On the Foundamental Theorem in Arithmetic. Progession of Primes

Transcription

1 On the Foundamental Theorem in Arithmetic roession of rimes Chun-Xuan ian. O. Box 9 Beijin 008. R. China jianchunxuan@vip.sohu.com Abstract Usin ian function we prove the foundamental theorem in arithmetic proression of primes [-]. The primes contain only < lon arithmetic proressions but the primes have no > lon arithmetic proressions.terence Tao is recipient of 006 Fields medal.green and Tao proved that the primes contain arbitrarily lon arithmetic proressions which is absolutely false[-9].they do not understand the arithmetic proression of primes [-].

2 Theorem. The fundamental theorem in arithmetic proression of primes. We define the arithmetic proression of primes [-]. i i i 0 L () where is called a common difference is called -th prime. We have ian function [-] ( ) ( X ( )) () X () denotes the number of solutions for the followin conruence where q L. If i ( q i) 0(mod ) then X ( ) 0 ; ( ) X otherwise. From () we have () ( ) ( ) ( ). () If then ( ) 0 0 there exist finite primes such that are primes. If then 0 there exist infinitely L < < lon arithmetic proressions but the primes have no > many primes such that are primes. The primes contain only arithmetio proressions. We have the best asymptotic formula [-] ( ) { i prime 0 i } lon φ ( o()) () where φ( ) ( ) IS called primorial φ () Euler function. Suppose. From () we have

3 From () we have [-] i i i 0 L. (6) ) ( ) ( ) as (7) ( We prove that there exist infinitely many primes such that L are primes for all. From () we have ( ) ( ( ) ) ( ) ( o()). (8) From (8) we are able to find the smallest solutions ( ) > for lare. Theorem is foundations for arithmetic proression of primes Example. Suppose. From (6) we have the twin primes theorem. (9) From (7) we have ( ) ( ) as (0) We prove that there exist infinitely many primes such that are primes. From (8) we have the best asymptotic formula ( ) ( o()). ( ) () Twin prime theorem is the first theorem in arithmetic proression of primes. Green and Tao do not prove the twin prime theorem. Therefore Green Tao theorem is absolutely false [-9]. The prime distribution is order rather than randomness. The arithmetic proreessions of primes are not directly related to erodic theory harmonic analysis

4 discrete eometry and additive combinatorics. Conjectures and theorems on arithmetic proressions of primes are absolutely false [-] because they do not understand the arithmetic proressions of primes. Example. Suppose 6. From (6) we have i 6i 0. () i From (7) we have ( ) ( ) as () We prove that there exist infinitely many primes such that and are primes. From (8) we have the best asymptotic formula ( ) ( ) 7 ( o()). () ( ) Example. Suppose From (6) we have i 09870i 0 7. () i L From (7) we have ( ) 6960 ( 8) as (6) 9 We prove that there exist infinitely many primes such that L are primes. From (8) we have the best asymptotic formula ( 8) ( ) ( ()) ( ) o (7) From (7) we are able to find the smallest solutions ( ) 0. 8 > On May Wroblewsi and Raanan Chermoni found the first nown case of primes: n for n 0 to. Theorem can help in findin for primes in arithmetic proressions of

5 primes. Corollary. Arithmetics proression with two prime variables Suppose d. From () we have d d L ( ) d ( d). (8) From (8) we obtain the arithmetic proression with two prime variables: and j ( j ) ( j ) j <. (9) We have ian function [] ( ) [( ) X ( )] (0) X () denotes the number of solutions for the followin conruence j [( j ) q ( j ) q ] 0(mod ) () where q L ; q L. From () we have ( ) ( ) ( )( ) as. () < We prove that there exist infinitely many primes and such that L are primes for < we have the best asymptotic formula {( j ) ( j ) prime j } ( ) φ ( o()) () From () we have the best asymptotic formula

6 ( ) ( ) ( o()). ( ) ( ) () < From () we are able to find the smallest solution ( ) 0 for lare <. Example. Suppose and >. From (9) we have >. () From () we have ( ) ( )( ) as (6) We prove that there exist infinitely many primes and such that are primes. From () we have the best asymptotic formula ( ) ( o()).0 ( o()). ( ) Example. Suppose and >. From (9) we have (7). (8) From () we have ( ) ( )( ) as (9) We prove that there exist infinitely many primes and such that and are primes. From () we have the best asymptotic formula ( 9 ( ) ) ( o()). (0) ( ) Example 6. Suppose and >. From (9) we have 6

7 . () From () we have ( ) ( )( ) as () We prove that there exist infinitely many primes and such that and are primes. From () we have the best asymptotic formula 7 ( ) ( ) ( o()). () ( ) Green and Tao study only corollary which is not the theorem [-9]. Corollary. Arithmetic proression with three prime variables From (8) we obtain the arithmetic proression with three prime variables: and j ( j ) ( j ) j < () We have ian function (( ) X ( )) () X () denotes the number of solutions for the followin conruence j where q i L i. ( q ( j ) q ( j ) q) 0(mod ) (6) Example 7. Suppose and >. From () we have. (7) From () and (6) we have ( )( ) as (8) We prove that there exist infinitely many primes and and such that are primes. we have the best asymptotic formula 7

8 For ( ) ( o()). ( ) (9) from () and (6) We have ian function < ( ) ( ) ( ) ( )[( ) ( )( )] as. (0) We prove that there exist infinitely many primes and and such that L are primes for <. we have the best asymptotic formula { ( j ) ( j ) prime j } ( ) φ ( o()). () From () we have ( ) ( ) < ( ) ( ) [( ) ( )( )] ( ) ( o()). () From () we are able to find the smallest solution ( 0 ) for lare <. Corollary. Arithmetic proression with four prime variables > From (8) we obtain the arithmetic proression with four prime variables: and j ( j ) ( j ) j < () 8

9 We have ian function [( ) X ( )] ( ) () X () denotes the number of solutions for the followin conruence where j [ q ( j ) q ( j ) q q] 0 q i L i (mod ) () Example 8. Suppose and >. From () we have From () and () we have. (6) ( ) ( )( 6 ) as. (7) We prove there exist infinitely many primes and such that are primes. We have the best asymptotic formula ( ) ( o()). (8) φ Example 9. Suppose 6 and > 6. From () we have 6 From () and () we have. (9) ( ) 0 ( )( 0 9) as. (0) We prove there exist infinitely many primes and such that and 6 are primes. We have the best asymptotic formula ( ) ( o()). (0) 6 6 φ 9

10 For 7 from () and () we have ian function ( ) 6 ( )( ) ( ) < as ( ) {( ) ( ) [( )( ) ] ( )( 9) } () We prove there exist infinitely many primes and such that are primes. We have best asymptotic formula L { ( j ) ( j ) prime j } ( ) L h φ ( o()). () I than professor Huan Yu-Zhen for compution of ian functions. References [] Chun-Xuan ian On the prime number theorem in additive prime number theory reprint 99. [] Chun-Xuan ian The simplest proofs of both arbitrarily lon arithmetic proressions of primes preprint 006. [] Chun-Xuan ian Foundations of Santiili s isonumber theory with applications to new cryptorams Fermat s theorem and Goldbach s conjecture Inter. Acad. ress MR 00c: 00 http: // [] B. Green and T. Tao The primes contain arbitrarily lon arithmetic proressions Ann. Math (008). [] T. Tao The dichotomy between structure and randomness arithmetic proressions and the primes. In: roceedins of the international conress of mathematicians (Madrid) Europ. Math. Soc. Vol (007). [6] T. Tao and V. Vu Additive combinatorics Cambride University ress (006). [7] T. Tao Lon arithmetic proressions in the primes Australian mathematical society meetin 6 September

11 [8] T. Tao What is ood mathematics? Bull. Amer. Soc. 6-6 (007). [9] B. Green Lon arithmetic proressions of primes arxiv: math. T/00806 v Au 00. [0] E. Szemerédi On sets of inteers containin no elements in arithmetic proressions Acta Arith (97). [] H. Furstenber Erodic behavior of diaonal measures and a theorem of Szemeré di on arithmetic proressions. Analyse Math. 0-6 (977). [] W. T. Gowers A new proof of Szemerédi s theorem GAFA 6-88 (00). [] B. Kra The Green-Tao theorem on arithmetic proressions in the primes: an erodic point of view Bull. Amer. Math. Soc. - (006). []. G. van der Corput Über Summen von rimzahlen und rimzahlquadraten Math. Ann. 6-0 (99). []. Erdös. Turán On some sequences of inteers. London Math. Soc. 6-6 (96)