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Automorphic Functions and Fermat's Last Theorem（3）

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Automorphic Functions and Fermat's Last Theorem（3） Jiang and Wiles Proofs on Fermat Last Theorem (3) Abstract D.Zagier(1984) and K.Inkeri(1990) said[7]:Jiang mathematics is true,but Jiang determinates the irrational numbers to be very difficult for prime exponent p.In 1991 Jiang studies the composite exponents n=15,21,33,…,3p and proves Fermat last theorem for prime exponenet p>3[1].In 1986 Gerhard Frey places Fermat last theorem at the elliptic curve that is Frey curve.Andrew Wiles studies Frey curve. In 1994 Wiles proves Fermat last theorem[9,10]. Conclusion:Jiang proof(1991) is direct and simple,but Wiles proof(1994) is indirect and complex.If China mathematicians had supported and recognized Jiang proof on Fermat last theorem,Wiles would not have proved Fermat last theorem,because in 1991 Jiang had proved Fermat last theorem.Wiles has received many prizes and awards,he should thank China mathematicians. - Automorphic Functions And Fermat’s Last Theorem（3） (Fermat’s Proof of FLT) 1 Chun-Xuan Jiang P. O. Box 3924, Beijing 100854, P. R. China [email protected] Abstract In 1637 Fermat wrote: “It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into powers of like degree: I have discovered a truly marvelous proof, which this margin is too small to contain.” This means: xyznnnn+ =>(2) has no integer solutions, all different from 0(i.e., it has only the trivial solution, where one of the integers is equal to 0). It has been called Fermat’s last theorem (FLT). It suffices to prove FLT for exponent 4 and every prime exponent P . Fermat proved FLT for exponent 4. Euler proved FLT for exponent 3. In this paper using automorphic functions we prove FLT for exponents 4P and P , where P is an odd prime. We rediscover the Fermat proof. The proof of FLT must be direct. But indirect proof of FLT is disbelieving. In 1974 Jiang found out Euler formula of the cyclotomic real numbers in the cyclotomic fields ⎛⎞41mm− 4 ii−1 （ ） exp⎜⎟∑∑tJii= SJ , 1 ⎝⎠ii==11 4m where J denotes a 4m th root of unity, J =1, m=1,2,3,…, ti are the real numbers. Si is called the automorphic functions(complex hyperbolic functions) of order 4m with 41m − variables [2,5,7]. m−1 ⎡⎤B 1(1)(1)A1 H ⎛⎞iij−−π j ⎛π ⎞ Seei =+⎢⎥2 cos⎜⎟βθ + + 2∑ e cos ⎜j + ⎟ 42mm⎣⎦⎝⎠j=1 ⎝ 2 ⎠ (1)i− m−1 (1)−−⎡⎤D ⎛⎞ (ij 1)π A2 j （ ） ++⎢⎥ee2cos∑ ⎜⎟φ j − 2 42mm⎣⎦j=1 ⎝⎠ where im=1,...,4 ; 41mm−− 41 21mm− 2 α α α AtAt12==−∑∑αα,(1), Ht= ∑∑221αα(1),−=β t− (1) −, αα==11 αα==11 41mm−−α jπαπ 41 j Btjj==−∑∑ααcos ,θ t sin , αα==1122mm 2 41mm−− 41 ααα jπαπj Dtjj=−∑∑αα(1)cos ,φ =− t (1)sin , αα==1122mm m−1 （ ） AA12++22( H +∑ BDjj + )0 =. 3 j=1 From (2) we have its inverse transformation[5,7] 44mm AA121+i eSeS==−∑∑ii,(1) ii==11 22mm H 1+iH i eSeScosββ=−∑∑21ii− ( 1) , sin =− 2 ( 1) , ii==11 41mm−− 41 BBj ijπ j ijπ eSSecosθθji=+11∑∑++ cos , sin ji =− S 1 sin , ii==1122mm 41mm−− 41 DDijπ ijπ j iij （ ） eSScosφφji=+11∑∑++ ( − 1) cos , e sin ji = S 1 ( − 1) sin . 4 ii==1122mm (3) and (4) have the same form. From (3) we have ⎡⎤m−1 （ ） exp⎢⎥AA12++ 2 H + 2∑ ( BDjj + ) = 1 5 ⎣⎦j=1 From (4) we have SS14m L S 2 m−1 ⎡⎤SS21L S 3 exp⎢⎥AA12++ 2 H + 2∑ ( BDjj + ) = ⎣⎦j=1 LLLL SS4411mm− L S SS111141()L () Sm− SS221241()L () Sm− = (6) LLLL SS441441mm()L () S mm− where ∂Si ()Sij= [7] ∂t j From (5) and (6) we have circulant determinant 3 SS14m L S 2 ⎡⎤m−1 SS S 21L 3 （ ） exp⎢⎥AA12++ 2 H + 2∑ ( BDjj + ) = = 1 7 ⎣⎦j=1 LLLL SS4411mm− L S Assume SSS12≠≠=0, 0,i 0 , where im= 3,...,4 . Si = 0 are (4m − 2) indeterminate equations with (4m − 1) variables. From (4) we have AA12222H eSSeSSe=+12,, =− 12 = SS 1 + 2 2B jπ 2D jπ eSSSSj =++222cos, eSSSSj =+−222cos （8） 12 122m 12 122m Example [2]. Let 412m = . From (3) we have Att11112103948576=+()()()()() ++ tt +++++++ tt tt ttt, Attttttttttt21112103948576=−()()()()() + + + − + + + − + + , Httttt=−()()21048 + + + − 6, π 2345ππππ B =+(tt )cos ++ ( tt )cos ++ ( tt )cos ++ ( tt )cos ++ ( tt )cos − t , 111166666 210 39 48 57 6 246810π πππ π Btt=+( ) cos ++ ( tt ) cos ++ ( tt ) cos ++ ( tt ) cos ++ ( tt ) cos + t , 211166666 210 39 48 57 6 π 2345ππππ D =−(tt + ) cos + ( tt + ) cos − ( tt + ) cos + ( tt + )cos − ( tt + ) cos − t , 111121066666 39 48 57 6 2π 46810πππ π D =−(tt + )cos + ( tt + )cos − ( tt + )cos + ( tt + )cos − ( tt + )cos + t , 211166666 210 39 48 57 6 AA12++2( HBBDD ++++ 12 1 2 ) = 0 , A22+ 23(Bttt=−+− 369 ). （9） From (8) and (9) we have 12 12 3 4 3 4 exp[AA12++++++=−=−= 2( HBBDD 12 1 2 )] S 1 S 2 ( S 1 ) ( S 2 ) 1. (10) From (9) we have 3 exp(A22+=−+− 2Bttt ) [exp( 369 )] . (11) From (8) we have 22 33 exp(A22+=−++=− 2B ) ( SSSSSSSS 12121212 )( ) . (12) From (11) and (12) we have Fermat’s equation 33 3 exp(A2212+ 2BSS ) =−= [exp( −+− ttt 369 )] . (13) Fermat proved that (10) has no rational solutions for exponent 4 [8]. Therefore we prove we prove that (13) has no rational solutions for exponent 3. [2] 4 Theorem . Let 44mP= , where P is an odd prime, (1)/2P − is an even number. From (3) and (8) we have P−1 44PP P 4 P 4 exp[AA12+++ 2 H 2∑ ( BDjj +=−=−= )] S 1 S 2 ( S 1 ) ( S 2 ) 1. (14) j=1 From (3) we have P−1 4 P exp[ABDttt2424++=−+− 2∑ (jj− )] [exp( PPP 23 )] . (15) j=1 From (8) we have P−1 4 PP exp[ABDSS242412++=− 2∑ (jj− )] . (16) j=1 From (15) and (16) we have Fermat’s equation P−1 4 PP P exp[ABDSSttt24241223++=−=−+− 2∑ (jj− )] [exp( PPP )] . (17) j=1 Fermat proved that (14) has no rational solutions for exponent 4 [8]. Therefor we prove that (17) has no rational solutions for prime exponent P . Remark. Mathematicians said Fermat could not possibly had a proof, because they do not understand FLT.In complex hyperbolic functions let exponent n be nP= Π ， nP=Π2 and nP=Π4 . Every factor of exponent n has Fermat’s equation [1-7]. Using modular elliptic curves Wiles and Taylor prove FLT [9,10]. This is not the proof that Fermat thought to have had. The classical theory of automorphic functions,created by Klein and Poincare, was concerned with the study of analytic functions in the unit circle that are invariant under a discrete group of transformation. Automorphic functions are the generalization of trigonometric, hyperbolic elliptic, and certain other functions of elementary analysis. The complex trigonometric functions and complex hyperbolic functions have a wide application in mathematics and physics. Acknowledgments. We thank Chenny and Moshe Klein for their help and suggestion. References [1] Jiang, C-X, Fermat last theorem had been proved, Potential Science (in Chinese), 2.17-20 (1992), Preprints (in English) December (1991). http://www.wbabin.net/math/xuan47.pdf. [2] Jiang, C-X, Fermat last theorem had been proved by Fermat more than 300 years ago, Potential Science (in Chinese), 6.18-20(1992). [3] Jiang, C-X, On the factorization theorem of circulant determinant, Algebras, Groups and Geometries, 11. 371-377(1994), MR. 96a: 11023, http://www.wbabin.net/math/xuan45.pdf [4] Jiang, C-X, Fermat last theorem was proved in 1991, Preprints (1993). In: Fundamental open problems in science at the end of the millennium, T.Gill, K. Liu and E. Trell (eds). Hadronic Press, 1999, P555-558. http://www.wbabin.net/math/xuan46.pdf. [5] Jiang, C-X, On the Fermat-Santilli theorem, Algebras, Groups and Geometries, 15. 319-349(1998) [6] Jiang, C-X, Complex hyperbolic functions and Fermat’s last theorem, Hadronic Journal 5 Supplement, 15. 341-348(2000). [7] Jiang, C-X, Foundations of Santilli Isonumber Theory with applications to new cryptograms, Fermat’s theorem and Goldbach’s Conjecture. Inter. Acad. Press. 2002. MR2004c:11001, http://www.wbabin.net/math/xuan13.pdf. http://www.i-b-r.org/docs/jiang.pdf [8] Ribenboim,P, Fermat last theorem for amateur, Springer-Verlag, (1999). [9] Wiles A, Modular elliptic curves and Fenmat’s last theorem, Ann of Math, (2) 141 (1995), 443-551. [10] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. of Math., (2) 141(1995), 553-572. Wiles' proof of Fermat's Last Theorem From Wikipedia, the free encyclopedia Jump to: navigation, search 6 Sir Andrew John Wiles Wiles' proof of Fermat's Last Theorem i s a proof of the modularity theorem for semistable elliptic curves, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized as having a serious gap. The widely accepted version of the proof was released by Andrew Wiles in September 1994, and published in 1995. The proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th century techniques not available to Fermat. The proof itself is over 100 pages long and consumed seven years of Wiles' research time. Among other honors for his accomplishment, he was knighted.
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