The Hardy-Littlewood prime k-tuple conjecture is false

Transcription

1 The Hardy-Lttlewood prme k-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 k -tuple theorem. We prove that the Hardy-Lttlewood prme k -tuple conjecture s false. Jang prme k -tuple theorem can replace the Hardy-Lttlewood prme k -tuple conjecture. (A) Jang prme k -tuple theorem [, ]. We defne the prme k -tuple equaton p p+, (), n where n, =, L k. we have Jang functon [, ] J ( ) ( ( )) ω = Π χ, () where ω =Π, χ ( ) s the number of solutons of congruence k 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 [, ] (,) { : } ~ ( ) k J ωω k n prme C( k) k k k = + = = () φ ( ω) log log φ( ω ) =Π( ), + χ( ) Ck ( ) =Π k ()

2 Example. Let k =,, +, twn prmes theorem. From () we have Substtutng (6) nto () we have χ() = 0, χ( ) = f >, (6) J ( ω) = Π( ) 0 (7) There exst nfntely many prmes such that + s prme. Substtutng (7) nto () we have the best asymptotc pormula k (,) = { : + = prme} ~ Π( ). ( ) log (8) Example. Let k =,, +, +. 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 k =,, + 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 k =,, + n, where n =,6,8,. From () we have Substtutng () nto () we have χ() = 0, χ() =, χ() =, χ( ) = f > ()

3 J ( ω) =Π( ) 0 () 7 There exst nfntely many prmes such that each of + n s prme. Substtutng () nto () we have the best asymptotc formula ( ) (,) = { : + n= prme} ~ Π 7 ( ) log (6) Example. Let k = 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 k -tuple conjecture[-]. Ths conjecture s generally beleved to be true,but has not been proved(odlyzko et al.999). We defne the prme k -tuple equaton where n, =, L, k., + n (8) In 9 Hardy and Lttlewood conjectured the asymptotc formula where k(,) = { : + n = prme} ~ H( k), (9) log k ν ( ) Hk ( ) =Π k (0) ν ( ) s the number of solutons of congruence k Π ( q+ n ) 0 (mod ), q=, L,. () = From () we have ν ( ) < and Hk ( ) 0. For any prme k -tuple equaton there exst nfntely many prmes such that each of + n s prme, whch s false. Conjectore. Let k =,, +, twn prmes theorem Frome () we have Substtutng () nto (0) we have ν ( ) = ()

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

5 H () = Π ( ) ( ) > () Substtutng () nto (9) we have asymptotc formula ( ) (,) = { : + n= prme} ~ Π > ( ) log Whch s false see example. Conjecutre. Let k = 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 k -tuple conjecture s false. The tool of addve prme number theory s bascally the Hardy-Lttlewood prme tuples conjecture. Jang prme k -tuple theorem can replace Hardy-Lttlewood prme k -tuple Conjecture. There cannot be really modern prme theory wthout Jang functon. 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 ( ) + ω n prme dstrbuton. ( www. wbabn. net/math/ xuan. pdf) ( n

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