ON THE DIOPHANTINE EQUATION

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1 Fudametal Joural of Mathematics ad Mathematical Scieces Vol., Issue 1, 01, Pages -1 This paper is available olie at Published olie July, 01 ON THE DIOPHANTINE EQUATION SARITA 1, HARI KISHAN,* ad MEGHA RANI 1 Departmet of Mathematics DCR Uiversity Murthal, Soipat, Haryaa Idia Departmet of Mathematics D.N. College Meerut, U. P. Idia Departmet of Mathematics RKGIT Ghaiabad, U. P. Idia Abstract I this paper, the surd euatios of the form x y = + have bee discussed for differet itegral values of > 1. This type of surd euatio has also bee discussed for ratioal values of of the form some itegral values of >. = for Keywords ad phrases: diophatie euatio, surd euatio ad itegral solutio. 010 Mathematics Subject Classificatio: 11D4. * Correspodig author Received May, 01; Revised Jue, 01; Accepted Jue, Fudametal Research ad Developmet Iteratioal

2 SARITA, HARI KISHAN AND MEGHA RANI 1. Itroductio Pierre de Fermat i 1 wrote i the margi of a copy of Arithmetica that o three positive itegers a, b ad c satisfy the euatio a + b = c for ay iteger value of greater tha. He claimed that he had a proof that was too large to be fitted i the margi. This result is kow as Fermat s Last Theorem or Fermat s Cojecture i the literature. Adrew Wiles i 1994 proved this result successfully. He formally published it i 199. This theorem is amog the most otable theorems i the history of mathematics ad was i the Guiess Book of World Records for most difficult mathematical problems prior to its proof. Billioaire baker Adrew Beal (199) while ivestigatig the geeraliatio of Fermat s Last Theorem proposed a cojecture. This cojecture is kow as Beal s Cojecture. Gregorio (01) preseted a proof for the the Beal s cojecture ad a ew proof for the Fermat s last theorem. Gola, L.W. (014) preseted a proof of Beal s cojecture. Ghaouchi (014) preseted a elemetary proof of Fermat-Wiles theorem ad geeraliatio to Beal s cojecture. Thiagraja (014) provided computatioal results ad a proof of Beal s cojecture. Gopala, Sumathi ad Vidhyalakshmi (01) discussed the trascedetal euatio with five ukows 4 x + y ( r + s ). Padichelvi, V. (01) discussed a exclusive trascedetal euatio x + y + + w = ( k + 1) R. Gopala, Vidhyalakshmi ad Usha Rai (01) discussed the itegral solutios of the surd euatio x y + X + Y + + w = p. Here, the surd euatios of the form + y have bee discussed for x = differet itegral values of > 1. This type of surd euatio has also bee discussed for ratioal values of of the form = for some itegral values of >.. Aalysis (a) For =, the give surd euatio reduces to

3 ON THE DIOPHANTINE EQUATION x + y = If we take x = ( a b ), y = 1a b ad = ( a + b ) 4 the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (b) For =, the give surd euatio reduces to If we take x = ( a b ), y = 4a b ad = ( a + b ) the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (c) For = 4, the give surd euatio reduces to If we take x = ( a b ), y = a b ad = ( a + b ) 8 the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (d) For =, the surd euatio is If we take x = ( a b ), y = 4 a b ad = ( a + b ) 10 the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (e) For =, the surd euatio is If we take x = ( a b ), y = 4 a b ad = ( a + b ) 1 the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (f) For =, the surd euatio is If we take x = ( a b ), y = 4 a b ad = ( a + b ) 14 the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (g) For geeral, the surd euatio is

4 8 SARITA, HARI KISHAN AND MEGHA RANI If we take x = ( a b ), y = 4 a b ad = ( a + b ) the above surd euatio is satisfies. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. The followig trasformatios may also be cosidered for the solutio x = α, y = β ad ( u + 1) = γ. These trasformatios reduces the give geeral surd euatio to α u+ 1 + β = γ This euatio is satisfied by ( ) u α = A A + B, β = B( A + B ) u ad γ = ( A + B ). Thus ( ) u x A A B y B = +, = ( A + B ) u ad ( A B ) ( u + 1 = + ) is the reuired solutio. Puttig values of A, B, u ad differet solutios ca be obtaied. Similarly, with the help of the trasformatio. x = α, y = β ad ( u + ) = γ, γ = a + b. It ca be show that 1 1 x = F ( u + 1), y = G ( u + 1) i ad ( a b ) ( u 1 = + ) + where ( ) ( ) ( u + 1 1) ( ) ( u+ F u + = a + ib + a ib 1) ad ( ) ( ) ( u +1 G u + 1 = a + ib ) u +1 ( a ib) ( ). Puttig differet suitable values of a, b, u ad differet reuired solutios ca be obtaied, 4 4 Example. Takig u = 1, we get F ( u + 1) = ( a a b + b ) ad G ( u + 1) = 4 4 4ab( a b ). Therefore x = ( a a b + b ), y = ( ab ( a b )) ad 4 = ( a + b ). Puttig suitable values of a ad b, we may get the reuired

5 solutios. ON THE DIOPHANTINE EQUATION x + y = 9 (h) For =, the surd euatio is x + y = If we take x = ( a b ), y = 8a b ad = ( a + b ) the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (i) For =, the surd euatio is x + y = If we take x = ( a b ), y = a b ad = ( a + b ) the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (j) For =, the surd euatio is x + y = If we take x = ( a b ), y18a b ad = ( a + b ) the above surd euatio is satisfied. Thus the above values of x, y ad provide the solutio of the surd euatio uder cosideratio. (k) For =, > the surd euatio is x + y = Cosiderig the trasformatios s x = α, y = β ad = γ, γ = a + b. 1 1 It ca be show that x = F ( s), y = G ( s) ad = ( a + b ) s is the i reuired solutio. By takig differet suitable values of a, b, s ad the reuired solutio ca be obtaied. 4 4 Example. Takig s =, we get F ( s) = ( a a b + b ) ad G ( s) =

6 0 SARITA, HARI KISHAN AND MEGHA RANI 4 4 4ab( a b ). Therefore ( ) x = a a b + b, y = ( ab( a b ) ad = ( a + b ). Puttig suitable values of a ad b, we may get the reuired solutios.. Cocludig Remarks Here the give surd euatio has bee discussed for =,, 4,,, ad geeral value of. Solutios are also obtaied for =,, ad, >. Ifiite solutios of these surd euatios are possible. Solutios for other values of ca also be obtaied. 4. Ackowledgemet The authors are extremely thakful to the ukow referee for his fruitful commets ad suitable suggestios for improvemet of the paper. Refereces [1] Adrew Wiles, Modular elliptic curve ad Fermat s Last Theorem, A. Math. 141() (199), [] J. Ghaouchi, A elemetary proof of Fermat-Wiles theorem ad geeraliatio to Beal cojecture, Bull. Math. Sci. & Appl. (4) (014), [] M. A. Gopala, G. Sumathi ad S. Vidhyalakshmi, O the trascedetal euatio 4 with five ukows x + y X + Y = ( r + s ), Glo. J. Math. Math. Sci. () (01), -. [4] M. A. Gopala, S. Vidhyalakshmi ad T. R. Usha Rai, Itegral solutios of the surd euatio x y + X + Y + + w = p, Arch. J. Math. () (01), -4. [] L. W. Gula, The proof of Beal,s cojecture, BOSMSS (4) (014), -1. [] L. T. D. Gregorio, Proof for the the Beal s cojecture ad a ew proof for the Fermat s last theorem, Pure & Appl. Math. J. () (01),

7 ON THE DIOPHANTINE EQUATION x + y = 1 [] V. Padichelvi, A exclusive trascedetal euatio x + y + + w = ( k + 1) R, Iter. J. Egg. Sci. Res. Tech. () (01), [8] Pierre de Fermat, Margi of a copy of Mathematica, 1. [9] R. C. Thiagraja, A proof of Beal s cojecture, Bull. Math. Sci. & Appl. () (014), 89-9.