In 1991 Fermat s Last Theorem Has Been Proved(II)

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1 In 99 Fermat s Last Theorem Has Been Proved(II) Chun-Xuan Jang P. O. Box 9, Beng 0085, P. R. Chna cxuan00@sna.com Abstract In 67 Fermat wrote: It s mpossble to separate a cube nto two cubes, or a bquadrate nto two bquadrates, or n general any power hgher than the second nto powers of lke degree: I have dscovered a truly marvelous proof, whch ths margn s too small to contan. n n n Ths means: x y z ( n> ) has no nteger solutons, all dfferent from 0(.e., t has only the trval soluton, where one of the ntegers s equal to 0). It has been called Fermat s last theorem (FLT). It suffces to prove FLT for exponent and every prme exponent P. Fermat proved FLT for exponent. Euler proved FLT for exponent. In ths paper usng automorphc functons we prove FLT for exponents P and P, where P s an odd prme. We redscover the Fermat proof. The proof of FLT must be drect. But ndrect proof of FLT s dsbelevng. In 97 Jang found out Euler formula of the cyclotomc real numbers n the cyclotomc felds exp tj SJ, () where J denotes a th root of unty, J, m,,,, t are the real numbers. S s called the automorphc functons(complex hyperbolc functons) of order wth varables [,5,7]. where,...,; ( ) π ( ) π cos cos θ m m A B H S e e β e m ( ) m ( ) A D ( ) π e e cos φ m () 000 Mathematcs subect classfcaton:d

2 Key words: complex hyperbolc, Fermat proof., A t, A t ( ) m m β, H t ( ), t ( ) π π B t cos, θ t sn, m m π π D t ( ) cos, t( ) sn m φ m m A A H ( B D) 0. () From () we have ts nverse transformaton[5,7] A A, ( ) e S e S m m H H cos β ( ), sn β ( ) e S e S B π π e cosθ S S cos, e sn S sn m B θ m D π π e cos φ S S ( ) cos, e sn S ( ) sn m D φ m. () () and () have the same form. From () we have m exp A A H ( B D) (5) From () we have m exp A A H ( B D)

3 where S ( S) [7] t (6) From (5) and (6) we have crculant determnant m exp A A H ( B D) (7) Assume S 0, S 0, S 0, where,..., m. S 0 are ( ) ndetermnate equatons wth ( ) varables. From () we have B π e S S SS cos, m Example []. Let. From () we have e S S, e S S, e S S A A H A ( t t ) ( t t ) ( t t ) ( t t ) ( t t ) t, D π e S S SS cos (8) m A ( t t ) ( t t ) ( t t ) ( t t ) ( t t ) t, H ( t t ) ( t t ) t, π π π π 5π B ( t t)cos ( t t0)cos ( t t9)cos ( t t8)cos ( t5 t7)cos t6, π π 6π 8π 0π B ( t t)cos ( t t0)cos ( t t9)cos ( t t8)cos ( t5 t7)cos t6, π π π π 5π D ( t t)cos ( t t0)cos ( t t9)cos ( t t8)cos ( t5 t7)cos t6, π π 6π 8π 0π D ( t t)cos ( t t0)cos ( t t9)cos ( t t8)cos ( t5 t7)cos t6, A A ( H B B D D) 0, A B ( t t6 t9). (9) From (8) and (9) we have exp[ A A ( H B B D D )] S S ( S ) ( S ). (0) From (9) we have

4 exp( A B ) [exp( t t t )] 6 9. () From (8) we have exp( A B ) ( S S )( S S S S ) S S. () From () and () we have Fermat s equaton exp( A B ) S S [exp( t t t )]. () 6 9 Fermat proved that (0) has no ratonal solutons for exponent [8]. Therefore we prove we prove that () has no ratonal solutons for exponent. [] Theorem. Let P, where P s an odd prme, ( P ) / s an even number. From () and (8) we have From () we have From (8) we have P P P P P exp[ A A H ( B D )] S S ( S ) ( S ). () P P exp[ A ( B D )] [exp( t t t )]. (5) P P P From (5) and (6) we have Fermat s equaton P P P exp[ A ( B D )] S S. (6) P P P P P P P exp[ A ( B D )] S S [exp( t t t )]. (7) Fermat proved that () has no ratonal solutons for exponent [8]. Therefor we prove that (7) has no ratonal solutons for prme exponent P. Remark. Mathematcans sad Fermat could not possbly had a proof, because they do not understand FLT.In complex hyperbolc functons let exponent n be n Π P, n Π P and n Π P. Every factor of exponent n has Fermat s equaton [-7]. Usng modular ellptc curves Wles and Taylor prove FLT [9,0]. They has not proved FLT[,] The classcal theory of automorphc functons,created by Klen and Poncare, was concerned wth the study of analytc functons n the unt crcle that are nvarant under a dscrete group of transformaton. Automorphc functons are the generalzaton of trgonometrc, hyperbolc ellptc, and certan other functons of elementary analyss. The complex trgonometrc functons and complex hyperbolc functons have a wde applcaton n mathematcs and physcs. References [] Jang, C-X, Fermat last theorem had been proved, Potental Scence (n Chnese),.7-0 (99), Preprnts (n Englsh) December (99). [] Jang, C-X, Fermat last theorem had been proved by Fermat more than 00 years ago, Potental

5 Scence (n Chnese), 6.8-0(99). [] Jang, C-X, On the factorzaton theorem of crculant determnant, Algebras, Groups and Geometres,. 7-77(99), MR. 96a: 0, [] Jang, C-X, Fermat last theorem was proved n 99, Preprnts (99). In: Fundamental open problems n scence at the end of the mllennum, T.Gll, K. Lu and E. Trell (eds). Hadronc Press, 999, P [5] Jang, C-X, On the Fermat-Santll theorem, Algebras, Groups and Geometres, (998) [6] Jang, C-X, Complex hyperbolc functons and Fermat s last theorem, Hadronc Journal Supplement, 5. -8(000). [7] Jang, C-X, Foundatons of Santll Isonumber Theory wth applcatons to new cryptograms, Fermat s theorem and Goldbach s Conecture. Inter. Acad. Press. 00. MR00c:00, [8] Rbenbom,P, Fermat last theorem for amateur, Sprnger-Verlag, (999). [9] Wles A, Modular ellptc curves and Fenmat s last theorem, Ann of Math, () (995), -55. [0] Taylor, R. and Wles, A., Rng-theoretc propertes of certan Hecke algebras, Ann. of Math., () (995), [] Zhvolov,Y,Fermat last theorem and mstakes of Andrew Wles,(006) [] Zhvolov,Y, Fermat last theorem and Kenneth Rbet mstakes,(006). 5