An Elementary Proof of Fermat-Wiles Theorem and Generalization to Beal Conjecture
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1 Iteratioal Joural of Sciece ad Research (IJSR ISSN (Olie: Impact Fact (2012: A Elemetar Proof of Fermat-Wiles Theem ad Geeralizatio to Beal Cojecture Jamel Ghaouchi RIME Departmet of Mathematics, Marsa, Tuisie Abstract: A proof of Fermat theem is preseted ad a geeralizatio to Beal cojecture is proposed. F this, e begi ith Fermat ad Fermat-Catala equatios ad solve them. Keds: Diophatie; Fermat; Fermat-Catala; Resolutio 1. The Fermat Equatio Fermat equatio is x z x az Let x x Aa 4 4 If A 0 x GCD x but (, 1 4 impossible, there is o solutio ad Aaz If A z ; 3 x x Z impossible ( x x z Aaz A A ( x z A ( z Ax x ( z A ( xz A But GCD(x,=1 the four cases: x u( x z A ; u( z Ax x v( x z A ; v( z Ax vx x z A ; v z Ax x u( x z A ; u( z Ax vx x z A ; v z Ax x v( x z A ; v( z Ax ith uv, Z First case x u( x z A ; u( z Ax x v( x z A ; v( z Ax uv z A x Az x x uv z A x A az uv A z uv( A z ( ( ( ( Impossible because A, u, v are itegers Paper ID: OCT
2 Secod case vx x z A ; v z Ax Iteratioal Joural of Sciece ad Research (IJSR ISSN (Olie: Impact Fact (2012: uv z A x Az ( x x z A x A az ( A z uv A z But uv( auva v( x Au( x A au x ; av x x aa ( z ( x( x aa ( z x ( x( x uv A z 0 a Aa( 4 4 va( x auvAv( x ad if x aa ( z ( x( x aa ( z x ( x( x uv A z 0 a Aa( 4 4 va( x auva v( x It is impossible because 2 Third case x u( x z A ; u( z Ax vx x z A ; v z Ax v u z A x Az x x u z A x A az u A z vu( A z ( 2 2 ( ( ( vx ( x vauv( x A( x A u( x 1 u ; 2 Impossible because u, A are itegers Fourth case x v( x z A ; v( z Ax Paper ID: OCT
3 Iteratioal Joural of Sciece ad Research (IJSR ISSN (Olie: Impact Fact (2012: u v z A x Az x x v z A x A az v A z u v( A z ( ( ( ( ux ( x ua( x Auv ( x A vx ( 1v1; x 1 Impossible! Because v, A are itegers! The ol solutio is A=1 ad =2 2. The Fermat-Catala equatio p q c q c x z x az q 2 p2 x x aa The equatio o is Let If q2 p4 A0 x GCD( x, 1 p 4 It meas that p=2 is the prime solutio! c q p3 z A z ; p3 x x Z Impossible because GCD(,z=1, thus p= ( x x z aaz A A q c p2 2 p2 c q ( x z A ( z Ax x 2 c q c 2 p2 ( z A ( xz A GCD( x, 1 the four cases: First case q p c c p q x u x z A u z Ax x v( x z A ; v( z Ax q c p2 2 p2 c q ( ; ( q c 2 p2 2 c uv z A x Az x x uv z A x A az uv A z uv( A z 1 p p 2c 2 q c p2 2 q p 2c 2 q 2 c p ( ( ( ( Impossible because A,u,v are itegers Secod case q c 2 p2 2 c v z A ; v z Ax q c p2 2 p2 c q uv z A x Az ( x x z A x A az ( A z uv A z p p 2c 2 q c p2 2 q p 2c 2 q 2 c p Paper ID: OCT
4 if a>0 ad the proof is the same f a<0, e have a 2 a p2 a c A ( x z ( p2 2 a A 0 0 p2 2 Iteratioal Joural of Sciece ad Research (IJSR ISSN (Olie: Impact Fact (2012: x ; x x x z x a b2 2 a 2 b a c b2 x z x a a b c 2 a x x z x z x 2 b2 a a c a c b a 0 ( x x x x 2 a a 2 It is impossible! if 2 a b2 a 2 b a c 2 c a x x x x z 1 x z x a c x z it is impossible! if a b2 2 a 2 b a c 2 c a x x x x z 0 x z x 0 Thus if It is impossible! Let us suppose o a A 0 0 p2 2 If x ; x x z x a a x b c 2 a z x 2 b2 a b a 0 ( x x c z 0 Impossible! if a b2 2 b2 2 b a c b2 2 a b2 a 2 b a c 2 c a x x x x z 0 x z x 0 Impossible! if x x x x z 1 x z x a c if x z it is impossible! Aother proof : 2 q p2 2p4 2p2 4 2 uv( auva v( x A u( x A 2p4 2q2 4 2 au x ; av x 4 p 1 2 q c 2 q p2 aa ( z ( x( x c 2 q p2 p q 2 q p2 aa ( z x ( x( x uv A z 0 p q x a 0 2 q p2 Aa( v a( x 2q2 2p4 0 auva v( x a b2 2 a 2 b a c 2 c a if Paper ID: OCT
5 Iteratioal Joural of Sciece ad Research (IJSR ISSN (Olie: Impact Fact (2012: p 1 2 q c 2 q p2 aa ( z ( x( x c 2 q p2 p q 2 q p2 aa ( z x ( x( x uv A z 0 p q x a0 2 q p2 Aa( 4 2 va( x 2q2 2p4 0 auva v( x Impossible because : p 2 Third case x u x z A u z Ax v z A ; v z Ax q c p2 2 p2 c q ( ; ( q c 2 p2 2 c v u z A x Az x x u z A x A az u A z vu( A z p ( p 2c 2 q c ( p2 2 q ( p 2c 2 q 2 c ( 2 2c p 2 q p2 2p4 2q2 4 2 vx ( x vauv( x A( x A 2p4 2q2 u( x 1 p2 Impossible because u, A are itegers Fourth case x v x z A v z Ax q c p2 2 p2 c q q c 2 p2 2 c ( ; ( u v z A x Az x x v z A x A az v A z u v A z p p 2c 2 q c p2 2 q p 2c 2 q 2 c p ( ( ( ( ( 2 q p2 2p4 2q2 4 2 ux ( x ua x Auv ( x A 4 2 v ( x 1 Impossible, because v,a are itegers! I the Fermat-Catala equatio, oe of the expoets must be equal to 2! The Beal cojecture has bee proved! I fact, i the three precedet equatios studied here, oe of the expoet greater equal to 2 must be miimum, hich meas that it must be 2! 3. Coclusio solved all three equatios b the same method ad proved to theems ad oe cojecture. Refereces [1] Paolo Ribeboïm, The Catala s cojecture, Academic Press, 1994 [2] Robert Tijdema, O the equatio of Catala, ActaArith, 1976 Paper ID: OCT
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