The Simplest Proofs of Both Arbitrarily Long. Arithmetic Progressions of primes. Abstract

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1 The Simplest Proofs of Both Arbitrarily Lo Arithmetic Proressios of primes Chu-Xua Jia P. O. Box 94, Beiji P. R. Chia Abstract Usi Jia fuctios J ( ω ), J ( ω ) ad J ( ) 4 ω we prove both arbitrarily lo arithmetic proressios of primes: () P = i P + + di, ( Pd, ) =, i =,, L,,, which have the same Jia fuctio; () P = i P + + ωi, i =,, L,,, ω = Π P ad eeralized arithmetic P P proressios of primes Pi = P+ iω ad P = i P + + iω, i=,,, L. The Gree-Tao theorem is false, because they do ot prove the twi primes theorem ad arithmetic proressios of primes [].

2 I prime umbers theory there are both well-ow cojectures that there exist both arbitrarily lo arithmetic proressios of primes. I this paper usi Jia fuctios J ( ω ), J ( ω ) ad J ( ) 4 ω we obtai the simplest proofs of both arbitrarily lo arithmetic proressios of primes. Theorem. We defie arithmetic proressios of primes: We rewrite () P, P = P + d, P = P + d, L, P = P + ( ) d,( P, d) =. () P = P P, P = ( j ) P ( j ) P, j. () j We have Jia fuctio [] J ω =Π P X P, () ( ) ( ) ( ) X ( P ) deotes the umber of solutios for the followi coruece j= [ ] Π ( j ) q ( j ) q 0(mod P), (4) where q =,, L, P ; q =,, L, P. From (4) we have J ( ω) = Π ( P ) Π( P )( P + ) as ω. (5) P< P We prove that there exist ifiitely may primes P ad P such that P, are all primes for all. It is a eeralizatio of Euclid ad Euler proofs for the existece of ifiitely may primes []. We have the best asymptotic formula [] (,) = ( j ) P ( j ) P = prime, j, P, P { } J( ωω ) = ( + o()), φ ( ω) lo L, P (6)

3 where ω =Π P, φω ( ) =Π( P ), (7) P P ω is called primorials, φ( ω ) Euler fuctio. (6) is a eeralizatio of the prime umber theorem ( ) = ( + o()) []. lo Substituti (5) ad (7) ito (6) we have the best asymptotic formula P P ( P + ) (, ) = Π Π ( + o()). (8) ( P ) ( P ) lo P< P From (8) we are able to fid the smallest solutio ( 0,) > for lare. Grosswald ad Zaier obtai heuristically eve asymptotic formulae []. Let = ad d =. From () we have twi primes theorem: P = P +. The Gree-Tao theorem is false, because they do ot prove the twi primes theorem ad arithmetic proressios of primes []. Example. Let =. From () we have From (5) we have P = P P. (9) J ( ω) =Π( P )( P ) as ω. (0) We prove that there exist ifiitely may primes P ad P such that P are primes. From (8) we have the best asymptotic formula (,) =Π ( o()) ( o()) P + = +.() ( P ) lo lo Example. Let = 4. From () we have P = P P, P4 = P P. ()

4 From (5) we have J ( ω) = Π( P )( P ) as ω. () 5 P We prove that there exist ifiitely may primes P ad P such that P ad P 4 are all primes. From (8) we have the best asymptotic formula 9 P ( P ) (, ) = Π ( + o()). (4) P ( P ) lo Example. Let = 5. From () we have P = P P, P4 = P P, P5 = 4P P. (5) From (5) we have J ( ω) = Π( P )( P 4) as ω. (6) 5 P We prove that there exist ifiitely may primes P ad P such that P, P 4 ad P 5 are all primes. From (8) we have the best asymptotic formula Theorem. From () we obtai 7 P ( P 4) 4(, ) = Π ( + o()). (7) P ( P ) lo P4 = P + P P, Pj = P + ( j ) P ( j ) P, 4 j. (8) We have Jia fuctio [] J =Π P X P (9) 4( ω) (( ) ( )), X ( P ) deotes the umber of solutios for the followi coruece Π ( q + ( j ) q ( j ) q ) 0(mod P), (0) j= 4 where q =,, L, P, i =,,. i 4

5 From (0) we have J4( ω) = Π ( P ) Π ( P ) ( P ) ( P )( ) P< ( ) ( ) P asω. () We prove there exist ifiitely may primes P, P ad P such that P 4, are all primes for all 4. We have the best asymptotic formula [] (, 4) = P + ( j ) P ( j ) P = prime, 4 j, P, P, P { } J4( ωω ) = ( + o()). 6 φ ( ω) lo Substituti (7) ad () ito () we have L, P () (,4) P P [( P ) ( P )( )] () = Π Π ( + o()). ( ) ( ) 6 P< ( P ) P ( P ) lo From () we are able to fid the smallest solutio ( 0,4) > for lare. Example 4. Let = 4. From (8) we have From () we have P4 = P + P P (4) J =Π P P P+ as ω. (5) 4( ω) ( )( ) We prove there exist ifiitely may primes P, P ad P such that P 4 are primes.from () we have (, 4) = Π + ( o()). + 4 P ( P ) lo From () We obtai the followi equatios: (6) 5

6 { L } (,5) = P + ( j ) P ( j ) P + P = prime,5 j, P,, P J5( ωω ) = + o 4 φ ( ω) lo ( ()) (7) { L } (,6) = P + ( j 4) P ( j 4) P P + P = prime,6 j, P,, P J6( ωω ) = + o 0 φ ( ω) lo Theorem. We defie arithmetic proressios of primes: From (9) we have We have Jia fuctio [] ( ()) (8) P P di i L. (9) = +, = i+,,, P = P P, P = ( j ) P ( j ) P, j. (0) j J ω P X P, () ( ) =Π ( ) ( ) X ( P ) deotes the umber of solutios for the followi coruece Π ( j ) q ( j ) q 0(mod P), () j= where q =,, L, P, q =,, L, P. From () we have J ( ω) = Π ( P ) Π( P )( P + ) as ω. () P< P We prove that there exist ifiitely may primes P ad P such that P, are all primes for all. We have the best asymptotic formula [] { } (,) = ( j ) P ( j ) P = prime, j, P, P L, P J( ωω ) = ( + o()). (4) φ ( ω) lo 6

7 Substituti (7) ad () ito (4) we have P P ( P + ) (,) = Π Π ( + o()). (5) ( P ) ( P ) lo P< P Theorem 4. We defie arithmetic proressios of primes: From (6) we have P P di i L. (6) = 5 +, = i+,,, P = P + P P, P = P + ( j ) P ( j ) P, 4 j. (7) j We have Jia fuctio [] J4( ω) = Π ( P ) Π ( P ) ( P ) ( P )( ) P< ( ) ( ) P asω. (8) We prove that there exist ifiitely may primes P, P ad P such that P,, 4 L P are all primes for all 4.. We have the best asymptotic formula 5 { } (, 4) = P + ( j ) P ( j ) P = prime, 4 j, P, P, P J4( ωω ) = ( + o()). 6 5 φ ( ω) lo Theorem 5. We defie arithmetic proressios of primes: From (40) we have We have Jia fuctio [] (9) P = j P + di, i =,, L +,,. (40) P = P P, P = ( j ) P ( j ) P. (4) j J ( ω) = Π ( P ) Π( P )( P + ) as ω. (4) P< P 7

8 We prove that there exist ifiitely may primes P ad P such that P, are all primes for all. We have the best asymptotic formula [] (,) = ( j ) P ( j ) P = prime, j, P, P { } J( ωω ) (4) =. φ ( ω) lo Substituti (7) ad (4) ito (4) we have P< P P P ( P + ) (, ) = Π Π ( + o()). (44) ( P ) ( P ) lo Theorem 6. We defie arithmetic proressios of primes: From (45) we have L, P P = j P + di, i =,, L +,,. (45) P = P + P P, P = P + ( j ) P ( j ) P, 4 j. (46) 4 j We have Jia fuctio [] J4( ω) = Π ( P ) Π ( P ) ( P ) ( P )( ) P< ( ) ( ) P asω. (47) We prove that there exist ifiitely may primes P, P ad P such that P,, 4 L P are all primes for all 4.. We have the best asymptotic formula [] (,4) ( ) ( ) = P + j P j P = prime,4 j, P, P, P { } J4( ωω ) = ( + o()). 6 φ ( ω) lo (48) 8

9 Substituti (7) ad (47) ito (48) we have (,4) P P [( P ) ( P )( )] = Π Π ( + o()). ( ) ( ) 6 P< ( P ) P ( P ) lo (49) Theorem 7. We defie aother arithmetic proressios of primes [, 4]: P = i P + ωi, i = +,, L, (50) where ω = Π is called a commo differece, P is called th prime. P P We have Jia fuctio [, 4] J ( ω) = Π( P X( P)), (5) X ( P ) deotes the umber of solutios for the followi coruece where q=,, L, P. Π ( q+ ω i) 0(mod P), (5) i= If P ω, the X( P ) = 0; X ( P) = otherwise. From (5) we have J ( ω) = Π ( P ) Π ( P ). (5) P P+ P If = P + the J ( P + ) = 0, ( ) 0 J ω =, there exist fiite primes P such that P,, L P are all primes. If < P + the J ( ω) 0, there exist ifiitely may primes P such that P, L, P are all primes. We have the best asymptotic formula [,4] 9

10 { ω } (, ) = P + i= prime, i, P J ( ωω ) ( o()). φ ( ω) lo = + (54) Let = P +. From (50) we have From (5) we have [, 4] P = P + ω i, i=,, L, P. (55) i+ + J ( ω) = Π ( P ) Π ( P P + ) as ω (56) + P P+ P We prove that there exist ifiitely may primes P such that P, L, P + are all primes for all P +. P Substituti (7) ad (56) ito (54) we have (,) = P + P P + + P P ( P P + + ) P P P P P P ( ) (lo ) Π Π = ( + o()). P P (57) From (57) we are able to fid the smallest solutios P 0 + (,) > for lare P +. Example 5. Let P =, ω =, P =. From (55) we have the twi primes theorem From (56) we have P = P + (58). J ( ω) =Π( P ) as ω, (59) We prove that there exist ifiitely may primes P such that P are primes. From 0

11 (57) we have the best asymptotic formula (, ) = Π ( o()) +. (60) ( P ) lo Example 6. Let P =, ω = 6, P = 5. From (55) we have P = i P + + 6, i i =,,. (6) From (56) we have J ( ω) = Π( P 4) as ω. (6) 5 P We prove that there exist ifiitely may primes P such that P, P ad P 4 are all primes. From (57) we have the best asymptotic formula P ( P 4) 4 5 P 4 4 (, ) = 7 Π ( + o()). (6) ( P ) lo Example 7. Let P 9 =, ω 9 = 09870, P 0 = 9. From (55) we have From (56) we have P = i P i, i = +,, L, 7. (64) J ( ω) = Π ( P 8) as ω (65) 9 P We prove that there exist ifiitely may primes P such that P, L, P 8 are all primes. From (57) we have the best asymptotic formula 7 7 P P ( P 8) 8(, ) = Π Π ( + o()). (66) P P P ( P ) lo From (66) we are able to fid the smallest solutios 8( 0,) >. Theorem 8. We defie aother arithmetic proressios of primes:

12 P = P + ω i, i =,, L +,,. (67) i We have Jia fuctio [] J ( ω) = Π( P X( P)), (68) X ( P ) deotes the umber of solutios for the followi coruece where q =,, L P. Π ( q + ω i) 0(mod P), (69) i= If X ( P) = P ad J ( P ) = 0, the there exist fiite primes P such that P L P are primes. If X( P) < P ad J ( ω) 0, the there exist ifiitely, may primes P such that P, L, P are all prime for all P. We have the best asymptotic formula [] { ω } (,) = P + i= prime, i, P J( ωω ) = ( + o()). φ ( ω) lo (70) Example 8. Let =, = ad ω = 6. From (67) we have P P 6 = +, We have Jia fuctio [] P P = +, P 4 P 8 = + (7) 6 J( ω) = Π P 4 5 P P P P as ω (7) where 6, P P ad P deote the Leedre symbols.

13 We prove that there exist ifiitely may primes P such that P, P ad P 4 are all primes. We have the best asymptotic formula [] { } (,) = P + 6i= prime, i=,,, P 4 J( ωω ) = ( + o()) φ ( ω) lo We shall move o to the study of the eeralized arithmetic proressio of cosecutive primes [5]. A eeralized arithmetic proressio of cosecutive primes is defied to be the sequece of primes, PP, + ω, P+ ω, L, P+ ω ad P + ω, P + ω, L, P + ω, where P is the first term,. For example, 5,, 7,, ad, 7, 4, is a eeralized arithmetic proressio of primes with P = 5, ω = 6, = ad =. Theorem 9. We defie the eeralized arithmetic proressios: (7) P = P+ iω ad i P i P iω + = + (74) where i =, L,,. We have Jia fuctio [] ( ) J ( ω) =Π P X( p), (75) X ( P ) is the umber of solutios of coruece q=,, L P. Π ( q+ iω )( q + iω ) 0(mod P), (76) i= If X ( P) = P ad J ( P ) = 0, the there exist fiite primes P such that P, P, L, P are primes. If X( P) < P, J ( ) 0 ω, the there exist ifiitely

14 may primes P such that P, P, L, P are all primes. If J ( ) 0 ω, we have the best asymptotic formula of the umber of primes P [] J ( ωω ) (, ) = ( + o()). (77) φ ( ω) (lo ) Example 9. Let ω = 6, =, ad =. From (74) we have P = P+ 6, P = P+, P = P+ 8 ad P = P + 6, P = P +, P = P + 8. (78) We have Jia fuctio [] 6 J( ω) = 67 Π P 7 0 P P P (79) Sice J ( ) ω as ω, there exist ifiitely may primes P such that P, L, P are all primes. 6 From (77) we have 6 J( ωω ) 7 (, ) = ( o()) φ ( ω) lo + (80) Remar. Theorems, ad 5 have the same Jia fuctio J ( ) ω ad theorems, 4 ad 6 the same Jia fuctio J 4 ( ω ) which have the same character. All irreducible prime equatios have the Jia fuctios ad the best asymptotic formulas []. I our theory there are o almost primes, for example P = PP + ad 4

15 = P+ PP are theorems of three euie primes. Usi the sieve method, circle method, erodic theory, harmoic aalysis, discrete eometry, ad combiatories they are ot able to attac twi primes cojecture, Goldbach cojecture, lo arithmetic proressios of primes ad other problems of primes ad to fid the best asymptotic formulas. The proofs of Szemerédic s theorem are false, because they do ot prove the twi primes theorem ad arithmetic proressios of primes [, 6-0]. Acowledemet the Author would lie to tha Zuo Mao-Xia for helpful coversatios. Refereces [] Chu-Xua, Jia, Foudatios of Satiili s isoumber theory with applicatios to ew cryptorams, Fermat s theorem ad Goldbach s cojecture, Iter. Acad. Press, pp.68-74, 00, MR 004c: 00, http: // [] E. Grosswald, Arithmetic proressios that cosist oly of primes, J. umber Theory, 4, 9- (98). [] B. Gree ad T. Tao, The primes cotai arbitrarily lo arithmetic proressios, to appear i A. Math. [4] Chu-Xua, Jia, O the prime umber theorem i additive prime umber theory, Yuxihe mathematics worshop, Fujia ormal uiversity, October 8 to 0, preprits, 995. [5] Chu-Xua, Jia, Geeralized Arithmetic Proressios Pi = P+ iω ad m P+ i P iω = +, preprits, 00. [6] E. Szemerédi, O sets of iteers cotaii o elemets i arithmetic proressios, Acta Arith., 7, 99-45(975). [7] H. Fursteber, Erodic behavior of diaoal measures ad a theorem of Szemerédi o arithmetic proressios, J. Aalyse Math.,, (977). [8] W. T. Gowers, A ew proof of Szemerédi s theorem, GAFA,, (00). [9] B. Kra, The Gree-Tao theorem o arithmetic proressios i the primes: a erodic poit of view, Bull. Amer. Math. Soc., 4, - (006). [0] Yu. I. Mai, A. A. Pachishi, Itroductio to Moder umber Theory, d ed, 5

16 pp. 8-8, Sprier, 年 0 月完成于北京 6