Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig, Ara Brach, Islamic Azad Uiversity, Ara, Ira Email: j.babaee@gaemiau.ac.ir Abstract The mai aim of the preset paper is to represet a exact ad simple proof for Fermat s Last Theorem by usig properties of the algebra idetities ad liear algebra.. Itroductio ad prelimiaries I umber theory, Fermat s Last Theorem states that: o three positive itegers a, b ad c ca satisfy the equatio for ay iteger value of x greater tha two. This theorem was first cojectured by Pierre de Fermat i 637 ad the first successful proof was released i 994 by Professor Adrew Wiles ad fially published i 995 after 358 years of efforts by mathematicias but the proof was well over 00 pages log ad complex utilizig the most moder 0 th cetury aalytical although Fermat claimed that: I ve foud a remarable proof of this fact, but there is ot eough space i the margi [of the boo] to write it. Therefore a questio has preoccupied the mids of mathematicias always ad that is, is there a simple proof of the Theorem? The author believes that based o the Lemmas ad provig preseted i this article, maybe he achieved to a simple provig actually. Lemma.. It is suffices to prove Fermat s Last Theorem for 4 ad for every odd prime p 3. It is well ow that if the Last Theorem ca be proved for = 4, the it is also prove for all multiples of = 4, because all of the remaiig umbers ca be reduced to a multiple of the prime umbers, it is therefore oly ecessary to prove Fermat s Last theorem for all the primes. Lemma.. I equatio + + a b c + Proof. I equatio + =, the expressios ( a b) +,( c b), ad ( c a) are coprime. + + + umbers a, b, ad c are relatively prime i pairs ad because that a + b, equatio ca be writte i the form of three relatios () to (3), so it ca be cocluded that the terms ( c a), ad ( c b) must be relatively prime i pairs. a + b = a + b a ab + a b ± ab = c + + + c a = c a a + ac + a c + ac = b + + + c b = c b b + bc + b c + bc = a + + + Theorem.3. Cosider the equatio + + a b c + + = which i that gcd a, b, c = ad + is prime umber ad let =, =, ad =, if set up above equatio the will be set up three followig couple-relatios simultaeously: a a a a b b b b c c c c + + c, a + b ab ba ± ab = c c + + c a = b b, c + a + ac + ca + ac = b b + + c b = a a, c + b + cb + bc + bc = a a () () (3)
That we have two case i above relatios, case I: a = b = c = ( + abc ) ad case II: exactly oe of the umbers a, b or c is equal to + ( + abc ). Proof. Cosider the equatio + + a b c + + = which i that positive umbers a, b, ad c are relatively prime i pair ad + is prime umber. Now let c c c c c c + + a + b based o the properties of algebra idetities ca be writte as follows: =, therefore the expressio a + b = a + b ( a + b +... ± ( ab) ) = c = c c c c... c + + + + + + + + (4) The equatio (4) ca be writte i the two followig forms: c c... c + m m m + + m + m + m + ± = c c... c a b ab ba ab c (5) (6) That i the above relatios, expoet c a b ab ba ( ab) [ a b ab ba ( ab) ] but c ( a b) [ a b ab ba ( ab) ] c is multiples of c that term ( a b) + just dividable by them ( c ( a b) + but [ + ± ] ) ad expoet c is multiples of c that term c c are multiples of c that both term ( a b) + ± just dividable by them ( [ a b ab + ba ± ab ] + ) ad expoets c to + ad + ± are dividable by them ad i other words are the commo factors of gcd a+ b, a + b ab ba ab = c c c metioed terms ( are relatively prime. Relatio (6) ca be expaded by the use of ( a b) writte i the form of relatio (9): ± ), therefore c ad c + of relatio (5) ad after summarizig ca be 4 a b ab ba ( ab) a b q a b q a b + +... ± = + + + + + +... ( ) [ ( + ) 4 6 + q a + b ± + ab = a + b a + b + q a + b + q a + b + + + m + m + m ].... + q ± + ab = c c c c ± ab = ( c c + + m + m + m + m m m 4 6 [( a + b) + q ( a + b) + q ( a + b) +... + q ]} M ( a+ b) c...c ) {( c c c... c ) m m m m m + + + c c ( + + ab = c c ) c c c ±...... M (7) (8) (9) That i above relatios, M is iteger ad amouts q to q are defiite iteger umbers (see the example.4). I both right term of the relatio (9) there are the commo factors c to c with a differet that i oe of the expressios have the powers ( + m ) to ( m ) m to + respectively ad i other expressios have the powers m respectively. Sice this factors( c ) themselves are a multiple of c, so i compariso with the term i
ab i the left side of the relatio (9) are prime ( gcd ( ab, c c... c ) = ) ad as the umber K + cosidered to be a prime umber, therefore this umber have just a factor (itself oly that is K + ) ad thus amog the commo factors c to c just oe of those umbers have a defiite amout ad the rest of them must be equal to ecessarily. Suppose that c has a defiite amout ad therefore always c to c are equal to. Thus the equatio (9) ca be writte as follows: m m ( ) + + c + c ± + ab = c c M (0) Now it is tired to declare that c is equal toor equal to K +. I relatio (0) for the power c, two states ca be cosidered: or state I: ( + m ) > m ad or state II:( + m ) < m. I the first state that is + m > m the relatio (0) ca be writte accordig to the followig relatio: m 3m ( ) = c c + + ( + ) ± + ab [( c ) + ( c M )] gcd ab, c = I relatio () as it referred before ( ) m, therefore c ca be equal to or +. Because + m is prime thus it s power always is equals to. To mae a equality of ( + ) = c there must be c = + ad m = that is a cotradictio, so i the first case there must be c =. Now the secod state for the powers is supposed, it s meas ( ) + m < m. I this case the relatio () ca be writte as follows: () +-m --+ 3m [ + + ( + ) ± + ab = c c ( c c M )] () +-m I relatio () based o above cocepts there must be: + = c that to set up this equality we must have c = + ad m = (because + m = ). So to study the relatio (0) by assumig that + is a prime umber it was idicated that firstly the amouts c to c must be equal to ad secodly by cosiderig the relatio (0) ad the two states which ca be for the power of c it was proved that c is equal toor + that i the secod state ( c = + ) the amout of power m must be equal to ad so c ad c are coprime. Thus the couple-relatios (5)-(6) ca be writte i the fial form of the followig couple-relatios: c + + + ± = a b ab ba ab c c (3) (4) That as it metioed before, i the above couple-relatios c that is greatest commo divider (gcd) terms ( a b) ( a b ab ba ab + ad + ± is equal to or equal to +. Now if the equatio + + + will be writte i the form of equatios + + c b a + + = + + + ad c + a = b based o the reasoig metioed above ad coductig the same procedure it ca be idicated that above equatios ca be writte accordig to the couple-relatios (5)-(6) ad (7)-(8) respectively: 3
c a = b b + + + + + + = c a ac ca ac b c b = a a + + + + + + = c b bc cb bc a b a (5) (6) (7) (8) I each of the above couple-relatios the greatest commo dividers a ad b are either equal toor equal to +. a b c a are relatively prime i pairs, so exactly oe of the Based o Lemma. as the terms ( + ), ( c b ), ad commo factors a, b, or c ca be equal to + ad the other two umbers are equal to ecessarily, i other words two states ca be cosidered for the above couples-relatios: case I: ( + abc ) or oe of the above couple-relatios has o commo factor that is a = b = c = ad or case II: ( + abc ) or exactly oe of the above couple-relatios has a commo factor that is or a = + (ad b = c = ) or b = + (ad a = c = ) ad or c = + (ad a = b = ). Example.4. Example of Theorem.3 for case K + = 7 ( ) 7 7 7 7 c a = b = b b b b b Thatb, b, ad b to b are relatively prime i pair ad expoet 4 b just couts term ( c a), (but + + + + + + = ) ad expoet b just couts term 6 6 5 5 4 4 3 3 gcd( a c a c c a a c c a a c, b ) 6 6 5 5 4 4 3 3 ( a c a c c a a c c a a c ) c + + + + + +, (but gcd( a, b ) = ) ad of both term. 7 m m m c a = b b b... b m to m are idefiite powers 6 6 5 5 4 4 3 7 7m 7m 7m + + + + + + = b b... b a c ac ca a c c a ac b ( c a) + 7 ac ( c a) + 4( ac) ( c a) + 7( ac) 6 4 3 7 7m 7m 7m ( ac) b... ( c a) ( c a) ab( c a) ( ac) b to b are the commo factors 3 4 7 = b b b + 7 + 4 = ( ) + ( ) + 7 7m 7m 4 m m m b b... b ( b b b... b ) c a 7 ab c a 4 ac 3 7 m 7 m 7 m m m m 7 ac b b 4 = 4 7 = b b... b ( b b... b ) M Because expoets b i ad ac are coprime ad 7 is prime umber, thus either b are equals to ecessarily. 3 7 m m 7 b 4 b 7 ac = b ( b ) M b = ad or b = + ad so m Either b = or 7 If so just b 7m = ad or b = or 7 b = or b = 7 (I this case: 6 m = ) c a b c a b b = = ad 3 7 7 7 7 6 a + c + ac + ca + a c + c a + ac = b b 6 6 5 5 4 4 7 b to 7 m = or
7 7 7 That b is equal or 7. If i relatio it is supposed that a = aa a ad b = bb b the accordig to 7 6 provig process metioed above ad i a similar way it ca be declared that the couple-relatios., 3 3 a + b + ba + ab + b a + a b + ba = c c ad so couple-relatios 7 6 c b = a., 6 6 5 5 4 4 7. b + c + bc + cb + b c + c b + bc = a a must be set up. 6 6 5 5 4 4 7. + = whe + is a prime umber, ca be Therefore it is geerally represeted that i relatio + + a b c + writte i the form of there couple-relatios (3)-(4), (5)-(6), ad (7)-(8) ad i other words if that relatio is set up, the above three couple-relatios must be set up simultaeously that ecessarily two states ca exist for the commo umbers a, b, ad c, case I: ( + abc ) or a = b = c = ad or case II: ( + abc ) or exactly oe of the umbers a or b or c is equal to + (ad the other two umbers should be equal to oe ecessarily). Lemma.5. I equatio + + + Proof., amout expressio ( a + b c) is always positive + + + + a + b a + b c a + b c a + b c 0 Lemma.6. I equatio + + + i the lieu of each positive a, b, ad c the relatios a + b c = ( ) ( ) = ( ) a a a a b b b b c c c c a = b c 0 Proof. Accordig to Theorem.4 if the relatio b = bb b ad c c c c a b = c + + + is always set up + is set up assumig c a a = a a a ad =, the the relatios (3), (5), ad (7) will be set up simultaeously ad i this case the amout of the expressio ( a + b c) ca be writte as follows: c a ( b c) a a a a a a a ( a ) + b + ( a c) = b b b b b = b ( b b b ) a + b c = c c c c c = c c c c 0 c c > 0 thus c c c 0 + + = = a a 0 a a > 0 thus a a a 0 + b 0 b b > 0 thus b b b 0 + + + Sice based o lemma.5 i equatio the amout of a + b c is always positive ad o the other had the amouts of a, b ad c are cosidered positive as well, so it ca be cocluded that i relatios (9) to () the expressio c c c, a a a, ad b b b are always positive. Theorem.7. If the determiat of the system of three homogeous equatios with three uows is equal to zero the each of the equatios ca be writte i the form of a liear combiatio of two other equatios. Cosider the system of three equatios with three uows a x + b y + c z = 0, ax + b y + cz = 0, a3x + b3 y + c3z = 0, assumig that its determiat is zero. By computig the determiat of system we will have: 3 3 3 3 3 3 Det = a ( b c b c ) b ( a c a c ) + c ( a b a b ) = 0 Now it will be idicated that each of the above equatios ca be writte i the form of a liear combiatio of two other equatios. Suppose that for example the equatio a x + b y + c z = 0 obtaied from liear combiatio of two other equatios. It s meas we have: (9) (0) () () 5
a x + b y + c z = 0 a x + b y + c z = 0 b = m b + b a x + b y + c z = 0 3 3 3 a = m a + a 3 3 c = m c + c 3 (3) (4) (5) Now we replace the amouts of a, b, ad c obtaied from the relatios (3) to (5) i the relatio (), we will have: Det = ( m a + a )( b c b c ) ( m b + b )( a c a c ) + ( m c + c )( a b a b ) = 3 3 3 3 3 3 3 3 3 m.( a b c a b c b a c + b a c + c a b c a b ) +.( a b c a b c b a c + b a c + c a b c a b ) = 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 As it is observed the coefficiet of m ad became equal to zero ad it meas that the amouts of the above matrix determiat is idepeded of the amouts m ad ad therefore the primary assumptio is correct ad i fact each equatio ca be writte i the form of a liear combiatios of two other equatios. This subject ca be show for the other equatios i similar ways. Lemma.8. I equatio a b c + = that m gcd a, b, c = ad c is eve, it will ecessarily be m =. r Proof. m + + + = 4 p + + 4q + = p + q + = u = c m = Lemma.9. I equatio a b c + = that gcd a, b, c = ad a, b, ad c are coprime, ca be writte i form of couple-relatios c b = A ad c + b = A that A ad A are coprime ad =.. Proof of Fermat s Last Theorem Cosider the followig equatio: m a A A (6) I relatio (6), a, b, ad c are all positive iteger umbers ad relatively prime i pairs. We goig to prove that equatio oly i lieu of x there will be a aswer. The above equatio will be examied i two states separately whe x is odd ad eve... Examiig the equatio whe x is odd Cosider the followig equatio: + + + (7) Based o Lemma. it is suffices to prove Fermat s Last Theorem for 4 ad for every odd prime p 3. Therefore assume + is a odd prime umber. Now by usig of the fudametal Theorem.3 ad Lemmas.4 to.7, it will be proved that equatio (7) will have a aswer just i the lieu of = 0. Accordig to Theorem.3, if equatio (7) set up, the will be set up three couple-relatios (3)-(4), (5)-(6), ad (7)-(8) simultaeously that i metioed equatios is a = a a a, b = b b b ad c = c c c, ad umbers a, b, b, c, c, a, b ad c are relatively prime i pairs ad are all positive iteger umber ad we have two state: or a = b = c = ad or exactly oe of them is equal to + ad the other two expoets should be equal to ecessarily. I other words at least two of the umbers amog commo factors a,b, ad c should 6
be equal to. Without loss of geerality, it is cosidered that a b +. = = ad therefore c is equal to or equal to Now based o Theorem.3 ad owig that the relatios (3), (5) ad (7) are set up simultaeously ad with respect to the relatios ad cocepts metioed above, it ca be writte that: + c = aa + bb a a + b b c ( c c ) = 0 + + c a = b b = b = cc c aa a a + b b c cc = 0 + + c b = a c = a = cc c bb a ( a ) + b b c cc = 0 (8) (9) (30) Therefore accordig to Theorem.3, if the relatio (7) is set up, if so three relatios (8), (9), ad (30) must be set up simultaeously. Now we try to study the aswers of three equatios with three uows metioed above that are i terms of the uow s a ad b ad c ad prove that the system of above equatios will be set up just i lieu of = 0. If the determiat of the system of above equatios will ot be zero the the equatios have a trivial solutio a = b = c = that is subsequetly cocluded that a = b = c = 0 that there is a cotradictio. Thus i order to 0 have a aswer for the above equatios system the matrix determiat of coefficiet must be equal to zero that if so the system will have ifiite aswerers. Through direct calculatio of determiat of above equatios ((8) to (30)) it ca be foud out that the determiat of coefficiets is always equal to zero. If the determiat of coefficiets of equatios are computed there will be: Determiat = a Eq. 5,6,7 b cc + bcc b a cc + a cc c c ab a b ( ) ( ) = a c c b b + b c c a a c c a b a b Now we multiply ad the divide the terms of the relatio (3) to expressio ( ab c ) ad we will have: (3) + + + + + ( ) ( ) ( ) Det = a a c c c b b b + b b c c c a a a c c a b a b a b a b c Base o Lemma.6 the amouts a ( a a ), b ( b b ), ad cc ( c c c ) are equal to (3) ( a + b c) ad also it is ow that, c = cc c, a = aa, ad b = bb, therefore by replacig these amouts i relatio (3) fially will have: { } Det = ac a + b c + bc a + b c a + b ab c a c b = a b c { ( a + b c) [ ac + bc ac bc] } = 0 a b c (33) Sice matrix determiat of the coefficiets is equals to zero, therefore based o Theorem.7, if we cosider each cosider the equatio (30), it is obtaied from the liear combiatio of equatios (8) ad (9) that we represet that i this state just i the lieu of = 0 the system of above equatios has a aswer. Because from the liear combiatio of the equatios (8) ad (9) i terms of desired coefficiets m, ad equalizig it with the equatio (30) we will have: 7
m a + a = a a : b : m b + b = b c : m ( cc ) + cc = c c Accordig to Lemma.6 if the equatio ( m + ). a = a ( m) b =. b ( ) c = m. c c a a thus should m + b b thus should m c c c thus should m (36) ( a a ) + ( b b ) = ( c c c ) will be set up i the lieu of each + + + positive a, b, or c, the the iequalities a a, b b, ad c c c will be always set up, so i order to set up the equalities (34) to (36) the iequalities m +, m, ad m should be set up simultaeously that it is cocluded the equatio m + = which is always the oly commo part of the above iequalities should be set up. Therefore the metioed equatios will be writte i the form of followig equatios: a = a ( b b ) = 0 Sice gcd ( a, a ) = gcd ( b, b ) = gcd ( c, c, c ) = therefore must = 0 m( c c c) = 0 So just i lieu of = 0 the system of three homogeous equatios with three uows metioed above will have a aswer. By studdig the other states of liear combiatio i above equatios we come to similar coclusios. I geeral firstly it was idicated that if the equatio (7) is set up the the relatios (3), (5), ad (7) (that are same relatios (8) to (30)) should be set up simultaeously as well ad sice it was show the determiat of coefficiets matrix of the system of above equatios is equal to zero, by usig of properties of the liear combiatios ad by applyig Lemmas.5 ad.6 ad so Theory.7, it was proved that the system of above equatios will have a aswer just i lieu of = 0 ad subsequetly it ca be cocluded that the equatio a aswer just i lieu of = 0. + + + will have Geerally based o Theorems.3 ad.7 ad Lemmas.4 to.6, it was proved that the equatio + + + whe + is prime umber, oly i recogitio of = 0 will have a aswer ad based o Lemma. we ca expad this statemet for all odd umbers +... Examiig the equatio whe x is eve ad equals to 4 Cosider the followig equatio that a, b, ad c are itegers ad relatively prime i pairs: (34) (35) 4 4 4 Equatio (37) ca be writte as follows: a + b c = a b (37) (38) Accordig to Lemma.8 i relatio (37), c is always odd, so a must be odd ad b must be eve or vice versa a must be eve ad b must be odd. Suppose that a is odd ad equals to a = a. a ad b is eve ad equals to b = p b b ( b ad ( a x b x c x ) b are odd ad p is idefiite). Thus both expressios ( a x b x c x ) + are eve. Therefore relatio (38) ca be writte i form of the relatios (39): + + ad p p+ + + + = = a b c a b c a a b b a a b b (39) 8
I relatio (39), base o Lemma.8, expressio ( a b c ) ( a + b c ) has factor the relatio (39) ca be writte as follows: a + b + c = a b a + b c = a b p + + has factor (therefore expressio p ), so because a, b, ad c are coprime therefore base o Lemmas.8 ad.9, That accordig to Lemma.9 i above relatios parameters a, b, a, ad b must be relatively prime i pair. By addig the relatios (40) ad (4) we will have: (40) (4) a + b = a b + a b p As assume a = a. a ad b = p. b. b, so the relatio (4) ca be writte as follows: ( ) = ( ) a a b b a b p I the above equatio as a ad ( ) t a b ad will have: a b = b t p a b = a t b are odd ad coprime, thus if assume ( ) t a b, the should (4) (43) (44) (45) By solvig two equatios (44) ad (45), the amouts ( ) ( ) b = b a t p a = b a t p a ad b will be equal to: (46) (47) As cd ( a, ) coditio of g b = ad we suppose b b b b + a = b = ad a p a a =, so gcd b, a =, the whe just t = the gcd b, a = will be set up, therefore the relatios (46) ad (47) will be formed as follows: p a + a = b p Accordig to Lemma.8 i the relatio (48) there must be p ( p ) = = ad i relatio (49) there must be p = ( p = 0.5), firstly because p ad are both iteger umbers, so p caot be equal to 0.5ad this is a cotradictio ad secodly as two relatios metioed above must be set up simultaeously, p caot possess both amouts 0.5ad, therefore aother cotradictio is see. Geerally it was proved that the equatio a x + b x = c x whe x is eve ad equal to 4 will have o aswer..3. Examiig the equatio whe x is eve ad equals to 4 + Cosider the followig equatio: (48) (49) 9
(a ) + ( b ) = ( c ) 4 + 4+ 4+ + + + (50) As it was proved formerly i part (-), equatio (50) will have a aswer just i the lieu of = 0 that i this state, the above equatio will be i the followig form: Now we study the possible aswers for the above equatio. Accordig to Lemma.8, c must be always odd ad amog a ad b oe of them is eve ad the other oe is odd. Suppose a is eve ad b is odd ad equal to b b b = b b. I that case based o Lemmas.9, the equatio (5) ca be writte i the form of the followig couple-relatios: (5) c a = b c + a = b (5) (53) So the geeral aswer of the equatio (5) (b is odd ad equal tob b ) will be as follows: c = b + b /, a = b b /, b = b b + + + Geerally it was proved that the equatio oly i lieu of = 0 will have a aswer ad the equatio a + b = c just i lieu of = has a aswer. Thus it was cofirmed that the equatio a x + b x = c x just i lieu of x has a aswer ad this proves the correctess of Fermat s Last Theorem. Acowledgmet We would lie to express our gratitude to Dr. Ali Rajaei, Dr. Joh Hamma, ad Dr. Reza Taleb, for providig their support ad ecouragemet. 3. Coclusio Geerally for the equatio a + b + c = 0 by usig properties of the algebra idetities ad liear algebra ad presetig a ew theorem, it was represeted that: If x is odd ad equal to +, it was proved accordig to ey + + + Theorem.3 ad Lemma.7 that just i lieu of = 0 the equatio will have a aswer. So if if so the equatio a 4 + b 4 = c 4 will have o aswer ad if x is eve ad equal to x is eve ad equal to ( ) + + + +, based o the first result i above, it ca be cocluded that the equatio ( a ) + ( b ) = ( c ) will have a aswer just i lieu of = 0 ad its geeral aswer for odd p ad odd q will be: c = p + q /, a = p q /, b = p q. Based o the above coclusios it was proved that the equatio just i lieu of x will have a aswer. 4. Refereces A. Wiles, Modular elliptic curves ad Fermat s Last Theorem, Aals of Mathematics, 995, 443 55. 0