Santilli s Isoprime Theory

Transcription

1 Santilli s Isoprime Theory Chun-Xuan. Jiang P. O. Box 94, Beijing P. R. China jiangchunxuan@vip.sohu.com Dedicated to the 0-th anniversary of harmonic mechanics Abstract We establish the Santilli s isomathematics based on the generalization of the modern mathematics: isomultiplication a a atˆ a ˆ = b and isodivision a ˆ b= Iˆ, where I ˆ b is called isounit, TI ˆˆ =, T ˆ inverse of isounit. Keeping unchanged addition and subtraction ( +,, ˆ, ˆ) are four arithmetic operations in Santilli s isomathematics. Isoaddition a+ ˆ b= a+ b+ 0ˆ and isosubtraction a ˆ b= a b ˆ0 where ˆ0 0 is called isozero, ( + ˆ, ˆ, ˆ, ˆ) are four arithmetic operations in Santilli s new isomathematics. We establish Santilli s isoprime theory of the first kind, Santilli s isoprime theory of the second kind and isoprime theory in Santilli s new isomathematics. We give an example to illustrate the Santilli s isomathematics.

2 Introduction Santilli [] suggested the isomathematics based on the generalization of the multiplication division and multiplicative unit in modern mathematics. It is epoch-making discovery. From modern mathematics we establish the foundations of Santilli s isomathematics and Santilli s new isomathematics. We establish Santilli s isoprime theory of both first and second kind and isoprime theory in Santilli s new isomathematics. () Division and multiplication in modern mathematics. Suppose that a a a 0 = =, () where is called multiplicative unit, 0 exponential zero. From () we define division and multiplication a a b=, b 0, a b= ab, () b 0 a = a ( a a) = a a = a. () We study multiplicative unit a = a, a = a, a= / a. (4) n a/ b n n a/ b a/ b ( + ) =,( + ) =,( ) = ( ),( ) = ( ). (5) The addition +, subtraction -, multiplication and division are four arithmetic operations in modern mathematics which are foundations of modern mathematics. We generalize modern mathematics to establish the foundations of Santilli s isomathematics. () Isodivision and isomultiplication in Santilli s isomathematics. We define the isodivision ˆ and isomultiplication ˆ [-5] which are generalization of division and multiplication in modern mathematics.

3 We suppose that a ˆ a= a = I, 0 0, (6) 0 ˆ where Î is called isounit which is generalization of multiplicative unit, 0 exponential isozero which is generalization of exponential zero. We have Suppose that ˆa a ˆb= I, b 0, a ˆb= atb ˆ. b (7) 0 a= a ˆ( a ˆ a) = a ˆa = ati ˆˆ = a. (8) From (8) we have TI ˆˆ = (9) where T ˆ is called inverse of isounit Î. We conjectured [-5] that (9) is true. Now we prove (9). We study isounit Î ˆ ˆ ˆ ˆ I ˆ a ˆ I = a, a ˆ I = a, I ˆ a= a = I / a, (0) aˆ aˆ a nˆ b nˆ n b b ( + Iˆ) = Iˆ,( + Iˆ) = Iˆ,( Iˆ) = ( ) Iˆ,( Iˆ) = ( ) Iˆ. () Keeping unchanged addition and subtraction ( +,, ˆ, ˆ) are four arithmetic operations in Santilli s isomathematics, which are foundations of isomathematics. When I ˆ =, it is the operations of modern mathematics. () Addition and subtraction in modern mathematics. We define addition and subtraction x = a+ b, y = a b. () a+ a a= a. () a a = 0. (4) Using above results we establish isoaddition and isosubtraction (4) Isoaddition and isosubtraction in Santilli s new isomathematics.

4 We define isoaddition ˆ+ and isosubtraction ˆ. From (6) we have a+ ˆ b= a+ b+ c, a ˆ b= a b c. (5) a= a+ ˆ a ˆ a= a+ c c = a. (6) c = c. (7) Suppose that c = c = ˆ0, where ˆ0 is called isozero which is generalization of addition and subtraction zero. We have a+ ˆ b= a+ b+ 0, ˆ a ˆ b= a b 0ˆ. (8) When ˆ0= 0, it is addition and subtraction in modern mathematics. From above results we obtain foundations of santilli s new isomathematics ˆ = T ˆ, + ˆ = + 0ˆ + ; ˆ = Iˆ, ˆ = 0ˆ ; a ˆ b = atb ˆ, a + ˆ b = a + b + 0ˆ ; a a ˆb= Iˆ, a ˆ b= a b 0; ˆ a= a ˆa ˆ a= a, a= a+ ˆ a ˆ a= a; b a ˆa= a T, a+ ˆ a= a+ 0; ˆ a ˆ a= Iˆ, a ˆ a= 0ˆ 0; TI ˆˆ=. (9) ( + ˆ, ˆ, ˆ, ˆ) are four arithmetic operations in Santilli s new isomathematics. Remark, a ˆ( b+ ˆ c) = a ˆ( b+ c+ 0) ˆ, From left side we have a ˆ( b+ ˆ c) = a ˆb+ a ++ ˆ ˆ a ˆc) = a ˆ( b+++ ˆ c) = a ˆ ( b+ 0 ˆ + c), where += ˆ 0ˆ also is numbers. a ˆ( b ˆ c) = a ˆ( b c 0) ˆ. From left side we have a ˆ( b ˆ c) = a ˆb a ˆ ˆ a ˆc) = a ˆ( b ˆ c) = a ˆ( b 0 ˆ c), where ˆ = ˆ0 also is numbers. It is the mathematical problems in the st century and a new mathematical tool for studying and understanding the law of world. Santilli s isoprime theory of the first kind 4

5 Let Fa+ (,, ) be a conventional field with numbers a equipped with the conventional sum a+ b F, multiplication ab F and their multiplicative unit F. Santilli s isofields of the first kind Fˆ = Fˆ( aˆ, +, ˆ) are the rings with elements â = aiˆ (0) called isonumbers, where a F, the isosum aˆ + bˆ = ( a+ b) Iˆ () with conventional additive unit 0= 0Iˆ = 0, aˆ+ 0= 0+ aˆ = aˆ, â Fˆ and the isomultiplications is aˆ ˆbˆ= atb ˆ ˆˆ= aitbi ˆˆ ˆ= ( ab) Iˆ. () Isodivision is ˆ ˆ ˆ ˆ a a b= I. () b We can partition the positive isointegers in three classes: ()The isounit: Î ; ()The isonumbers: ˆ = I ˆ,, ˆ, ˆ 4, ˆ 5, ˆ ; ()The isoprime numbers:, ˆ, ˆ 5, ˆ 7,. ˆ Theorem. Twin isoprime theorem Pˆ = Pˆ + ˆ. (4) J( ω) =Π( P ) 0, (5) P where ω =Π P is called primorial. P, there exist infinitely many isoprimes ˆP such that ˆP is an isoprime. We have the best asymptotic formula of the number of isoprimes less than ˆN ˆ J ( ω) ω N ( N, ) = ( o()). φ ( ω) log N + (6) 5

6 where φ( ω) =Π( P ). P Let I ˆ =. From (4) we have twin prime theorem P = + (7) P Theorem. Goldbach isoprime theorem Nˆ = Pˆ + Pˆ (8) P J( ω) =Π( P ) Π 0 (9) P P N P every isoeven number ˆN greater than ˆ4 is the sum of two isoprimes. We have ˆ J( ω) N ( N, ) = ( o()). φ ( ω) log N + (0) Let I ˆ =. From (8) we have Goldbach theorem N = P+ P () Theorem. The isoprimes contain arbitrarily long arithmetic progressions. We define arithmetic progressions of isoprimes: Pˆ, Pˆ ˆ ˆ ˆ ˆ ˆ ˆ = Pˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ + d, P = P+ d,, Pk = P+ ( k I) d,( P, d) = I. () Let I ˆ =. From () we have arithmetic progressions of primes[6-0] P, P = P + d, P = P + d,, P = P + ( k ) d,( P, d) =. () k We rewrite () P = P P, P = ( j ) P ( j ) P, j k. (4) j J =Π P P, (5) ( ω) [( ) χ( )] P 6

7 χ ( P) denotes the number of solutions for the following congruence k Π[( j ) q ( j ) q ] 0(mod P), (6) j= where q =,,, P ; q =,,, P. From (6)we have J ( ω) = Π ( P ) Π( P )( P k+ ) 0. (7) P< k k P We prove that there exist infinitely many primes P and P such that P, primes for k. We have the best asymptotic formula k ( N,) = { P, P N : j k,( j ) P ( j ) P prime} ( ) k J ωω N = ( + o()) k k φ ( ω) log N, Pk are k k P P ( P k+ ) N = Π Π ( + o()) k k k P< k ( P ) K P ( P ) log N (8) Theorem4. From () we obtain P = P + P P, P = P + ( j ) P ( j ) P,4 j k. (9) 4 j J =Π P P (40) 4( ω) (( ) χ( )), P χ ( P) denotes the number of solutions for the following congruence k Π [ q + ( j ) q ( j ) q ] 0(mod P), (4) j= 4 where q =,,, P, i =,,. i From (4) we have J = Π P Π P P P k. (4) 4( ω) ( ) ( )[( ) ( )( )] 0 P< ( k ) ( k ) P We prove there exist infinitely many primes P, P and P such that P 4,, Pk are 7

8 primes for k 4. We have the best asymptotic formula ( N,4) = { P, P, P N : 4 j k, P + ( j ) P ( j ) prime} k P ( ) k J 4 ωω N = ( + ( o)) k k 6 φ ( ω) log N k k P P [( P ) ( P )( k )] N = Π Π ( + o()).(4) ( ) k ( ) k k 6 P< k ( P ) k P ( P ) log N Theorem5. Isoprime equation P = Pˆ + = PIˆ+. (44) Let Î be the odd number. P J( ω) =Π( P ) Π 0. (45) P PIˆ P, there exist infinitely primes P such that P is a prime. We have J ( ω) ω N ( N, ) = ( o()). φ ( ω) log N + (46) Theorem6. Isomprime equation P = ( Pˆ) + = P Iˆ+. (47) ˆ Let Î be the odd number. where J ( ω) =Π( P X( P)), (48) P 8

9 I ( ) X( P) = P if PIˆ I If ( ) =, there infinitely many primes P such that P is a prime. If I ( ) =, J() = 0, there exist finite primes P such that P is a prime. Santilli s Isoprime theory of the second kind Santilli s isofields of the second kind F ˆ = F ˆ ( a, +, ˆ) (that is, a F is not lifted to â = aiˆ ) also verify all the axioms of a field. The isomultiplication is defined by a ˆ b= atb ˆ. (49) We then have the isoquotient, isopower, isosquare root, etc., ˆ / ˆ ˆ n n ˆ n / ˆ / a ˆb= ( a/ b) I, a = a ˆ ˆa(ntimes) = a ( T), a = a ( I). (50) Theorem7. Isoprime equations ˆ ˆ ˆ 4 P = P + 6, P = P +, P = P + 8. (5) Let T =. From (5)we have P = P + 6, P = P +, P = P + 8, (5) 4 6 J( ω) = Π( P 4 ( ) ( ) ( )) 0, (5) 5 P P P P 6 where ( ),( ) and ( ) denote the Legendre symbols. P P P, there exist infinitely many primes P such that P, P and P 4 are primes. J( ωω ) N 4( N, ) = ( o()) φ ( ω) log N + (54) 9

10 Let T ˆ = 5. From (5) we have P = 5P + 6, P = 5P +, P = 5P + 8. (55) J( ω) = 8 Π( P 4 ( ) ( ) ( )) 0. (56) 7 P P P P, there exist infinitely many primes P such that P, P and P 4 are primes. We have Let T ˆ = 7. From (5) we have J( ωω ) N 4( N, ) = ( o()) φ ( ω) log N +. (57) P = 7P + 6, P = 7P +, P = 7P + 8. (58) 4 We have Jiang function J (5) = 0. (59) There exist finite primes P such that P, P and P 4 are primes. Theorem8. Isoprime equations ˆ ˆ ˆ ˆ 4 5 P = P + 0, P = P + 60, P = P + 90, P = P + 0. (60) Let T ˆ = 7. From (60) we have P = 7P + 0, P = 7P + 60, P = 7P + 90, P = 7P + 0. (6) j J( ω) = 48 Π ( P 5 ( )) 0. (6) P P j=, there exist infinitely many primes are primes. We have P such that P, P, P 4 and P 5 0

11 4 J( ωω ) N 5( N, ) = ( o()) φ ( ω) log N +. (6) Let Tˆ 7 be the odd prime. From (60) we have P = P T + k k =. (64) k ˆ 0( ),,,4,5 J ( ω) = 8 Π( P 5 χ( P)) 0. (65) 7 P 4 If ˆ 0Tj ˆ PT, χ ( P) = 4; χ( P) = ( ) otherwise. P j=, there exist infinitely many primes are primes. We have P such that P, P, P 4 and P 5 4 J( ωω ) N 5( N, ) = ( o()) φ ( ω) log N +. (66) Theorem9. Isoprime equation Let T ˆ = P = P ˆ ( P + b) b. (67) J = Π P + P+ P, (68) ( ω) ( χ( )) 0 P P i where χ ( P) = P+ if Pb; χ ( P) = 0 otherwise., there exist infinitely many primes P and P such that P is a prime. The best asymptotic formula is

12 ( J ( ω) ω N N,) = { P, P : P, P N; P prime} = ( + o()). (69) 4φ ( ω) log N Theorem0. Isoprime equation Let T ˆ = ˆ P = P ˆ ( P + b) b. (70) J = Π P P+ + P, (7) ( ω) ( χ( )) 0 P P i b where χ ( P) = P if Pb; χ( P) = ( ) otherwise. P, there exist infinitely many primes P and P such that P is a prime. The best asymptotic formula is ( J ( ω) ω N N,) = { P, P : P, P N; P prime} = ( + o()).. (7) 6φ ( ω) log N Theorem. Isoprime equation Let T ˆ =. ˆ P = P ( P + ). (7) J = Π P P+. (74) ( ω) ( 4) 0 P P i, there exist infinitely many primes P and P such that P is a prime. The best asymptotic formula is ( N,) = { P, P : P, P N; P prime} = J( ωω ) N ( o()) 6 φ ( ω) log N +. (75) 4 Isoprime theory in Santilli s new isomathematics

13 Theorem. Isoprime equation P = P ˆ ˆ + P = P+ P + 0. (76) Suppose ˆ0=. From (76) we have P = P+ P +. (77) J =Π P P+. (78) ( ω) ( ) 0 P, there exist infinitely many primes P and P such that P is a prime. We have the best asymptotic formula is J( ωω ) N ( N, ) = ( o()) φ ( ω) log N +. (79) Theorem. Isoprime equation P = ( P + ˆ ) ˆ ( P ) + ˆ P = Tˆ [ P (+ 0) ˆ ] + P + 0ˆ. (80) ˆ Suppose T ˆ = 6 and ˆ0= 4. From (80) we have P = 6( P 6) + P + 4 (8) J =Π P P+. (8) ( ω) ( ) 0 P, there exist infinitely many primes P and P such that P is a prime. We have the best asymptotic formula J( ωω ) N ( N, ) = ( o()) 4 φ ( ω) log N +. (8) 5 An Example We give an example to illustrate the Santilli s isomathematics.

14 Suppose that algebraic equation y = a ( b + c ) + a ( b c ). (84) We consider that (84) may be represented the mathematical system, physical system, biological system, IT system and another system. (84) may be written as the isomathematical equation yˆ = a ˆ( b+ ˆ c ˆ ˆ ˆ ˆ ˆ ˆ ˆ ) + a ( b ˆ c) = at ( b+ c+ 0) a / T( b c 0). (85) If T ˆ = and ˆ0= 0, then y = yˆ. Let T ˆ = and ˆ0=. From (85) we have the isomathematical subequation yˆ = a( b+ c+ ) + + a / ( b c ). (86) Let T ˆ = 5 and ˆ0= 6. From (85) we have the isomathematical subequation yˆ = 5 a( b+ c+ 6) a / 5( b c 6). (87) Let T ˆ = 8 and ˆ0= 0. From (85) we have the isomathematical subequation yˆ = 8 a( b+ c+ 0) a / 8( b c 0). (88) From (85) we have infinitely many isomathematical subequations. Using T ˆ and ˆ0 we study stability and optimum structures of algebraic equation (84). Acknowledgements The author would like to express his deepest appreciation to A. Connes, R. M. Santilli, L. Schadeck, G. Weiss and Chen I-Wan for their helps and supports. References [] R. M. Santilli, Isonumbers and genonumbers of dimension,, 4, 8, their isoduals and 4

15 pseudoduals, and hidden numbers of dimension, 5, 6, 7, Algebras, Groups and Geometries 0, 7- (99). [] Chun-Xuan Jiang, Foundations of Santilli s isonumber theory, Part I: Isonumber theory of the first kind, Algebras, Groups and Geometries, 5, 5-9(998). [] Chun-Xuan Jiang, Foundations of Santilli s isonumber theory, Part II: Isonumber theory of the second kind, Algebras Groups and Geometries, 5, (998). [4] Chun-Xuan Jiang, Foundations of Santilli s isonumber theory. In: Fundamental open problems in sciences at the end of the millennium, T. Gill, K. Liu and E. Trell (Eds) Hadronic Press, USA, 05-9 (999). [5] Chun-Xuan Jiang, Foundations of Santilli s isonumber theory, with applications to new cryptograms, Fermat s theorem and Goldbach s conjecture, International Academic Press, America- Europe- Asia (00) (also available in the pdf file http: // www. i-b-r. org/jiang. Pdf) [6] B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Ann. Math.,67,48-547(008). [7] E. Szemer é di, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 7, 99-45(975). [8] H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math.,, (977). [9] W. T. Gowers, A new proof of Szemerédi s theorem, GAFA,, (00). [0] B. Kra, The Green-Tao theorem on arithmetic progressions in the primes: an ergodic 5

16 point of view, Bull. Amer. Math. Soc., 4, - (006). 6