A New Sifting function J ( ) n+ 1. prime distribution. Chun-Xuan Jiang P. O. Box 3924, Beijing , P. R. China

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1 A New Siftig fuctio J ( ) + ω i prime distributio Chu-Xua Jiag. O. Box 94, Beijig 00854,. R. Chia jiagchuxua@vip.sohu.com Abstract We defie that prime equatios f (, L, ), L, f (, L ) (5) are polyomials (with iteger coefficiets) irreducible over itegers, where,, L are all prime. If siftig fuctio J ( ) 0 + ω = the (5)has fiite prime solutios. If J ( ) 0 + ω the there are ifiitely may primes,, L such that f, L f are primes. We obtai a uite prime formula i prime distributio + ( N, + ) = {, L, N : f, L, f are primes} J + ( ωω ) N = (deg fi ) ( + o()). (8) + +! φ ( ω) log N i= A ew siftig fuctio is accurate sieve fuctio. Usig siftig fuctio we prove about 600 prime theorems [6]. Siftig fuctio provides proofs of the prime theorems which are simple eough to uderstad ad accurate eough to be useful.

2 Mathematicias have tried i vai to discover some order i the sequece of prime umbers but we have every reaso to believe that there are some mysteries which the huma mid will ever peetrate. Leohard Euler It will be aother millio years, at least, before we uderstad the primes. aul Erdös Suppose that Euler totiet fuctio φ( ω) =Π( ) = as ω, () where ω =Π is called primorial. Suppose that ( ω, h i ) =, where i =, L, φ( ω). We have prime equatios = ω+, L, = ω+ h () φ ( ω) φ( ω) where = 0,,, L. ()is called ifiitely may prime equatios (IME). Every equatio has ifiitely may prime solutios. We have ( N) h = ( o()). i = +, () φω ( ) i N i hi(mod ω) where h i deotes the umber of primes i N i i = ω+ h i = 0,,, L, ( N) the umber of primes less tha or equal to N. We replace sets of prime umbers by IME. () is the fudametal tool for provig the prime theorems i prime distributio. Let ω = 0 ad φ (0) = 8. From () we have eight prime equatios = 0+, = 0+ 7, = 0+, 4 = 0+, 5 = 0+ 7, 6 = 0+ 9, 7 = 0+, 8 = 0+ 9, = 0,,, L (4) Every equatio has ifiitely may prime solutios. THEOREM. We defie that prime equatios f (, L, ), L, f (, L, ) (5)

3 are polyomials (with iteger coefficiets) irreducible over itegers, where,, L are primes. If siftig fuctio J ( ) = 0 the (5) has fiite prime solutios. If J + ( ω) 0 the there exist ifiitely may primes, L, such that each f is a prime. ROOF. Firstly, we have siftig fuctio [-] + + ω J ( ω) =Π[( ) χ( )], (6) where χ ( ) is called sieve costat ad deotes the umber of solutios for the followig cogruece Π f ( q, L, q ) 0 (mod ), (7) i= where q =, L,, L, q =, L,. i J ( ) + ω deotes the umber of sets of, L, prime equatios such that f(, L, ), L, f(, L, ) are prime equatios. If J ( ) 0 + ω = the (5) has fiite prime solutios. If J ( ) 0 + ω usig χ ( ) we sift out from () prime equatios which ca ot be represeted,, L, the residual prime equatios of () are,, L prime equatios such that f(, L, ), L, f(, L, ) are prime equatios. Therefore we prove that there exist ifiitely may primes,, L such that f (, L, ), L, f(, L, ) are primes. Secodly, we have the best asymptotic formula [,,4,6] + ( N, + ) = {, L, N : f, L, f are primes} J + ( ωω ) N = (deg fi ) ( + o()). (8) + +! φ ( ω) log N i= (8)is called a uite prime formula i prime distributio. Let =, = 0, J ( ω) φω ( ) =. From (8) we have prime umber theorem N ( N, ) = { N: is prime } = ( + o()).. (9) log N

4 Number theorists believe that there are ifiitely may twi primes, but they do ot have rigorous proof of this old cojecture by ay method. All prime theorems are cojectures except the prime umber theorem, because they do ot prove that prime equatios have ifiitely may prime solutios. We prove the followig cojectures by this theorem. Example. Twi primes +, (00BC). From (6) ad (7) we have siftig fuctio J ( ω) = Π( ) 0. Sice J ( ) 0 ω i () exist ifiitely may prime equatios such that + is a prime equatio. Therefore we prove that there are ifiitely may primes such that + is a prime. Let ω = 0 ad J (0) =. From (4) we have three prime equatios = 0+, = 0+ 7, = J( ω) ω N ( N,) = { N : + prime } = ( o()) φ ( ω) log N + N = Π ( o()). + ( ) log N I 996 we proved twi primes cojecture [] Remar. J ( ) ω deotes the umber of prime equatios, ω N ( + o()) the umber of solutios of primes for every prime φ ( ω) log N equatio. Example. Eve Goldbach s cojecture N = +. Every eve umber N 6 is the sum of two primes. 4

5 From (6) ad (7) we have siftig fuctio J( ω) =Π( ) Π 0. N Sice J ( ω) 0 as N i () exist ifiitely may prime equatios such that N is a prime equatio. Therefore we prove that every eve umber N 6 is the sum of two primes. J( ω) ω N ( N, ) = { N, N prime } = ( o()). φ ( ω) log N + N = Π ( o()) Π +. ( ) N log N I 996 we proved eve Goldbach s cojecture [] Example. rime equatios, +, + 6. From (6) ad (7) we have siftig fuctio J ( ω) =Π( ) 0, 5 J ( ) ω is deotes the umber of prime equatios such that + ad + 6 are prime equatios. Sice J ( ω) 0 i () exist ifiitely may prime equatios such that + ad + 6 are prime equatios. Therefore we prove that there are ifiitely may primes such that + ad + 6 are primes. Let ω = 0, J(0) =. From (4) we have two prime equatios = 0+, = J ( ω) ω N ( N,) = { N : +, + 6are primes} = ( + o()). φ ( ω) log N Example 4. Odd Goldbach s cojecture N = + +. Every odd umber N 9 is the sum of three primes. 5

6 From (6) ad (7) we have siftig fuctio J( ω) =Π( + ) ) Π 0 N. + Sice J ( ω) 0 as N i () exist ifiitely may pairs of ad prime equatios such that N is a prime equatio. Therefore we prove that every odd umber N 9 is the sum of three primes. J( ωω ) N ( N,) = {, N : N prime } = ( o()) φ ( ω) log N +. N =Π + ( ()) Π o +. ( ) N + log N Example 5. rime equatio = +. From (6) ad (7) we have siftig fuctio ( ) J ( ω) =Π + 0 J ( ω ) deotes the umber of pairs of ad prime equatios such that is a prime equatio. Sice J ( ) 0 ω i () exist ifiitely may pairs of ad prime equatios such that is a prime equatio. Therefore we prove that there are ifiitely may pairs of primes ad such that is a prime. J( ωω ) N ( N, ) = {, N : + prime } = ( o()). 4 φ ( ω) log N + Note. deg ( ) =. Example 6 []. rime equatio = +. From (6) ad (7) we have siftig fuctio J, ( ω) =Π ( ) χ( ) 0 6

7 where χ ( ) = ( ) if χ ( ) = otherwise. (mod ) ; χ ( ) = 0 if / (mod ) ; Sice J ( ω) 0 i () there are ifiitely may pairs of ad prime equatios such that is a prime equatio. Therefore we prove that there are ifiitely may pairs of primes ad such that is a prime. ( J ( ω) ω N N,) = {, N : + prime} = ( + o()). 6φ ( ω) log N 4 Example 7 []. rime equatio = + ( + ). From (6) ad (7) we have siftig fuctio J ( ω) =Π ( ) χ( ) 0 where χ ( ) = ( ) if (mod 4) ; χ ( ) = ( ) if (mod8) ; χ ( ) = 0 otherwise. Sice J ( ω) 0 i () there are ifiitely may pairs of ad prime equatios such that is a prime equatio. Therefore we prove that there are ifiitely may pairs of primes ad such that is a prime. J( ωω ) N ( N,) = {, N : prime } = ( o()). 8 φ ( ω) log N + Example 8 [4-0]. Arithmetic progressios cosistig oly of primes. We defie the arithmetic progressios of legth., = + d, = + d, L, = + ( ) d,(, d) =. (0) ( N,) = { N :, + d, L, + ( ) d are primes} 7

8 ( ) J ωω N = ( + o()).. φ ( ω) log N If J ( ω ) = 0 the (0) has fiite prime solutios. If J ( ) 0 ω the there are ifiitely may primes such that,, L are primes. To elimiate d from (0) we have =, = ( j ) ( j ), j. j From (6) ad (7) we have siftig fuctio J ( ω) = Π ( ) Π( )( + ) 0 < Sice J ( ω) 0 i () there are ifiitely may pairs of ad prime equatios such that, L, are prime equatios. Therefore we prove that there are ifiitely may pairs of primes ad such that,, L are primes. { } = ( N,), N : ( j ) ( j ) prime, j ( ) J ωω N = ( + o()) φ ( ω) log N ( + ) N = Π Π ( + o()). < ( ) ( ) log N Example 9. It is a well-ow cojecture that oe of, +, + is always divisible by. To geeralize above to the primes, we prove the followig cojectures. Let be a square-free eve umber.., +, +, where ( + ). From (6) ad (7) we have J () = 0, hece oe of, +, + is always divisible by. 4., +, +, L, +, 8

9 where 5( + b), b=,. From (6) ad (7) we have J (5) = 0, hece oe of 4, +, +, L, + is always divisible by 5. 6., +, +, L, +, where 7( + b), b=,4. From (6) ad (7) we have J (7) = 0, hece oe of 6, +, +, L, + is always divisible by , +, +, L, +, where ( + b), b=,4,5,9. From (6) ad (7) we have J () = 0, hece oe of 0, +, +, L, + is always divisible by. 5., +, +, L, +, where ( + b), b=,6,7,. From (6) ad (7) we have J () = 0, hece oe of, +, +, L, + is always divisible by. 6 6., +, +, L, +, where 7 ( + b), b=,5,6,7,0,,,4,5. From (6) ad (7) we have J (7) = 0, hece oe of 6, +, +, L, + is always divisible by , +, +, L, +, where 9 ( + b), b= 4,5,6,9,6.7. From (6) ad (7) we have J (9) = 0, hece oe of 8, +, +, L, + is always divisible by 9. Example 0. Let be a eve umber. i., +, i=,,5, L,+, From (6) ad (7) we have J ( ω) 0. Therefore we prove that there exist i ifiitely may primes such that, + are primes for ay. i., +, i=, 4,6, L,. From (6) ad (7) we have J ( ω) 0. Therefore we prove that there exist i ifiitely may primes such that, + are primes for ay. 9

10 Example. rime equatio = + From (6) ad (7) we have siftig fuctio J ω =Π +. ( ) ( ) 0 Sice J ( ω) 0 i () there are ifiitely may pairs of ad prime equatios such that is prime equatios. Therefore we prove that there are ifiitely may pairs of primes ad such that is a prime. J( ωω ) N ( N,) = {, N : prime } = ( o()). φ ( ω) log N + I the same way we ca prove = + which has the same siftig fuctio. Siftig fuctio is accurate sieve fuctio. Usig it we ca prove ay irreducible prime equatios i prime distributio. There are ifiitely may twi primes but we do ot have rigorous proof of this old cojecture by ay method [0]. As strog as the umerical evidece may be, we still do ot eve ow whether there are ifiitely may pairs of twi primes []. All prime theorems are cojectures except the prime umber theorem, because they do ot prove the simplest twi primes. They cojecture that the prime distributio is radomess [-5], because they do ot uderstad theory of prime umbers. Acowledgemets The Author would lie to express his deepest appreciatio to M. N. Huxley, R. M. Satilli, L. Schadec ad G. Weiss for their helps ad supports. Refereces [] Chu-Xua Jiag, O the Yu-Goldbach prime theorem, Guagxi Scieces (Chiese) (996), 9-. [] Chu-Xua Jiag, Foudatios of Satilli s isoumber theory, art I, Algebras Groups ad Geometries, 5(998), 5-9. [] ChuXua Jiag, Foudatios of Satilli s isoumber theory, art II, Algebras Groups ad Geometries, 5(998),

11 [4] Chu-Xua Jiag, Foudatios Satilli s isoumber theory, I: Fudametal ope problems i scieces at the ed of the milleium, T. Gill, K. Liu ad E. Trell (Eds) Hadroic ress, USA, (999), [5] Chu-Xua Jiag, roof of Schizel s hypothesis, Algebras Groups ad Geometries, 8(00), [6] Chu-Xua Jiag, Foudatios of Satilli s isomuber theory with applicatios to ew cryptograms, Fermat s theorem ad Goldbach s cojecture, Iter. Acad. ress, 00, MR004c: 00, [7] Chu-Xua Jiag,rime theorem i Satilli s isoumber theory, 9(00), [8] Chu-Xua Jiag, rime theorem i Satilli s isoumber theory (II), Algebras Groups ad Geometries, 0(00), [9] Chu-Xua Jiag, Disproof s of Riema s hypothesis, Algebras Groups ad Geometries, (005), Riema.pdf [0] Chu-Xua Jiag, Fiftee cosecutive itegers with exactly prime factors, Algebras Groups ad Geometries, (006), 9-4. [] Chu-Xua Jiag, The simplest proofs of both arbitrarily log arithmetic progressios of primes, preprit, 006. [] D. R. Heath-Brow, rimes represeted by x + y, Acta Math., 86 (00), -84. [] J. Friedlader ad H. Iwaiec, The polyomial x + y 4 captures its primes, A. Math., 48(998), [4] E. Szemerédi, O sets of itegers cotaiig o elemets i arithmetic progressios, Acta Arith., 7(975), [5] H. Fursteberg, Ergodic behavior of diagoal measures ad a theorem of Szemerédi o arithmetic progressios, J. Aalyse Math., (997), [6] W. T. Gowers, A ew proof of Szemerédi s theorem, GAFA, (00), [7] B. Kra, The Gree-Tao theorem o arithmetic progressios i the primes: A ergodic poit of view, Bull. Amer. Math. Soc., 4(006), -. [8] B. Gree ad T. Tao, The primes cotai arbitrarily log arithmetic

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