( N) Chun-Xuan Jiang. P. O. Box 3924, Beijing , P. R. China

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1 ang s functon n ( ) n prme dstrbuton Chun-Xuan ang P O Box 94, Bejng 00854, P R Chna jcxuan@snacom Abstract: We defne that prme equatons f( P,, Pn ),, f ( P, Pn ) (5)are polynomals (wth nteger coeffcents) rreducble over ntegers, where P,, Pn are all prme If ang s functon ( ) 0 n then (5) has fnte prme solutons If ( ) 0 n then there are nfntely many prmes P,, Pn such that f, f are prmes We obtan a unte prme formula n prme dstrbuton N, n ) { P,, P N : f,, f are prmes} ( n n n( ) N (deg f) ( o()) (8) ang s functon s accurate seve functon Usng n n n! ( ) log N ang s functon we prove about 600 prme theorems [6] ang s functon provdes proofs of the prme theorems whch are smple enough to understand and accurate enough to be useful ( ) [Chun-Xuan ang ang s functon n (ISSN: ) 7 Keywords: functon; ang; Prme n prme dstrbuton Rep Opnon 0;4(8):8-4] 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 Paul Erdös Suppose that Euler totent functon ( ) ( ) as P, () where P s called prmoral P Suppose that (, h ), where,, ( ) We have prme equatons P n,, P ( ) n h ( ) () where n 0,,, ()s called nfntely many prme equatons (IMPE) Every equaton has nfntely many prme solutons We have ( N) h ( o()), () ( ) where P N P h (mod ) h denotes the number of prmes P N n P n h n 0,,,, ( N) the number of prmes less than or equal to N We replace sets of prme numbers by IMPE () s the fundamental tool for provng the prme theorems n prme dstrbuton Let 0 and (0) 8 From () we have eght prme equatons P 0n, P 0n 7, P 0n, P4 0n, P5 0n 7, 8

2 P6 0n 9, P7 0n, P8 0n 9, n 0,,, (4) Every equaton has nfntely many prme solutons THEOREM We defne that prme equatons f( P,, Pn ),, f ( P,, Pn ) (5) are polynomals (wth nteger coeffcents) rreducble over ntegers, where P,, Pn are prmes If ang s functon n( ) 0 then (5) has fnte prme solutons If n ( ) 0 then there exst nfntely many prmes P,, Pn such that each f s a prme PROOF Frstly, we have ang s functon [-] n n( ) [( P ) ( P)], (6) P where ( P) s called seve constant and denotes the number of solutons for the followng congruence f ( q,, q ) 0 (mod P), (7) where q,, P,, q,, P n n ( ) n denotes the number of sets of P,, Pn prme equatons such that f( P,, Pn ),, f ( P,, Pn ) are prme equatons If ( ) 0 n then (5) has fnte prme solutons If ( ) 0 n usng ( P) we sft out from () prme equatons whch can not be represented P,, Pn, then resdual prme equatons of () are P,, Pn prme equatons such that f( P,, P n ),, f ( P,, Pn ) are prme equatons Therefore we prove that there exst nfntely many prmes P,, Pn such that f( P,, P n ),, f ( P,, Pn ) are prmes Secondly, we have the best asymptotc formula [,,4,6] N, n ) { P,, P N : f,, f are prmes} ( n n n( ) N (deg f) ( o()) (8) n n n! ( ) log N (8)s called a unte prme formula n prme dstrbuton Let n, 0, ( ) ( ) From (8) we have prme number theorem N ( N, ) P N : P s prme ( o()) (9) log N Number 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 Twn prmes P, P (00BC) From (6) and (7) we have ang s functon ( ) ( P ) 0 P Snce ( ) 0 n () exst nfntely many P prme equatons s a prme equaton Therefore we prove that there are nfntely many prmes P s a prme Let 0 and (0) From (4) we have three P prme equatons P 0n, P5 0n 7, P8 0n 9 From (8) we have the best asymptotc formula 9

3 ( ) N ( N,) P N : P prme ( o()) ( ) log N N ( o()) P ( P ) log N In 996 we proved twn prmes conjecture [] Remar ( ) denotes the number of P prme equatons, N ( o()) the number of ( ) log N solutons of prmes for every P prme equaton Example Even Goldbach s conjecture N P P Every even number N 6 s the sum of two prmes From (6) and (7) we have ang s functon P ( ) ( P ) 0 P P N P Snce ( ) 0 as N n () exst nfntely many P prme equatons such that N P s a prme equaton Therefore we prove that every even number N 6 s the sum of two prmes From (8) we have the best asymptotc formula ( ) N ( N,) P N, N P prme ( o()) ( ) log N P N ( o()) P ( P ) P N P log N In 996 we proved even Goldbach s conjecture [] Example Prme equatons P, P, P 6 From (6) and (7) we have ang s functon ( ) ( P ) 0, 5P ( ) s denotes the number of P prme equatons 6 are prme equatons Snce ( ) 0 n () exst nfntely many P prme equatons 6 are prme equatons Therefore we prove that there are nfntely many prmes P 6 are prmes Let 0, (0) From (4) we have two P prme equatons P 0n, P5 0n 7 From (8) we have the best asymptotc formula ( ) N ( N,) { P N : P, P 6are prmes} ( o()) ( ) log N Example 4 Odd Goldbach s conjecture N P P P Every odd number N 9 s the sum of three prmes From (6) and (7) we have ang s functon ( ) P P ) 0 P P N P P Snce ( ) 0 as N n () exst nfntely many pars of P prme equatons such that N P P s a prme equaton Therefore we prove that every odd number N 9 s the sum of three prmes From (8) we have the best asymptotc formula ( ) N ( N,) P, P N : N P P prme ( o()) ( ) log N 0

4 N ( ()) P o ( P ) P N P P log N Example 5 Prme equaton P PP From (6) and (7) we have ang s functon ( ) P P 0 P ( ) denotes the number of pars of P prme equatons s a prme equaton Snce ( ) 0 n () exst nfntely many pars of P prme equatons s a prme equaton Therefore we prove that there are nfntely many pars of prmes P s a prme From (8) we have the best asymptotc formula ( ) N ( N,) P, P N : PP prme ( o()) 4 ( ) log N Note deg ( PP ) Example 6 [] Prme equaton P P P From (6) and (7) we have ang s functon ( ) ( P ) ( P) 0, P P where ( P) ( P ) f (mod P) ; ( P) 0 f (mod P) ; ( P) P otherwse Snce ( ) 0 n () there are nfntely many pars of P prme equatons s a prme equaton Therefore we prove that there are nfntely many pars of prmes P s a prme From (8) we have the best asymptotc formula ( ( ) N N,) { P, P N : P P prme} ( o()) 6 ( ) log N 4 Example 7 [] Prme equaton P P ( P ) From (6) and (7) we have ang s functon ( ) ( P ) ( P) 0 P where ( P) ( P ) f P (mod 4) ; ( P) ( P ) f P (mod8) ; ( P) 0 otherwse Snce ( ) 0 n () there are nfntely many pars of P prme equatons s a prme equaton Therefore we prove that there are nfntely many pars of prmes P s a prme From (8) we have the best asymptotc formula ( ) N ( N,) P, P N : P prme ( o()) 8 ( ) log N Example 8 [4-0] Arthmetc progressons consstng only of prmes We defne the arthmetc progressons of length P, P P d, P P d,, P P ( ) d,( P, d) (0) From (8) we have the best asymptotc formula N,) { P N : P, P d,, P ( ) are prmes} ( d ( ) N ( o()) ( ) log N P

5 If ( ) 0 then (0) has fnte prme solutons If ( ) 0 then there are nfntely many prmes P such that P,, P are prmes To elmnate d from (0) we have P P P, Pj ( j ) P ( j ) P, j From (6) and (7) we have ang s functon ( ) ( P ) ( P )( P ) 0 P P Snce ( ) 0 n () there are nfntely many pars of P prme equatons,, P are prme equatons Therefore we prove that there are nfntely many pars of prmes P such that P,, P are prmes From (8) we have the best asymptotc formula ( N,) P, P N : ( j ) P ( j ) P prme, j ( ) N ( o()) ( ) log N P P ( P ) N ( o()) P ( P ) P ( P ) log N Example 9 It s a well-nown conjecture that one of P, P, P s always dvsble by To generalze above to the prmes, we prove the followng conjectures Let n be a square-free even number P, P n, P n, where ( n ) From (6) and (7) we have () 0, hence one of P, P n, P n s always dvsble by 4 P, P n, P n,, P n, where 5 ( n b), b, From (6) and (7) we have (5) 0, hence one of 4 P, P n, P n,, P n s always dvsble by 5 6 P, P n, P n,, P n, where 7 ( n b), b, 4 From (6) and (7) we have (7) 0, hence one of 6 P, P n, P n,, P n s always dvsble by P, P n, P n,, P n, where ( n b), b, 4,5,9 From (6) and (7) we have () 0, hence one of 0 P, P n, P n,, P n s always dvsble by 5 P, P n, P n,, P n, where ( n b), b, 6, 7, From (6) and (7) we have () 0, hence one of P, P n, P n,, P n s always dvsble by 6 6 P, P n, P n,, P n, where 7 ( n b), b,5, 6, 7,0,,,4,5 From (6) and (7) we have (7) 0, hence one of 6 P, P n, P n,, P n s always dvsble by 7

6 8 7 P, P n, P n,, P n, where 9 ( n b), b 4,5, 6, 9,67 From (6) and (7) we have (9) 0, hence one of 8 P, P n, P n,, P n s always dvsble by 9 Example 0 Let n be an even number P, P n,,,5,,, From (6) and (7) we have ( ) 0 Therefore we prove that there exst nfntely many prmes P such that P, P n are prmes for any P, P n,, 4,6,, From (6) and (7) we have ( ) 0 Therefore we prove that there exst nfntely many prmes P such that P, P n are prmes for any Example Prme equaton P P P From (6) and (7) we have ang s functon ( ) ( P P ) 0 P Snce ( ) 0 n () there are nfntely many pars of P prme equatons s prme equatons Therefore we prove that there are nfntely many pars of prmes P s a prme From (8) we have the best asymptotc formula ( ) N ( N,) P, P N : P prme ( o()) ( ) log N In the same way we can prove P P P whch has the same ang s functon ang 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 [0] 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 randomness [-5], because they do not understand theory of prme numbers Acnowledgements The Author would le to express hs deepest apprecaton to M N Huxley, R M Santll, L Schadec and G Wess for ther helps and supports References [] Chun-Xuan ang, On the Yu-Goldbach prme theorem, Guangx Scences (Chnese) (996), 9- [] Chun-Xuan ang, Foundatons of Santll s sonumber theory, Part I, Algebras Groups and Geometres, 5(998), 5-9 [] ChunXuan ang, Foundatons of Santll s sonumber theory, Part II, Algebras Groups and Geometres, 5(998), [4] Chun-Xuan ang, 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 Press, USA, (999), 05-9 [5] Chun-Xuan ang, Proof of Schnzel s hypothess, Algebras Groups and Geometres, 8(00), 4-40 [6] Chun-Xuan ang, Foundatons of Santll s sonmuber theory wth applcatons to new cryptograms, Fermat s theorem and Goldbach s conjecture, Inter Acad Press, 00, MR004c: 00, [7] Chun-Xuan ang Prme theorem n Santll s sonumber theory, 9(00), [8] Chun-Xuan ang, Prme theorem n Santll s sonumber theory (II), Algebras Groups and Geometres, 0(00), [9] Chun-Xuan ang, Dsproof s of Remann s hypothess, Algebras Groups and Geometres, (005), -6

7 Remannpdf [0] Chun-Xuan ang, Ffteen consecutve ntegers wth exactly prme factors, Algebras Groups and Geometres, (006), 9-4 [] Chun-Xuan ang, The smplest proofs of both arbtrarly long arthmetc progressons of prmes, preprnt, 006 [] D R Heath-Brown, Prmes represented by x y, Acta Math, 86 (00), -84 [] Fredlander and H Iwanec, The polynomal 4 x y captures ts prmes, Ann Math, 48(998), [4] E Szemeréd, On sets of ntegers contanng no elements n arthmetc progressons, Acta Arth, 7(975), [5] H Furstenberg, Ergodc behavor of dagonal measures and a theorem of Szemeréd on arthmetc progressons, Analyse Math, (997), [6] W T Gowers, A new proof of Szemeréd s theorem, GAFA, (00), [7] B Kra, The Green-Tao theorem on arthmetc progressons n the prmes: An ergodc pont of vew, Bull Amer Math Soc, 4(006), - [8] B Green and T Tao, The prmes contan arbtrarly long arthmetc progressons, Ann Math, 67(08), [9] T Tao, The dchotomy between structure and randomness, arthmetc progressons, and the prmes, In: Proceedngs of the nternatonal congress of mathematcans (Madrd 006), Europ Math Soc Vol , 007 [0] B Green, Long arthmetc progressons of prmes, Clay Mathematcs Proceedngs Vol 7, 007,49-59 [] H Iwance and E Kowals, Analytc number theory, Amer Math Soc, Provdence, RI, 004 [] R Crandall and C Pomerance, Prme numbers a computatonal perspectve, Sprng-Verlag, New Yor, 005 [] B Green, Generalsng the Hardy-Lttlewood method for prmes, In: Proceedngs of the nternatonal congress of mathematcans (Madrd 006), Europ Math Soc, Vol II, 7-99, 007 [4] K Soundararajan, Small gaps between prme numbers: The wor of Goldston-Pntz-Yldrm, Bull Amer Math Soc, 44(007), -8 [5] A Granvlle, Harald Cramér and dstrbuton of prme numbers, Scand Actuar, 995() (995), -8 7//0 4