Joural of Pure ad Applied Mathematics: Advaces ad Applicatios Volume 17, Number, 017, Pages 77-87 Available at http://scietificadvaces.co.i DOI: http://dx.doi.org/10.1864/jpamaa_71001178 SOLVING THE CONGRUENT NUMBER PROBLEM IS SOLVING THE ELLIPTIC CURVE PROBLEM Istitute Poltechic of Leo Leo Spai e-mail: eulogiogar@gmail.com Abstract I this paper, we preset solutio of the cogruets umbers problem ad it is applied for solved elliptical curve. 1. Itroductio The cogruet umber problem was proposed b Al-Karaji (95-109) woderig what umbers aother square, i.e., Z b subtractig them from a square give b, the Z is cogruet. We did ot a refer to the areas of the triagles. However, we also ow that () is a b a umber cogruet if it is the area of triagle ad therefore are all the triagles that have b area half of the rectagle (all the triagles rectagles). The best ow achievemets were i 198 b Tuell with 010 Mathematics Subject Classificatio: 11Mxx. Kewords ad phrases: cogruet umbers, equatio of Efer, Tuell theorem, elliptic curve. Received Februar 5, 017; Revised March 5, 017 017 Scietific Advaces Publishers
78 his theorem ad fiall those achieved b (Robert Bradshaw, W. B. Hart, D. Harve, G. Toraria ad M. Watis i 009) with the help of computers. I this wor it is verified the mistae that have give b the computers.. Mai Results The achievemet of m results is a direct cosequece of the followig equatio of Efer [4]: 4 + ( m + 1) a. m (1) Example. 4 + 5 9; 9 + 7 16; 16 + 9 5. Theorem 1. All umbers squared are defied b equatio 4 + ( m + 1). Proof 1. Subtractig each of the last sums of Equatio (1), we have () cogruet odd. 4 + m 4 + m ( m + 1) b, ( m + 1) [ + 1] ( b ), ( ) 4 + b ( m + 1), m ad so o, we observe that: ( + 1 ); is cogruet umber. 1
SOLVE THE CONGRUENT NUMBER PROBLEM, IS 79 Proof. We tae, the last two sums of Equatio (1). We subtract betwee if ad we have () that is a cogruet eve. 4 + m ( m + 1) [( + 1) + ( ( 1) ) + 1] c < b, [ + 1 + ( ( 1) ) + 1] 4 + ( m + 1), m agai: [ ( + 1) + ( ( 1) ) + 1] ; ( ) is a eve cogruet. The method to ow the cogruet umbers, pairs or odd as quicl ad accuratel; is with the two expressios of cogruet [6], [7] followig: for: ( ; ; 4; 5 ). Therefore: [ + 1 5 mod ]; [ 4( + 1) 0 mod ], (1) A odd umber with > 1 is a cogruet umber. () A eve umber i the form of ( + 1) is ot cogruet; as 4 + 4( + 1); let us recall that 4( + 1) ( + 1) + ( ( 1) + 1) b the Equatio (1). [ ] ; () ( m ) 4( + 1), where ( m eve), cogruet umber if ad ol if, is the area of a triagle whose sides are ( a, b, c) Z, i.e., the Pthagoras theorem a + b c.. Discussio This sectio is composed of two parts related to the cogruet umbers. (i) Theorem of Tuell.
80 (ii) Cogruet umbers i the elliptic curves ad i the modular forms; Taiama-Shimura cojecture toda called theorem. Theorem of Tuell. Give () a umber square free positive iteger will be a cogruet umber if fulfilled the followig: f ( ) ( x,, z) Z x + + 8z, g ( ) ( x,, z) Z x + + 8z, h ( ) ( x,, z) Z x + + z, ( ) ( x,, z) Z x + 4 + z, If ( odd) ad () is cogruet the f ( ) g( ) ad for ( eve) if () is cogruet the h( ) ( ). I the previous poit () it was show that ( ) with ( odd) are ot cogruet ad therefore f ( ) g( ). For ( eve), h( ) ( ) alwas that () is the area of a rectagle triagle of sides ( a, b, c) Z ad i tur that () also is cogruet (the differece betwee two squares (poit ). Therefore h ( ) 4( + 1) 0 mod, ( ) ( + 1). With Tuell s theorem few cogruet umbers are defied; i relatio with that the exist. (ii) Cogruet umbers allow us to ow if a existig elliptic curve, (if the have solutios for ( Z ) ad how ma). Let be elliptic curve E Z. x x. () This is the same as: x( x ).
SOLVE THE CONGRUENT NUMBER PROBLEM, IS 81 B Pthagoras ( x b ) if ( x,, b) are Pthagorea terms; () is a cogruet umber because we ow that ( 1 ), where ( 1 ) is also cogruet; example 5 5 + 1 5 mod ad hece the Equatio () is reduced to the form: xb, which implies ( x ). ( ) with ( m, b, ) Z, b example: ( 5 ); ( 0) ; ( b 15); implies ( 75) 75 5 0 5. Thus the curve exists ad is ot a elliptic curve modular, it is reducible ad violates the law of additio (a sum of two poits of a lie that cuts to the elliptic curve) gives a third poit of the curve of the same group ( P, Q, R) Q, however here we have P + Q R for beig R Z ). Therefore this curve ca ot be parametrized with a modular fuctio. Oe elliptic curve exist if ( Z ) ad has i its coefficiets (a cogruet umber) ad hece will have ol oe solutio for ( Z ) as a cosequece of beig () the differece betwee two squares, this maes impossible the existece of aother pair of squares alog (x). At this poit we shall aalze some of the elliptic curves alread demostrated b Efer [4]; the reaso for its aalsis is to see that a curve that exist ( Z ) has a uique solutio for ( Z ). We start with the elliptic curve that is the cetral axis of stud of all for the elliptic curves b the Riema-Roch theorem: The statemet is that b meas of a series of trasformatio a elliptical curve arrives at the expressio. x + bx + c. ()
8 This is the basis of the article b Richar Talor et al. [] to affirm the demostratio of the Taiama-Shimura cojecture. Ufortuatel, the Riema-Roch theorem caot be corrected sice of beig true the there could be o elliptic curve. I this wor, we demostrate the existece of several of them that the have (solutio iteger), the first Equatio (); the followig is the Equatio () that the step to the form of the followig: x( x + b) + c, beig ( b; c) cogruet umbers the equatio exists; because we ow that: x + b with (b) cogruet ad therefore x ( m1 ) for ( x ( ) ). m Leavig the Equatio () as: (( m ) )( m ) + c, where (c cogruet umber) ( Z ). solutio: x 4 ; b 9; ( c 44) ad also (c 169) 4 + 9 4 + 44 144 1, 4 + 9 4 + 69 169 1. There are ma elliptical curves of this class with a sigle solutio i each oe of them, (() is a cogruet umber, differece betwee two squares) does ot admits others squares. It is eough that oe of the three poits above is ot fulfilled for the curve to be a modular elliptic curve. Amog the elliptical curves that exist some have their expressio of Galois; provided that (x) is a power. Two examples of this are give b Equatios () ad (), respectivel. 5 x + 80x + 115 0, ( x 5) ; x + 9x + + 7 0, ( x ). 5
SOLVE THE CONGRUENT NUMBER PROBLEM, IS 8 Now, we aalze the elliptic curve E Q of such form that: If ( E Q ) ( E Z ). For it we have the followig: x j + m x + i. j Ad as first orm of a elliptic curve: the coefficiet of the cube will alwas be the uit, that is to sa factorizig we have: ( j ) x + xm + i ( j ) x( x + m) +. At this poit we appl the rules established to that exist ( EQ ) : x ; j ad m, where cogruet umber ad i tur ( i ) also is a cogruet umber, therefore we have j 8 ; 5; x 4 i ( 8 ) 8 + 4m +. We ow that 4m + i ; with 5 ; 4m 0 ad 19; i.e., ( 8 ) 8 +, i i 8 + 51 64 9, solutio for the Equatio () with E Q is x j + mj x + j i, 4 4 + 5 4 16 + 19 64.
84 Other curve E Q. x x c, 9 4 9 4 9 64. Proof 6. The last elliptic curve to be aalzed is: The equatio of Fre [5], is relevat for the demostratio of Fermat theorem b Adrw Willes [1]. We will show that the existece of the curve of Fre does ot deped o the Fermat theorem (either this true or false), the curve is alwas modular. m m m x + ( c b ) x ( bc). We ow that there are Pthagorea terms ifiite to appl i the equatio ad see that the result is: there is o curve for (a, b 4, ad c 5) ad also for ( a, b, c); with ( 1; ; ; 4 ). for m [( c b ) a ]. If we factorize the curve we have: x[ x + ( c b )] ( bc). If we replace ( c b ) b a we have x[ x + a ] ( bc) ; at the same time x + a ( ) therefore the curve is reduced as x( ) ( bc). This maes it eas for us to aalze all the Pthagorea terms, we observe that (bc) is a icrease o the terms (b), i.e.,: ( b ) ( cb) ad requires doig the same with (a) therefore, that ( x c); we ow that ( x ) with which ( a) ( bc) ad also that ( a < b < c) ad ( c ) therefore ( a) ever is a Pthagorea term; does ot exist
SOLVE THE CONGRUENT NUMBER PROBLEM, IS 85 ( Z ) for a value of (x) ad either it is possible for a other value curve for (m ), we will chec if it is also for ( m > ). of () is to fulfill ( a) ( bc). Is alwas a modular elliptic Let ( m + ) ad (m eve). + + xa We have for our aalsis the optios. or + m + ( bc). ( bc) ( ; ); cogruet, + Let s see if ( bc) ( ). ( a ) ( ; ); beig x. + For this we factorize ( bc) ( bc) ( bc) ( ) for ( m ; 4; + m 6 ); i.e., ( ( bc) ( )) + is a square, ad the same is true for ( a), this implies that (bc) ad (a) are Pthagorea terms ad, as it is alread demostrated, it is ot possible to have ( Z ). Let s see what ( m 1 ; ; 5; 7 ). We have ( bc) + 1 ( bc) ( ) ( ) + ( m + 1) + 1 + 1 + 1 bc bc bc we ow that ( ) ( ) ( ) ( ) ( ); ad therefore we eeds that (bc) also be a square, however ( bc) ( ) ; alwas that ( c b) is required b the curve. I abstract: if Fermat theorem is true for + m + m + m c b a + + + + + + curve would be modular ad, if [ ( 1) 1 ( 1) 1 ] ( 1) 1 b a c the curve would also be modular; is verified that the curve of Fre does ot exist ad therefore is a elliptic modular curve. That is, all semi-stable elliptic curves are elliptic modular curves it is ot show i the Fermat theorem. the
86 4. Coclusio (1) I the Birch ad Swiero-Der cojecture it is demostrated that because we ca ow if a elliptic curve exists or ot ( Z or Z ) ad if exists the it has ol oe solutio. () It is to solve the cogruet umbers problem ad with it is verified that with the Tuell theorem ot we have all cogruet umbers. The questio is iverted, at resolve the cogruet umber problem is verified the existece of elliptic curve. () Goro Shimura asserted that EVERY elliptic curve is a elliptic curve modular; is icorrect, exist elliptic curves ad therefore, the are ot oe elliptic curve modular. All elliptic curve that exist is reduced to the form [ ( x ) + B ] ad therefore it is ot possible the have a fuctio modular ( f ( z) + B). Acowledgemets I tha Efer Diez for the publicatio of his article, without it, it would ot have bee possible to preset this demostratio. Refereces [1] Adrew Willes, Modular elliptic curves ad Fermat lest theorem, Aals of Mathematic 141() (1995), 44-551. [] Bra J. Birch ad Nelso M. Stephes, The parit of the ra of the mordell-weil group, Topolog 5(4) (1966), 95-99. [] Christope Breu, B. Corad, F. Diamod ad R. Talor, O the modularit of elliptic curves Q wild -adic exercises, J. America Mathematic Societ 14 (001), 84-99. [4] Efer Diez, Equatio for solve elliptical curve, U. J. Computatioal Mathematic 1 (01). [5] G. Fre, Lis betwee stable elliptic curves ad certai diophatie equatios, Aales Uiversitatis Saraviesis: Series Mathematicae 1(1) (1986), 1-40.
SOLVE THE CONGRUENT NUMBER PROBLEM, IS 87 [6] Nelso M. Stephes, Cogruet properties of cogruet umber, Bull. Lodo Math. Soc. 7 (1975). [7] Keqi Fig, No-cogruet, odd graphs ad BDS cojecture, Acta Arith. 75(1) (1996). [8] Keqi Figs ad Ya Xue, New series of odd o-cogruet umbers, Sciece i Chia Series a Mathematics 49(11) (006). g