Diophantine Equation. Has Infinitely Many Prime Solutions
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1 Diophatie Equatio + L+ λ + λ λ + L+ Has Ifiitely May rime Solutios Chu-Xua iag. O. Box 9, Beijig 0085,. R. Chia public. bta. et. c Abstract By usig the arithmetic fuctio we prove that Diophatie equatio ( ω + L+ λ + λ λ + L+ It is the Book proof. The ushers i a ew era i the prime umbers theory. ( ω
2 AMS Mathematics subject classificatio: rimary A ew brach of umber theory: Satilli s additive isoprime theory is itroduced. By usig the arithmetic fuctio (ω the followig prime theorems have bee proved [-8]. It is the Book proof.. There exist ifiitely may twi primes.. The Goldbach theorem. Every eve umber greater tha is the sum of two odd primes.. There exist fiitely may Mersee primes, that is, primes of the form where is prime.. There exist fiitely may Fermat primes, that is, primes of the form There exist fiitely may repuit primes whose digits (i base 0 are all oes. 6. There exist ifiitely may primes of the forms: x +, x +, x 8 +, x There exist ifiitely may primes of the form: x. 8. There exist ifiitely may prime m-chais, j + mj ± ( m, m,, L, icludig the Cuigham chais. 9. There exist ifiitely may triplets of cosecutive itegers, each beig the product of k distict primes, (Here is a example: , , Every iteger m may be writte i ifiitely may ways i the form m + k where k,,, L, ad are primes.. There exist ifiitely may Carmichael umbers, which are the product of three primes, four primes, ad five primes.. There exist ifiitely may prime chais i the arithmetic progressios.. I a table of prime umbers there exist ifiitely may k-tuples of primes, 5 where k,,, L, 0.
3 . roof of Schizel s hypothesis. 5. Every large eve umber is represetable i the form + L. It is the primes theorem which has o almost-primes. I this paper by usig the arithmetic fuctio (ω Diophatie equatios are studied. Theorem. Diophatie equatio λ b + L , λ λ + L+ has ifiitely may prime solutios, where b is the iteger. We rewrite ( λ λ λ ( + L + L b. The arithmetic fuctio [-8] is ( ( ( ω H 0, where ω is called the primorials. i Let H ( deote the umber of solutios of the cogruece i λ ( λ λ q L + q q q L q b (mod, where q j,, L,, j,, L,. Sice ( ω as ω, there exist ifiitely may primes,l, such that It is the Book proof. It is a geeralizatio of the Euler proof for the existece of ifiitely may primes. The best asymptotic formula [-8] is {, L, :, L, ; (, +
4 ( ω ω ( ω log (!( λ + λmax φ ( + O(. 5 where λ max is a maximal value amog ( λ, L, λ, φ ( ω ( is called the Euler fuctio of the primorials. i Theorem. Diophatie equatio 6 The arithmetic fuctio [-8] is ( ( ( ω + + χ 0, 7 i where χ ( + if b; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that The best asymptotic formula [-8] is {, :, ; (, ( ω ω φ ( ω log ( + O(. 8 Theorem. Diophatie equatio The arithmetic fuctio is 9 ( + + χ( 0 ( ω, 0 i
5 where χ ( if b; b χ ( otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 6φ ( ω log ( + O(. Theorem. Diophatie equatio The arithmetic fuctio is + + ( + 0 ( ω, i Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 6φ ( ω log ( + O(. Theorem 5. Diophatie equatio +, 5 + 5
6 The arithmetic fuctio is ( + + χ( 0 ( ω, 6 i where χ ( if ( ; χ ( otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 8φ ( ω log ( + O(. 7 Theorem 6. Diophatie equatio 8 0 has ifiitely may prime solutios, where 0 is a odd prime. The arithmetic fuctio is ( + χ( 0 ( ω, 9 i where χ ( + if b; χ ( + 0 if 0 b (mod ; 0 χ ( if b (mod ; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, 6
7 ( ω ω ( + φ ( ω log 0 ( + O(. 0 Theorem 7. Diophatie equatio b + 0, has ifiitely may prime solutios, where 0 is a odd prime. The arithmetic fuctio is ( ( ( ω + χ 0, i where χ ( + if b ; χ ( + 0 if ( ; χ ( 0 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω ( + φ ( ω log 0 ( + O(. Theorem 8. Diophatie equatio has ifiitely may prime solutios The arithmetic fuctio is +, + ( + χ( 0 ( ω, 5 where χ ( + 8 if ( ; χ ( otherwise. Sice i ( ω as ω, there exist ifiitely may primes ad 7
8 such that {, :, ; (, ( ω ω 0φ ( ω log Theorem 9. Diophatie equatio has ifiitely may prime solutios The arithmetic fuctio is ( + O(. 6 +, 7 + ( + χ( 0 ( ω, 8 where χ ( if ( ; χ ( otherwise. i Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 0φ ( ω log ( + O(. 9 Theorem 0. Diophatie equatio The arithmetic fuctio is +, 0 + 8
9 ( + χ( 0 ( ω, i where χ ( if ( ; χ ( if 8( ; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 0φ ( ω log Theorem. Diophatie equatio ( + O(. +, + 6 The arithmetic fuctio is ( ( ( ω + χ 0, i where χ ( 5 if ( ; χ ( if 6( ; χ ( ( otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω φ ( ω log Theorem. Diophatie equatio ( + O(. 5 9
10 +, The arithmetic fuctio is ( ( ( ω + χ 0, 7 i where χ ( 7 if 6( ; χ ( if 8( ; χ ( if ( ; χ ( otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that {, :, ; (, ( ω ω 8φ ( ω log Theorem. Diophatie equatio The arithmetic fuctio [6] is ( + O( , ( 5 + ( ω 0. 0 i 5 Sice ( ω 5 as ω, there exist ifiitely may primes,, ad such that 5 {, L, :, L, ; (,5 5 0
11 5( ω ω 5 5 8φ ( ω log ( + O(. Theorem. Diophatie equatio + + L+ + L+ + +, + The arithmetic fuctio [6] is ( + ( + ω 0, i Sice ( ω as ω, there exist ifiitely may primes L, such that {, L, :, L, ; (, + Theorem 5. Diophatie equatio The arithmetic fuctio [6] is ( ω ω (! φ ( ω log ( + O(. + + L+,, 5 + L+ ( + ( + ω + 0, 6 i Sice ( ω as ω, there exist ifiitely may primes L, such that
12 The best asymptotic formula [6] is {, L, :, L, ; (, + Theorem 6. Diophatie equatio The arithmetic fuctio [6] is ( ω ω (! φ ( ω log ( + O(. 7 λ λ m + + L+ + +, 8 λ λ + L+ ( H( 0 ( ω, 9 Let H ( deote is the umber of solutios of the cogruece λ ( q + b i λ m λ λ L + q q q L q 0 (mod, 50 where q j, L,, j, L,. Sice ( ω as ω, there exist ifiitely may primes L, such that The best asymptotic formula [6] is {, L, :, L, ; (, + ( ω ω (!( m + λ φ ( ω log Theorem 7. For every iteger m Diophatie equatio ( + O(. 5 m + L+ + L+, 5
13 The arithmetic fuctio [6] is ( ( ω ( χ 0, 5 i ( + where χ ( if m; χ ( otherwise. Sice ( ω as ω, there exist ifiitely may primes L, such that The best asymptotic formula [6] is {, L, :, L, ; (, ( ω ω (! φ ( ω log ( + O(. 5 Theorem 8. Diophatie equatio ± m +, ( m,, m, b, 55 The arithmetic fuctio [-8] is ( ( ( ω + χ 0, 56 i where χ ( if m (mod ; χ ( + if m (mod ; χ ( + if m; χ ( otherwise. Sice ( ω as ω, there exist ifiitely may primes ad such that It is the Book proof. The best asymptotic formula [-8] is {, :, ; (,
14 6φ ( ω ω ( ω log ( + O(. 57 Let m ad. From (55 we have + [9]. Theorem 9. Diophatie equatio a + c,( a, c, ac, a + c d, 58 The arithmetic fuctio [-8] is ( ( ( ω + χ 0, 59 where i ac ac χ ( if ; χ ( + if ad ac. Sice ( ω as ω, there exist ifiitely may primes ad such that It is the Book proof. The best asymptotic formula [-8] is {, :, ; (, 8φ ( ω ω ( ω log ( + O(. 60 Let a ad c. From (58 we have Remark. b c ( + + c, [0]. a + + ad a arithmetic fuctio ad the same property. If have the same a, b ad c have o prime factor i commo, they represet ifiitely may primes as ad ru through the positive primes. Gauss proved that there are ifiitely may primes of the form x x + y + y,. It is show that there are ifiitely may primes of the form [8]. Theorem 0. Two primes represeted by 6.
15 Suppose that 6 ( + 6( 6. 6 Form (6 we have two equatios + 6 ad 6. 6 The arithmetic fuctio [-8] is ( ( ω 0, 6 5 i Sice ( ω as ω, there exist ifiitely may primes such that ad are primes. It is the Book proof. The best asymptotic formula [-8] is { : ;, primes } (, ( ω ω φ ( ω O log ( + (. 6 Theorem. Two primes represeted by Suppose that. ( ( Form (65 we have two equatios ad The arithmetic fuctio [-8] is ( ω 0, 67 5 i Sice ( ω as ω, there exist ifiitely may primes such that ad are primes. The best asymptotic formula [-8] is 5
16 { : ;, (, ( ω ω φ ( ω O log ( + (. 68 Theorem. Two primes represeted by Suppose that ( + ( Form (69 we have two equatios + ad The arithmetic fuctio [-8] is ( ( ( ω χ 0, 7 i where χ ( if 5( ; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes such that ad are primes. The best asymptotic formula [-8] is { : ;, primes } (, ( ω ω φ ( ω O log ( + (. 7 Theorem. Three primes represeted by 0. Suppose that 0 ( + 0( 0( Form (7 we have three equatios +0, 0 ad
17 The arithmetic fuctio [-8] is ( 8 ( ( ω χ 0, 75 7 i where χ ( if ( ; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes such that, ad are primes. The best asymptotic formula [-8] is { : ;,, primes } (, ( ω ω φ ( ω O log ( + (. 76 Theorem. Four primes represeted by Suppose that ( + ( ( ( Form (77 we have four equatios +,, , The arithmetic fuctio [-8] is ( ( ( ω χ 0, 79 i where χ ( if ( ; χ ( 0 otherwise. Sice ( ω as ω, there exist ifiitely may primes such that,, ad are primes. The best asymptotic formula [-8] is { : ;,,, primes } 5 (, ( ω ω 5 φ ( ω O 5 log ( + (. 80 7
18 REFERECES [] C. X. iag. O the Yu-Goldbach prime theorem. Guagxi Scieces (i Chiese,, ( [] C. X. iag. Foudatios of Satilli s isoumber theory. Algebras, Groups ad Geometries 5 ( MR000c:. [] C. X. iag. Foudatios of Satilli s isoumber theory. Algebras, Groups ad Geometries 5 ( [] C. X. iag. Foudatios of Satilli s isoumber theory. I: Foudametal ope problems i scieces at the ed of the milleium, T. Gill, K. Liu ad E. Trell (Eds Hadroic ress, USA, ( [5] C. X. iag. roof of Schizel s hypothesis, Algebras Groups ad Geometries 8 (00-0. [6] C. X. iag. Foudatios of Satilli s isoumber theory with applicatios to ew cryptograms, Fermat s theorem ad Goldbach s cojecture. Iteratioal Academic ress, 00. www. i-b-r.org [7] C. X. iag. rime theorem i Satilli s isoumber theory. To appear. [8] C. X. iag. rime theorem i Satilli s isoumber theory (II. To appear. [9] D. R. Heath-Brow. rimes represeted by (00-8. MR00b:. [0]. Friedlader ad H. Iwaiec. The polyomial x + y. Acta Math. 86 x + y captures its primes. A. of Math. 8 ( MR000c: Cole prize i umber theory ad 00 Ostrowski prize are awarded for this paper. 8
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