The Hardy-Littlewood prime k-tuple conjecture is false

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1 The Hardy-Lttlewood prme -tuple conjecture s false Chun-Xuan Jang. O. Box 9, Bejng 008,. R. Chna Jangchunxuan@vp.sohu.com Abstract Usng Jang functon we prove Jang prme -tuple theorem. We prove that the Hardy-Lttlewood prme -tuple conjecture s false. Jang prme -tuple theorem can replace the Hardy-Lttlewood prme -tuple conjecture. Hardy-Lttlewood 论文作为数论圣经, 一百年来华罗庚 王元和一大批数论家必读, 这是当代最高数论水平, 数学天才陶哲轩拼命在学习他们论文, 下一步仔细研究他们论文, 到底有多少猜想是正确的, 素数太复杂, 蒋春暄已打开素数大门 (A) Jang prme -tuple theorem [, ]. We defne the prme -tuple equaton p p+, (), n where n, =, L. we have Jang functon [, ] J ( ) ( ( )) ω = Π χ, () where ω =Π, χ ( ) s the number of solutons of congruence Π ( q+ n ) 0 (mod ), q =, L, p. () = If χ ( ) < then J ( ω) 0. There exst nfntely many prmes such that each of + n s prme. If χ ( ) = then J ( ω ) = 0. There exst fntely many prmes such that each of + n s prme. J ( ω ) s a subset of Euler functon φ( ω )[]. If J ( ω) 0, then we hae the best asymptotc formula of the number of prme [, ] (,) { : } ~ ( ) J ωω = + n = prme = C( ) φ ( ω) log log ()

2 φ( ω ) =Π( ), + χ( ) C ( ) =Π () Example. Let =,, +, twn prmes theorem. From () we have Substtutng (6) nto () we have There exst nfntely many prmes χ() = 0, χ( ) = f >, (6) J ( ω) = Π( ) 0 (7) () we have the best asymptotc pormula such that + s prme. Substtutng (7) nto (,) = { : + = prme} ~ Π( ). ( ) log (8) Example. Let =,, +, +. From () we have From () we have χ() = 0, χ() = (9) J ( ω ) = 0. (0) It has only a soluton =, + =, + = 7. One of, +, + s always dvsble by. Example. Let =,, + n, where n =,6,8. From () we have Substtutng () nto () we have χ() = 0, χ() =, χ( ) = f >. () J ( ω) =Π( ) 0, () There exst nfntely many prmes such that each of + n s prme. Substtutng () nto () we have the best asymptotc formula 7 ( ) (,) = { : + n= prme} ~ Π ( ) log () Example. Let =,, + n, where n =,6,8,.

3 From () we have Substtutng () nto () we have χ() = 0, χ() =, χ() =, χ( ) = f > () J ( ω) =Π( ) 0 () 7 There exst nfntely many prmes () nto () we have the best asymptotc formula such that each of + n s prme. Substtutng ( ) (,) = { : + n= prme} ~ Π 7 ( ) log (6) Example. Let = 6,, + n, where n =, 6,8,,. From () and () we have χ() = 0, χ() =, χ() =, J () = 0 (7) It has only a soluton =, + = 7, + 6=, + 8=, + = 7, + = 9. One of + n s always dvsble by. (B)The Hardy-Lttlewood prme We defne the prme where n, =, L,. -tuple equaton, n -tuple conjecture[-8]. + (8) In 9 Hardy and Lttlewood conjectured the asymptotc formula where (,) = { : + n = prme} ~ H( ), (9) log ν ( ) H( ) =Π (0) ν ( ) s the number of solutons of congruence Π ( q+ n ) 0 (mod ), q =, L,. () = From () we have ν ( ) < and H( ) 0. For any prme -tuple equaton there exst nfntely many prmes Conjectore. Let Frome () we have such that each of =,, +, twn prmes theorem + n s prme, whch s false.

4 ν ( ) = () Substtutng () nto (0) we have H () =Π () Substtutng () nto (9) we have the asymptotc formula (,) = { : + = prme} ~ Π log () whch s false see example. Conjecture. Let h =,, +, +. From () we have Substtutng () nto (0) we have ν () =, ν ( ) = f > () H () = Π ( ) ( ) (6) Substtutng (6) nto (9) we have asymptotc formula { } (,) = : + = prme, + = prm ~Π ( ) whch s false see example. Conjecutre. Let =,, + n, where n =,6,8. ( ) log (7) From () we have ν () =, ν() =, ν( ) = f > (8) Substtutng (8) nto (0) we have H () = Π 7 ( ) > ( ) (9) Substtutng (9) nto (9) we have asymptotc formula 7 ( ) = { + n= prme} Π > (,) : ~ ( ) log Whch s false see example. Conjecture. Let =,, + n, where n =, 6,8, (0) From () we have ν () =, ν() =, ν() =, ν( ) = f > ()

5 Substtutng () nto (0) we have ( ) > ( ) H () = Π () Substtutng () nto (9) we have asymptotc formula ( ) (,) = { : + n= prme} ~ Π () > ( ) log Whch s false see example. Conjecutre. Let = 6,, + n, where n =,6,8,,. From () we have ν () =, ν() =, ν() =, ν( ) = f > () Substtutng () nto (0) we have ( ) H (6) = Π 6 > ( ) () Substtutng () nto (9) we have asymptotc formula ( ) 6(,) = { : + n= prme} ~ Π > ( ) 6 log 6 (6) whch s false see example. Concluson. The Hardy-Lttlewood prme -tuple conjecture s false. Jang prme -tuple theorem can replace Hardy-Lttlewood prme -tuple Conjecture.

6 References [] 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, MR00c:00, ( ( pdf). [] Chun-Xuan Jang, Jang s functon J ( ) n+ ω 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, (9), -70. [] B. Green and T. Tao, The prmes contan arbtrarly long arthmetc progressons, Ann. Math., 67(008), 8-7. [] D. A. Goldston, S. W. Graham, J. ntz and C. Y. Yldrm, Small gaps between 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., 6(009) 8-0. [7] D. A. Goldston, J. ntz and C. Y. Yldrm, rmes n tulpes I, Ann. Math., 70(009) [8]. Rbenbom, The new boo of prme number records, rd edton, Sprnger-Verlag, ew Yor, Y,