutng (1) nto (0) we have H (5) 4 ( 1) ( 4) () Substtutng () nto (19) we have asymptotc formula ( 4) 5(,) : n prme ~ ( 1) 5 log 5 Whch s wrong see example 4. () 17

18 Conjecture 5.Let 6,, n,where n,6,8,1,14. From (1) we have () 1, (), (5) 4, ( ) 5 f 5 (4) Substtutng (4) nto (0) we have 15 ( 5) H (6) ( 1) 5 5 (5) Substtutng (5) nto (19) we have asymptotc formula ( 5) 6(,) : n prme ~ 1 5 ( 1) 6 log 6 (6) whch s wrong see example 5. Concluson.From Hardy-Lttlewood(19) to 01 all prme papers are wrong. The Jang prme -tuples theorem has true sngular seres.the Hardy-Lttlewood prme -tuples conjecture has wrong sngular seres.. The tool of addtve prme number theory s bascally the Hardy-Lttlewood wrong prme -tuples conjecture [-14]. Usng Jang true sngular seres we prove almost all prme theorems. Jang prme -tuples theorem wll replace Hardy-Lttlewood prme -tuples conjecture. There cannot be really modern prme theory wthout Jang functon. References [1] Chun-Xuan Jang, Foundatons of Santll s sonumber theory wth applcatons to new cryptograms, Fermat s theorem and Goldbach s conjecture. Inter. Acad. ress, 00,MR004c:11001,( ( pdf). [] Chun-Xuan Jang, Jang s functon J ( ) 1 n n prme dstrbuton. ( www. wbabn. net/math/ xuan. pdf) ( [] G. H. Hardy and J. E. Lttlewood, Some problems of artton umerorum, III: On the expresson of a number as a sum of prmes, Acta Math, 44(19), [4] B. Green and T. Tao, The prmes contan arbtrarly long arthmetc progressons, Ann. Math., 167(008), [5] D. A. Goldston, S. W. Graham, J. ntz and C. Y. Yldrm, Small gaps between 18

19 products of two prmes, roc. London Math. Soc., () 98 (009) [6] D. A. Goldston, S. W. Graham, J. ntz and C. Y. Yldrm, Small gaps between prmes or almost prmes, Trans. Amer. Math. Soc., 61(009) [7] D. A. Goldston, J. ntz and C. Y. Yldrm, rmes n tulpes I, Ann.of Math., 170(009) [8]. Rbenbom, The new boo of prme number records, rd edton, Sprnger-Verlag, ew Yor, Y, [9] H.Halberstam and H.-E.Rchert,Seve methods, Academc ress,1974. [10] A.Schnzel and W.Serpns, Sur certanes hypotheses concernant les nombres premers,acta Arth.,4(1958) [11].T.Bateman and R.A.Horn,A heurstc asymptotc formula concernng the dstrbuton of prme numbers,math.comp.,16(196)6-67 [1] W.arewcz,The development of prme number theory,from Eucld to Hardy and Lttlewood,Sprnger-Verlag,ew Yor,Y,000,-5. [1] B.Green and T.Tao,Lnear equatons n prmes, Ann.of Math.171(010) [14] T.Tao,Recent progress n addtve prme number theory, Theewrmetheorem(5), j j( j 1,, 1) Chun-Xuan Jang. O. Box 94, Bejng ,. R. Chna Abstract Usng Jang functon we prove that there exst nfntely many prmes such that 19

20 each j j s a prme. Theorem. Let be a gven prme., j j( j 1,, 1) (1) There exst nfntely many prmes such that each of j j s a prme. roof. We have Jang functon[1] where, J ( ) [ 1 ( )], () ( ) s the number of solutons of congruence 1 ( jq j ) 0 (mod ), () j1 q1,, 1. From () we have () 0,f then ( ), ( ) 1,f ( ) 1. From () and () we have J ( ) ( ) ( ) 0. (4) then We prove that there exst nfntely many prmes such that each of j j s a prme We have the asymptotc formula [1] 1 (,) : ~ ( ) J j j prme, (5) ( ) log where ( ) ( 1). Reference [1] Chun-Xuan Jang, Jang s functon J ( ) n1 n prme dstrbuton. wbabn.net/math /xuan. pdf. 0

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