I 99 Frmat s Last Thorm Has B Provd Chu-Xua Jag P.O.Box 94Bg 00854Cha Jcxua00@s.com;cxxxx@6.com bstract I 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.o Octobr 599 w provd Frmat last thorm[]. I ths papr usg th complx hyprbolc fuctos w prov FLT for xpots P ad P whr P s a odd prm. 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 uty J = s a odd umbr t ar th ral umbrs. s calld th complx hyprbolc fuctos of ordr wth varabls [-7]. ( ) B ( ) = + ( ) cos θ + ( ) () 000 mathmatcs subct classfcato D4. Ky words complx hyprbolc Eulr proof.
whr = ; = t α α α B = tα ( ) cos α α = + α α θ = ( ) tα ( ) s α = () may b wrtt th matrx form + B = 0 () 0 0 ( ) cos s s ( ) = cos s s ( ) ( ) ( ) cos s s B cosθ B sθ xp B sθ (4) whr ( ) / s a v umbr. From (4) w hav ts vrs trasformato ( ) cos cos cos B cosθ B sθ ( ) = 0 s s s xp( B )s( θ ) ( ) ( ) ( ) 0 s s s From (5) w hav (5) = + ( ) cos = B cos θ + = = s ( ) ( ) s (6) B + θ = + = 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).
( ) cos cos cos B cosθ B sθ ( ) = 0 s s s xp( B )s( θ ) ( ) ( ) ( ) 0 s s s 0 0 ( ) cos s s B cosθ ( ) B sθ cos s s xp( B )s( θ ) ( ) ( ) ( ) cos s s 0 0 0 0 0 0 B cosθ B 0 0 0 sθ = xp( B )s( θ ) 0 0 0 cosθ B B sθ = (7) xp( B )s( θ ) whr + (cos ) = (s ) =. From () w hav From (6) w hav xp( + B ) =. (8)
( ) ( ) ( ) ( ) xp( + B ) = = ( ) ( ) (9) whr ( ) = [7]. t From (8) ad (9) w hav th crculat dtrmat xp( + B ) = = (0) If 0 whr = th (0) has ftly may ratoal solutos. ssum 0 0 = 0 whr = 4. = 0 ar dtrmat quatos wth varabls. From (6) w hav B = + = + + ( ) cos. () From (0) 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 ) 4 4 5 0 6 9 7 8 4 B = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos 5 5 5 5 5 6 7 ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos 5 5 5 4 6 8 B = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos 5 5 5 5 0 4 + ( t5 + t )cos + ( t06 + t9)cos + ( t7 + t8)cos 5 5 5 6 9 B = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos 5 5 5 5 5 8 ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos 5 5 5 4 8 6 B4 = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos 5 5 5 5 0 4 8 + ( t5 + t )cos + ( 06 t + t9)cos + ( t7 + t8)cos 5 5 5 4
5 0 5 0 B5 = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos 5 5 5 5 5 0 5 ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos 5 5 5 6 8 4 B6 = ( t + t4)cos + ( t + t)cos + ( t + t)cos + ( t4 + t)cos 5 5 5 5 0 6 4 + ( t5 + t )cos + ( t06 + t9)cos + ( t7 + t8)cos 5 5 5 7 4 8 B7 = ( t + t4)cos + ( t + t)cos ( t + t)cos + ( t4 + t)cos 5 5 5 5 5 4 49 ( t5 + t0)cos + ( t6 + t9)cos ( t7 + t8)cos 5 5 5 7 6 5 0. () + B = 0 + B + B = 5( t + t ) Form () w hav th Frmat quato From () w hav From () w hav 7 5 5 5 5. (4) xp( + B ) = + = ( ) + ( ) = xp( B B ) [xp( t t )] From (5) ad (6) w hav th Frmat quato 5 + + 6 = 5 + 0. (5) xp( + B + B ) = +. (6) 5 5 6 xp( + B + B ) = + = [xp( t + t )]. (7) 5 5 5 6 5 0 Eulr provd that (4) has o ratoal solutos for xpot [8]. Thrfor w prov that (7) has o ratoal solutos for xpot 5[]. Thorm. [-7]. 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 5
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]. Not. Wls had ot provd Frmat last thorm[9-] Rfrcs [] Jag C-X Frmat last thorm had b provd Pottal cc ( Chs).7-0 (99) Prprts ( Eglsh) Dcmbr (99). http://www.wbab.t/math/xua47.pdf. [] 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 http://www.wbab.t/math/xua45.pdf [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 P555-558. http://www.wbab.t/math/xua46.pdf. [5] Jag C-X O th Frmat-atll thorm lgbras Groups ad Gomtrs 5. 9-49(998) [6] Jag C-X Complx hyprbolc fuctos ad Frmat s last thorm Hadroc Joural upplmt 5. 4-48(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:00 http://www.wbab.t/math/xua.pdf. http://www.-b-r.org/docs/ag.pdf [8] Rbbom P Frmat last thorm for amatur prgr-vrlag (999). [9] Wls Modular llptc curvs ad Frmat last thorm. Math 4()44-55(995). [0] ZhvolovY Frmat last thorm ad mstaks of drw Wls.006. www.baoway.com/bbs/vwthrad.php?td=86&fpag= [] ZhvolovY Frmat last thorm ad Kth Rbt mstaks.006. www.baoway.com/bbs/vwthrad.php?td=864&fpag= 6