A METHOD TO SOLVE THE DIOPHANTINE EQUATION ax 2 by 2 c 0

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1 A METHOD TO SOLVE THE DIOPHANTINE EQUATION ax by c Floreti Smaradache, Ph D Associate Professor Chair of Departmet of Math & Scieces Uiversity of New Mexico College Road Gallup, NM 87, USA smarad@um.edu ABSTRACT We cosider the equatio () ax by c, with a,b * ad c It is a geeralizatio of the Pell s equatio: x Dy. Here, we show that: if the equatio has a iteger solutio ad a b is ot a perfect square, the () has a ifiitude of iteger solutios; i this case we fid a closed expressio for (x,y ), the geeral positive iteger solutio, by a origial method. More, we geeralize it for ay Diophatie equatio of secod degree ad with two ukows. INTRODUCTION If ab k is a perfect square ( k ) the equatio () has at most a fiite umber of iteger solutios, because () become: () (ax ky)(ax ky) ac If (a,b) does ot divide c, the Diophatie equatio does ot have solutios. METHOD TO SOLVE. Suppose that () has may iteger solutios. Let (x,y ), (x,y ) be the smallest positive iteger solutios for (), with x x. We costruct the recurret sequeces: () x x y y x y makig coditio () verify (). It results: a b () havig the ukows,,,. a b a (5) a b b (6) We pull out a ad a from (5), respectively (6), ad replace them i () at the square; we obtai a b a (7). We subtract (7) from (5) ad fid: (8). Replacig (8) i () we obtai: *.

2 b (9). a Afterwards, replacig (8) i (5), ad (9) i (6) we fid the same equatio: a b a (). Because we work with positive solutios oly, we take b x a x a y y x y where (a, ) is the smallest, positive iteger solutio of () such that a. Let A x' y' b a, A x' y' ( ). It is evidet that if (x', y') is a iteger solutio for () the is aother oe where A is the iverse matrix of A, i.e. A A A A I (uit matrix). Hece, if () has a iteger solutio it has a ifiity. (Clearly A ( ) ). The geeral positive iteger solutio of the equatio () is: GS with ' ' ( x, y) x, y x y A x y, for all, where by covetio A I ad A k A...A of k times. I problems it is better to write GS as: x ' y ' GS ad x " y " A x y, A x y, * We prove, by reductio at absurdum that GS is a geeral positive iteger solutio for (). Let (u,v) be a positive iteger particular solutio for (). If k :(u,v) A k x y, or k * :(u,v) A k x y the (u,v) GS. Cotrary to this, we calculate (u i,v i ) A u i v i, for i,,,... where u u, v v. Clearly u i u i for all i. After a certai rak x u i x it fids either u i x, but that is absurd. It is clear that we ca put

3 GS x y A x y,, where. Now we shall trasform the geeral solutio GS i a closed expressio. Let be a real umber. Det(A I) ivolves the solutios, ad the proper vectors V, (i.e., Av i i v i, i, The P AP ). Note, whece A P P ad doig the computatios we fid a closed expressio for GS. EXAMPLES. For the Diophatie equatio x y 5 we obtai x 5 6 y 5 whece a closed expressio for x ad y : x 6 v v i M ( ) P, ad replacig it i GS, ad, 5 6, v, ( 6, ), (5 6) 6 (5 6) for all 6 6 y (5 6) (5 6) 6 6. For equatio x y the geeral solutio i positive iteger is: x ( ) ( ) y ( for all, ) ( ) that is (,), (,), (,8), (5,), EXERCICES FOR RADERS. Solve the Diophatie equatios:. x y [Remark: x 7 y 7. x 6y [Remark: x 5 y 5 5. x y 9?, ]?, ]

4 [Remark: x 7 y 7?, ] 6. x y 8 GENERALIZATIONS If f (x,y) is a Diophatie equatio of secod degree ad with two ukows, by liear trasformatio it becomes () ax by c, with a,b,c. If ab the equatio has at most a fiite umber of iteger solutios which ca be foud by attempts. It is easier to preset a example: 7. The Diophatie equatio () 9x 6xy y 6x 6y becomes () u 7v 5, where (5) u x y ad v y We solve (). Thus: u 5u 8v (6), with (u v 8u 5v,v ) (, ) First solutio: By iductio we prove that for all we have that v is odd, ad u as well as v are multiple of. Clearly v, u. For we have: v 8u 5v eve odd odd, ad of course u,v are multiples of because u,v are multiple of too. Hece, there exist x,y i positive itegers for all : (7) x (u v ) / 6 y (v ) / (from (5)). Now we ll fid the GS for () as closed expressio, ad by meas of (7) it results the geeral iteger solutio of the equatio (). Secod solutio: Aother expressio of the GS for () will be obtaied if we trasform (5) as u x y ad v y for all. Whece, usig (6) ad doig the computatio, we fid 5 x (8) x x y, with (x,y ) (,) or (, ) y x 9y (two ifiitude of iteger solutios).

5 Let A 5 9, the x y A or (9) x y A, always. From (8) we have always y y... y (mod), hece always x. Of course, (9) ad (7) are equivalet as geeral iteger solutio for (). [The reader ca calculate A (by the same method liable to the start o this ote) ad fid a closed expressio for (9).]. More geerally: This method ca be geeralized for the Diophatie equatios: () a i X i i b, with all a i,b. If always a i a j, i j, the equatio () has at most a fiite umber of iteger solutios. Now, we suppose i, j,..., for which a i a j (the equatio presets at least a variatio of sig). Aalogously, for () x h ( ) i ihx i (), h, we defie the recurret sequeces: cosiderig (x,..., x ) the smallest positive iteger solutio of (). Replacig () i (), it idetifies the coefficiets ad it looks for ukows ih,,i,h. (This calculatio is very itricate, but it ca be doe by meas of a computer.) The method goes o similarly, but the calculatios become more ad more itricate for example to calculate A, oe must use a computer. (The reader will be able to try this for the Diophatie equatio ax by cz d, with a,b,c * ad d ) REFERENCES [] M. Becze - Applicaţii ale uor şiruri de recureţă î teoria ecuaţiilor Diophatie - Gamma (Braşov), XXI-XXII, Aul VII, Nr. -5, 985, pp [] Z. I. Borevich, I.R. Shafarevich - Teoria umerelor - EDP, Bucharest, 985. [] A. Kestam - Cotributios to the Theory of the Diophatie Equatios Ax By C. 5

6 [] G. H. Hardy ad E. M. Wright - Itroductio to the theory of umbers - Fifth editio, Claredo Press, Oxford, 98. [5] N. Ivăşchescu - Rezolvarea ecuaţiilor î umere îtregi - This is his work for obtaiig the title of professor grade, (coordiator G. Vraciu), Uiv. Craiova, 985. [6] E. Ladau - Elemetary Number Theory - Chelsea, 955. [7] Calvi T. Log - Elemetary Itroductio to Number Theory - D. C. Heath, Bosto, 965. [8] L. J. Mordell - Diophatie equatios - Lodo, Academia Press, 969. [9] C. Staley Ogibvy, Joh T. Aderso - Excursios i umber theory - Oxford Uiversity Press, New York, 966. [] W. Sierpiski - Oeuvres choisiers - Tome I, Warszawa, [] F. Smaradache - Sur la resolutio d équatio du secod degré a dex icoues das Z i the book Gééralizatios et gééralités Ed. Nouvelle, Fes, Marocco; MR: 85, H:. [Published i Gazeta Matematică, Serie, Vol., Nr., 988, pp. 5-7; Traslated i Spaish by Fracisco Bellot Rasado, U metodo de resolucio de la ecuacio diofatica, Madrid. 6