Aotomorphic Functions And Fermat s Last Theorem(4)
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1 otomorphc Fuctos d Frmat s Last Thorm(4) Chu-Xua Jag P. O. Box 94 Bg P. R. Cha agchuxua@sohu.com bsract 67 Frmat wrot: It s mpossbl to sparat a cub to two cubs or a bquadrat to two bquadrats or gral ay powr hghr tha th scod to powrs of lk dgr: I hav dscovrd a truly marvlous proof whch ths marg s too small to cota. Ths mas: x + y = z ( > ) has o tgr solutos all dffrt from 0(.. t has oly th trval soluto whr o of th tgrs s qual to 0). It has b calld Frmat s last thorm (FLT). It suffcs to prov FLT for xpot 4. ad vry prm xpot P. Frmat provd FLT for xpot 4. Eulr provd FLT for xpot. I ths papr usg automorphc fuctos w prov FLT for xpots P ad P whr P s a odd prm. W fd th Frmat proof. Th proof of FLT must b drct. But drct proof of FLT s dsblvg.. I 974 Jag foud out Eulr formula of th cyclotomc ral umbrs th cyclotomc flds xp tj = J () = = whr J dots a th root of gatv uty J = s a odd umbr t ar th ral umbrs. s calld th automorphc fuctos(complx trgoomtrc fuctos) of ordr wth varabls [-7]. ( ) B ( ) ( ) = [ + ( ) cos( + ( ) )] () whr = ; α = tα ( ) α = B α ( ) α tα ( ) cos () α = = α = ( ) ( ) s + ( ) α tα α = () may b wrtt th matrx form + B = 0
2 0 L 0 ( ) cos s L s ( ) = cos s L s L L L L L L ( ) ( ) ( ) cos s L s B cos B s L xpb s (4) whr ( )/ s a v umbr. From (4) w hav ts vrs trasformato L ( ) cos cos L cos cos B B s ( ) = 0 s s L s L xp( B )s( ) L L L L L 0 s s L s ( ) ( ) ( ) From (5) w hav L (5) + ( ) = = cos ( ) cos B = + + ( ) = s ( ) ( ) s (6) B + = + ( ) = I () ad (6) t ad hav th sam formulas. (4) ad (5) ar th most crtcal formulas of proofs for FLT. Usg (4) ad (5) 99 Jag vtd that vry factor of xpot has th Frmat quato ad provd FLT [-7].ubsttutg (4) to (5) w prov (5). L ( ) cos cos L cos cos B B s ( ) = 0 s s L s L xp( B )s( ) L L L L L 0 s s L s ( ) ( ) ( )
3 0 L 0 ( ) cos s L s B cos ( ) B cos s s s L L L L L L L xp( B )s( ) ( ) ( ) ( ) cos s L s 0 0 L L 0 B cos B s = L L K L L L L xp( B )s( ) L cos B B s = (7) L xp( B )s( ) whr + (cos ) = (s ) =. From () w hav From (6) w hav xp( + B ) =. (8) L ( ) L ( ) L ( ) L ( ) xp( + B ) = = L L L L L L L L L ( ) L ( ) (9) whr ( ) = [7]. t From (8) ad (9) w hav th crculat dtrmat
4 L L xp( + B ) = = L L L M L (0) If 0 whr = L th (0) has ftly may ratoal solutos. ssum 0 0 = 0 whr = 4 L. = 0 ar dtrmat quatos wth varabls. From (6) w hav B = = + + ( ) cos. () From () ad () w hav th Frmat quato xp( + B ) = ( ) Π ( + + ( ) cos ) = = () Exampl[]. Lt = 5. From () w hav =( t t ) + ( t t ) ( t t ) + ( t t ) ( t t ) + ( t t ) ( t t ) B = ( t t4)cos + ( t t )cos + ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos + ( t7 t8)cos B =( t t4)cos + ( t t)cos ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos ( t7 t8)cos B = ( t t4)cos + ( t t)cos + ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos + ( t7 t8)cos B4 =( t t4)cos + ( t t)cos ( t t)cos + ( t4 t)cos ( t5 t0) cos + ( t6 t9) cos ( t7 t8) cos B5 = ( t t4)cos + ( t t)cos + ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos + ( t7 t8)cos B6 =( t t4)cos + ( t t)cos ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos ( t7 t8)cos
5 7 4 8 B7 = ( t t4)cos + ( t t)cos + ( t t)cos + ( t4 t)cos ( t5 t0)cos + ( t6 t9)cos + ( t7 t8)cos () + B = 0 + B + B = 5( t + t ) Form () w hav th Frmat quato From () w hav From () w hav (4) xp( + B ) = = ( ) ( ) = xp( B B ) [xp( t t )] From (5) ad (6) w hav th Frmat quato = (5) xp( + B + B ) =. (6) xp( + B + B ) = = [xp( t + t )]. (7) Eulr provd that (4) has o ratoal solutos for xpot [8]. Thrfor w prov that (7) has o ratoal solutos for xpot 5[]. Thorm. Lt = Pwhr P > s odd prm. From () w hav th Frmat s quato From () w hav From () w hav P P P P P. (8) xp( + B ) = = ( ) ( ) = P P P P. (9) xp( + B ) = [xp( t + t )] P P P. (0) xp( + B ) = From (9) ad (0) w hav th Frmat quato P P P P P P. () xp( + B ) = = [xp( t + t )] Eulr provd that (8) has o ratoal solutos for xpot [8]. Thrfor w prov that () has o ratoal solutos for P > [ -7]. Thorm. W cosdr th Frmat s quato w rwrt () P P P x y = z () 5
6 P P P ( x ) ( y ) = ( z ) () From (4) w hav P P P P P P P ( x y )( x + x y + y ) = z (4) x y Lt = =. From (0) ad (4) w hav th Frmat s quato z z P P P P P P ( x x y y z [xp( tp tp)] + + = (5) x y = [ z xp( t + t )] (6) P P P P P Eulr provd that () has o tgr solutos for xpot [8]. Thrfor w prov that (6) has o tgr solutos for prm xpot P. Frmat Thorm. It suffcs to prov FLT for xpot 4. W rwrt () P P P ( x ) ( y ) ( z ) = (7) Eulr provd that()has o tgr solutos for xpot [8]. Thrfor w prov that (7) has o tgr solutos for all prm xpot P [-7]. W cosdr Frmat quato W rwrt (8) 4P 4P 4P x y = z (8) P 4 P 4 P 4 ( x ) (( y ) = ( z ) (9) 4 P 4 P 4 P ( x ) ( y ) ( z ) = (0) Frmat provd that (9) has o tgr solutos for xpot 4 [8]. Thrfor w prov that (0) has o tgr solutos for all prm xpot P [57].Ths s th proof that Frmat thought to hav had. Rmark. It suffcs to prov FLT for xpot 4. Lt = 4P whr P s a odd prm. W hav th Frmat s quato for xpot 4P ad th Frmat s quato for xpot P [57]. Ths s th proof that Frmat thought to hav had. I complx hyprbolc fuctos lt xpot b =Π P = Π P ad = 4Π P. Evry factor of xpot has th Frmat s quato [-7]. I complx trgoomtrc fuctos lt xpot b =Π P = Π P ad = 4Π P. Evry factor of xpot has Frmat s quato [-7]. Usg modular llptc Curvs Wls ad Taylor prov FLT[90]. Ths s ot th proof that Frmat thought to hav had. Th classcal thory of automorphc fuctos cratd by Kl ad Pocar was cocrd wth th study of aalytc fuctos th ut crcl that ar varat udr a dscrt group of trasformato. utomorphc fuctos ar th gralzato of trgoomtrc hyprbolc llptc ad crta othr fuctos of lmtary aalyss. Th complx trgoomtrc fuctos ad complx hyprbolc fuctos hav a wd applcato mathmatcs ad physcs. ckowldgmts. W thak Chy ad Mosh Kl for thr hlp ad suggsto. Rfrcs 6
7 [] Jag C-X Frmat last thorm had b provd Pottal cc ( Chs).7-0 (99) Prprts ( Eglsh) Dcmbr (99). [] Jag C-X Frmat last thorm had b provd by Frmat mor tha 00 yars ago Pottal cc ( Chs) 6.8-0(99). [] Jag C-X O th factorzato thorm of crculat dtrmat lgbras Groups ad Gomtrs. 7-77(994) MR. 96a: 0 [4] Jag C-X Frmat last thorm was provd 99 Prprts (99). I: Fudamtal op problms scc at th d of th mllum T.Gll K. Lu ad E. Trll (ds). Hadroc Prss 999 P [5] Jag C-X O th Frmat-atll thorm lgbras Groups ad Gomtrs (998) [6] Jag C-X Complx hyprbolc fuctos ad Frmat s last thorm Hadroc Joural upplmt (000). [7] Jag C-X Foudatos of atll Isoumbr Thory wth applcatos to w cryptograms Frmat s thorm ad Goldbach s Coctur. Itr cad. Prss. 00. MR004c: [8] RbbomP Frmat last thorm for amatur prgr-vrlag (999). [9] WlsModular llptc curvs ad Frmat last thorm. of Math.()4(995) [0] TaylorRad Wls Rg-thortc proprts of crta Hck algbras. of Math. ()4(995)
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