The margin is too narrow to contain a truly remarkable proof.

Transcription

1 The mrgin is too nrrow to contin trul remrkble proof. Pierre de Fermt B For HP & KP Abstrct. The im of this pper is to tr for Fermt s lost proof. Introduction In bout 637 Pierre de Fermt, French mthemticin, nnotted in his cop of Diophntus Arithmetic s follows: There re no nonzero integers, b, c, n with n such tht n n n b c. I hve discovered trul remrkble proof of this proposition which the mrgin is too nrrow to contin. Unfortuntel, his proof ws lost. Despite of mn mthemticins ttempts, it remined unsolved for over 35 ers until in 994 Professor Andrew John Wiles proved successfull specil cse of the Shimur-Tnim conjecture tht implies Fermt s Lst Theorem (FLT). Since Wiles is too long nd extremel complicted, it is believed tht Fermt hd wrong proof in mind. However, contrr to the stndrd view, mteur mthemticins still tr hrd to find n elementr one. For this purpose, we present here such n elegnt proof of FLT tht we hope the method used will work effectivel for certin plne smooth curves.. A short proof of FLT n n Our pproch is Fermt s curve x. In the proof, we use the method of contrdiction b ssuming tht Fermt s curve hs certin nonzero rtionl point, we then show tht this will led to the existence of qudruple being non-trivil integrl solution of the eqution x z w. On the other hnd, we prove tht the existence of this qudruple will led to the contrr of the supposition. 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge

2 Lemm Given is even nd b is odd. If there exist nonzero integers e, f, g, h such tht ( fg eh) ( b) ( eg fh) b e f g h b b ( ) e f g h b b ( fg eh) b ( eg fh) ( b) e f g h b b ( ) e f g h b b, then there exist two different prit integers i, j nd GCD( i, j) such tht i j is common divisor of nd b. Proof First, let us consider sstem of equtions (.) (.4). ( fg eh) ( b) (.) ( eg fh) b (.) e f g h b b ( ) (.3) Adding equtions (.3) & (.4) term b term, e f g h b b (.4) ( e f ) (.5) 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge

3 ( e f ) ( e f ) (.6) Solving eqution (.6), e f k i j ( ) (.7) e f k( ij) (.8) k i j ( ) (.9) where i, j re two different prit integers, GCD( i, j) nd k GCD( e f, e f ). It follows tht Adding equtions (.) & (.) term b term, k e i j ij ( ) (.) k f i j ij ( ) (.) Wherefrom follows ( e f ) g ( e f ) h (.) h ( e f ) g ( e f ) (.3) Replcing the vlues of e f, e f nd from (.7),(.8) nd (.9) into (.3), k( i j ) ( j i ) g h (.4) 4ij From (.), ( eg fh) b (.5) Replcing the vlues of e, f, nd h from (.),(.),(.9) nd (.4) into (.5), 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 3

4 ( i j )( kj kij ki g) b (.6) 4ij Since GCD( i, j) nd i, j re two different prit integers, we hve GCD i j ij integer. Then (, 4 ). It follows tht kj kij ki g l ij (4 ), where l is nonzero b l i j ( ) (.7) From (.9) nd (.7), it follows tht i j is common divisor of nd b. Next, considering the following sstem of equtions will led to the sme result. ( fg eh) b ( eg fh) ( b) e f g h b b ( ) Hence, lemm is proved. Theorem e f g h b b For n integer n greter thn, there re no nonzero rtionl points on the n n curve x. Proof It is cler tht we re onl necessr to prove the theorem for two cses: Cse : n is n odd number greter thn. Cse : n 4. Suppose tht for n odd number n greter thn or equl to 4, the curve ( ) : n n L x hs certin nonzero rtionl point M ( x, ). Without loss of 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 4

5 b generlit, ssume tht x,, where, b, c re nonzero integers nd reltivel c c prime in pirs. x Let m. It is es to see tht m, m nd m. b In the plne Ox, we introduce the circle ( C) with center t the origin nd pssing through the point, M x. Then, the eqution of the tngent ( ) to the circle ( C ) t the point M ( x, ) is m( x x ). Moreover, the eqution of the tngent ( ) to the curve ( L ) t the point M ( x, ) is n m ( x x ). If the stright line ( ) coincides with the stright line ( ), then n m m (.8) n m (.9) The rtionl roots of eqution (.9) m be m or m, if exist. (see reference []). For this reson, the stright line ( ) intersects the circle ( C) t two distinct points M ( x, ) nd M( x, ) such tht ( m ) x m m n n n x m x ( m ) m n n n (.) (.) From (.) & (.) nd, n n x m m m n m (.) n n m m n m (.3) If x, then m m. This eqution hs no rtionl roots. n n 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 5

6 If, then m n m. This eqution hs no rtionl roots. n Therefore, x nd. Replcing m into equtions (.) & (.3), b x b b b b n n n n ( ) nn ( ) (.4) b b nn b n n nn (.5) Since M ( x, ) nd M( x, ) re on the circle ( C ), we hve x x (.6) Dividing both members of eqution (.6) b, x x (.7) ( ) ( ) ( ) x Replcing the vlues of nd from (.4) nd (.5) into eqution (.7), p r (.8) bq q b ( ) ( ) ( ) n n n n where p b b, nonzero integers. (It is es to see this). q b nn nd r b b n n nn re Since ( bq), from (.8) it follows tht ( p) ( br) ( b ) q (.9) Multipling both members of eqution (.9) b b, ( ) ( ) ( ) ( ) b p br b q (.3) 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 6

7 Then, dding both members of eqution (.3) to b( p) ( br) From (.9) & (.3), ( b b) ( p) ( br) ( b ) q b ( p) ( br) (.3) ( b) ( b ) q ( b ) q b ( b ) q (.3) ( b) ( b ) q ( b b) q ( b) q (.33) Since q, from (.33) we find ( b) ( b) b( b) ( b b) (.34) Since GCD(, b), there onl exist the two following cses: Cse : The numbers, b re of different prit (one is odd, one is even). Cse b: The numbers, b re both odd. In cse, for n odd number n greter thn or equl to 4, without loss of generlit, ssume tht is even nd b is odd. Then ( b), b re even nd re odd. Thus, from (.34), it is cler tht the qudruple is non-trivil integrl solution of the eqution GCD( b( b), b b), the qudruple primitive integrl solution of the eqution ( b), b( b), b, b b b( b), b b ( b), b, b( b), b b x z w. Moreover, since ( b), b, b( b), b b is x z w. As known, if the qudruple is primitive integrl solution of the eqution x z w, then there exist nonzero integers e, f, g, h such tht (see reference []). ( fg eh) ( b) (.35) ( eg fh) b (.36) e f g h b b ( ) (.37) 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 7

8 e f g h b b (.38) ( fg eh) b (.39) ( eg fh) ( b) (.4) e f g h b b ( ) (.4) e f g h b b (.4) On the other hnd, b lemm, from this fct it follows tht there exist two different prit integers i, j nd GCD( i, j) such tht i j is common divisor of nd b. This is contrr to the bove supposition GCD(, b). In cse b, if n is n odd number greter thn, then from n n n c ( b). Let A c, B b, C. Then of different prit. This contrdicts cse. In cse b, if n 4, then from From here it follows tht 4 8l is even b c b n n n b c we hve n n n A B C, where the integers A, B re we hve c l, where l is nonzero integer. 8l 4. This is contrdiction becuse Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 8 b is odd nd In short, there re no nonzero rtionl points on the curve ( L ). The theorem is proved.. Generliztion for plne smooth curve One question rises from the bove proof nturll. To which curve cn the bove method be pplied when determining whether n rtionl point on it? To nswer little bout this question, we will show tht plne smooth curve hs lws specific eqution the form of which is Lemm. Given in the plne Ox circle x z w.. Suppose tht the circle ( C ) ( C) : x r hs two distinct nonzero points M ( x, ), M( x, ) such tht ) x x

9 ) x x 3) Then Proof ( ) ( ) ( ) ( ) x x x x Let M be the smmetric point of M through the xis O nd k, l be the slopes of the stright lines MM, M M. Then k x x (.) l x x (.) Let the circle ( C) intersect the xis O t A(, x ) nd B(, x ). Then the point M is either on the rc MAM or on the rc MBM nd the slopes of the stright lines M A, M B re, respectivel x h (.3) x x g (.4) x We strt with the cse the point M is on the rc MBM. From the supposition, it follows tht the point M does not coincide with the points M, M, A. If kh, i.e. the stright line M A is perpendiculr to the stright line M M, then the point M coincides with the point supposition. Hence, kh. B(, x ) nd x. This is contrr to the 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 9

10 If lh, i.e. the stright line M A is perpendiculr to the stright line M M, then the point M coincides with the point supposition. Hence, lh. B(, x ) nd x. This is contrr to the Since kh, lh nd the rc MA equls to the rc AM, we find h l k h hl kh (.5) ( k l) h ( kl) h ( k l) (.6) x( ) Let us consider (.6) s qudrtic eqution for h. Since k l, we get x x h, kl k l k l / ( )( ) (.7) h, ( ) / ( x ) ( x x ) x ( ) (.8) From (.3) & (.8), if h h, then x ( ) ( x ) ( x x ) x x ( ) (.9) ( ) x ( x ) ( x x ) (.) Squring both members of (.) nd rerrnging, From (.3) & (.8), if h h, then ( ) ( ) ( ) ( ) x x x x (.) 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge

11 x ( ) ( x ) ( x x ) x x ( ) (.) ( ) x ( x ) ( x x ) (.3) Wherefrom follows equlit (.). In the cse the point M is on the rc MAM, b smmetr of A, B nd h, g we esil obtin equlit (.). Hence, lemm. is proved. Theorem. Given in the plne Ox smooth curve ( L) : f ( x, ). Suppose tht the curve ( L ) hs certin nonzero rtionl point M ( x, ) such tht ) ) 3) f ( x, ) ' x k is nonzero rtionl number. ' f ( x, ) ( k ) x k kx ( k ) 4) kx 5) x k Then kx( x k) k ( x k) ( k ) x kx ( k ) x kx k Proof In the plne Ox, we introduce the circle ( C) with center t the origin nd pssing through the point, M x. Then, the eqution of the tngent ( ) to the circle ( C ) t 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge

12 x the point M ( x, ) is ( x x ). Moreover, the eqution of the tngent ( ) to the curve ( L ) t the point M ( x, ) is k( x x ), where k is its slope. x If the stright line ( ) coincides with the stright line ( ), then k. This is contrr to the supposition. For this reson, the stright line ( ) intersects the circle ( C) t two distinct points M ( x, ) nd M( x, ) such tht ( k ) x k x k kx ( k ) k From these equlities, we find k( kx ) x x k ( x k) x x k k( x k ) k ( k ) x kx k x x ( k ) x kx k k x Then, we hold theorem. b lemm. Anlogousl, considering M s the smmetric point of M through the xis Ox will ield the following lemm nd theorem. Lemm. 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge

13 Given in the plne Ox circle. Suppose tht the circle ( C ) ( C) : x r hs two distinct nonzero points M ( x, ), M( x, ) such tht ) ) x x 3) Then Theorem. ( ) ( ) ( ) ( ) x x x x x x x x Given in the plne Ox smooth curve ( L) : f ( x, ). Suppose tht the curve ( L ) hs certin nonzero rtionl point M ( x, ) such tht ) ) 3) f ( x, ) ' x k is nonzero rtionl number. ' f ( x, ) ( k ) x k kx ( k ) 4) kx 5) x k Then x ( x k) ( x k) kx ( k ) x kx ( k ) 3. Appliction To be continued, we will ppl theorem. to prove FLT gin. To do this, we obviousl need nother lemm. Lemm Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 3

14 Given two different prit integers, b. If GCD(, b), then there do not exist nonzero integers d, e, f, g, h with d odd such tht d fg eh b b n n n ( ) ( ) d eg fh b b n n n ( ) ( ) d( e f g h ) b b n n n n d( e f g h ) b b b nn4 n n n d fg eh b b n n n ( ) ( ) d eg fh b b n n n ( ) ( ) d( e f g h ) b b n n n n d( e f g h ) b b b nn4 n n n where n is odd greter thn or equl to 4. Proof Assuming tht there exist nonzero integers d, e, f, g, h with d odd such tht d( fg eh) u (3.) d( eg fh) v (3.) n n n n d( e f g h ) b b (3.3) nn4 n n n d( e f g h ) b b b (3.4) where u b b n n n ( ), v b b n n n ( ). Adding equtions (3.3) & (3.4) term b term, n4 d( e f ) ( b ) (3.5) 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 4

15 It is cler tht d is positive integer. Let us consider (3.) & (3.) s sstem of two liner equtions in two unknowns g, h nd solve it n n ev fu b( b )( eb f) n ( ) ( ) g d e f b (3.6) n n eu fv b( b )( e fb) n ( ) ( ) h d e f b (3.7) n4 n4 n4 n4 In the cse n is odd greter thn, since GCD( b, b ), divisor of b nd n4 n4 b is divisor of n n b n4 n4, we hve b is n n n n n GCD( b, b ), then GCD( ( b ), b( b )). From (3.6),(3.7), it follows tht where i, i re nonzero integers. Squring both members of eqution (3.8), eb f i b (3.8) n ( ) e fb i b (3.9) n ( ) Squring both members of eqution (3.9), ( eb) ebf ( f) i ( b ) (3.) n4 Adding equtions (3.) & (3.)term b term, ( e) efb ( fb) i ( b ) (3.) n4 From equtions (3.5) & (3.), we get This is contrdiction. ( b )( e f ) ( i i ) ( b ) (3.) n4 i i (3.3) d 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 5

16 In the cse n 4, (3.6) & (3.7) become ev fu b( b )( eb f) ( ) ( ) g d e f b (3.4) eu fv b( b )( e fb) ( ) ( ) h d e f b (3.5) Since GCD( b, b ), from (3.4),(3.5) it follows tht eb f j b (3.6) ( ) e fb j b (3.7) ( ) where j, j re nonzero integers. It is cler tht this cse lso leds to contrdiction. Next, considering the following sstem of equtions in the sme w will lso led to contrdiction. d( fg eh) v d( eg fh) u d( e f g h ) b b n n n n Hence, lemm 3. is proved. Theorem d( e f g h ) b b b nn4 n n n For n integer n greter thn, there re no nonzero rtionl points on the n n curve x. Proof It is cler tht we re onl necessr to prove the theorem for two cses: Cse : n is n odd number greter thn. Cse : n Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 6

17 Suppose tht for n odd number n greter thn or equl to 4, the curve ( ) : n n L x hs certin nonzero rtionl point M ( x, ). Without loss of b generlit, ssume tht x,, where, b, c re nonzero integers nd reltivel c c prime in pirs. x Let m. It is es to see tht m, m nd m. b The slope of the tngent ( ) to the curve ( L ) t the point M ( x, ) is k n m. We hve ( k ) x k ( m ) x m (3.8) n n If ( m ) x m, then n n x or n n ( m ) m m m (3.9) n n The rtionl roots of eqution (3.9) m be m or m, if exist. kx ( k ) m x ( m ) (3.) n n If m x ( m ), then n n x n n m ( m ) or m n m (3.) n The rtionl roots of eqution (3.) m be m or m, if exist. kx m x (3.) n If, then n m x x n m or n m (3.3) The rtionl roots of eqution (3.3) m be m or m, if exist. 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 7

18 k x m x (3.4) n If, then n m x x n m or n m (3.5) The rtionl roots of eqution (3.5) m be m or m, if exist. Therefore, b theorem. kx( x k) k ( x k) ( k ) x kx ( k ) x kx k (3.6) 4 x x 4 x 4 x x 4 x x ( ) ( ) ( ) ( ) k k k k k k k k k (3.7) Since 4 x nd m, from (3.7) we find n n n n n n n n n ( ) ( ) m m m m m m m m m m m m m (3.8) Since 4 m, from (3.8)we hve n n n nn nnn4 n m ( m ) m ( m ) m m m m m (3.9) Replcing m into (3.9) nd simplifing, b n n n n n nn n n nnn4 n n n b( b ) b ( b ) b b b b b (3.3) Since GCD(, b), there onl exist the two following cses: Cse : The numbers, b re of different prit (one is odd, one is even). Cse b: The numbers, b re both odd. 54 Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 8

19 In cse, for n odd number n greter thn or equl to 4, n n n b ( b ) re even nd n n n n b b, n n n b( b ), nn4 n n n b b b re odd. Thus, from (3.3), it is cler tht the qudruple b( b ), b ( b ), b b, b b b n n n n n n n n n n n n 4 n n n is non-trivil integrl solution of the eqution x z w. As known, if so, then there exist nonzero integers d, e, f, g, h with d odd such tht (see reference []). n n n d( fg eh) b( b ) (3.3) n n n d( eg fh) b ( b ) (3.3) n n n n d( e f g h ) b b (3.33) nn4 n n n d( e f g h ) b b b (3.34) n n n d( fg eh) b ( b ) (3.35) n n n d( eg fh) b( b ) (3.36) n n n n d( e f g h ) b b (3.37) nn4 n n n d( e f g h ) b b b (3.38) On the other hnd, b lemm 3., from this fct it follows tht GCD(, b). This is contrr to the bove supposition GCD(, b). In cse b, if n is n odd number greter thn, then from n n n c ( b). Let A c, B b, C. Then of different prit. This contrdicts cse. In cse b, if n 4, then from From here it follows tht 4 8l is even b c b n n n b c we hve n n n A B C, where the integers A, B re we hve c l, where l is nonzero integer. 8l 4. This is contrdiction becuse Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge 9 b is odd nd In short, there re no nonzero rtionl points on the curve ( L ). The theorem is proved.

20 References [] R.D. Crmichel, Diophntine nlsis, Dover Publictions Inc, New York (959), p 8-34 [] Sierpinski.Wclw, Elementr theor of numbers, PWN-Polish Scientific Publishers, Wrszw (964), p Le Di Hnh street, wrd 7, district, Ho Chi Minh cit, Viet Nm Pge