Science and Technology RMUTT Journal

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1 Science nd Technolog RMUTT Journl Vol7 No (017) : wwwscirmuttcth/stj Online ISSN Possile Solutions of the Diophntine Eqution x Pinut Pungjump Deprtment of Mthemtics nd Sttistics, Fcult of Science, Surindr Rjht Universit, Surin 3000 E-mil: pinut@mthsutcth Astrct This pper is to identif the Diophntine eqution stisfies; cse 1:, hs no integer solution if 4m x where nd re integers is odd, nd hve integer solutions ( x,, ) is x,,,, where m is n integer, is squre numer if is even, cse : m 1, hve integer solutions ( x,, ) is,, where m is n integer, is n odd squre numer if is odd, nd,, where m is n integer, is squre numer if is even, cse 3: 4m, hve integer solutions ( x,, ),, where m is n integer, is squre numer, 4 4 Kewds: Diophntine eqution, Congruence, Integer solutions, Divisiilit is 1 Introduction Nowds, there re mn studies in the literture tht concern the Diophntine eqution In 1995, Wiles showed tht the Diophntine eqution x n n n, x 0 hs no integer solution when n 3 [1] In 1999, Bruin studied the 4 6 Diophntine eqution x nd 8 3 x [] In 004, Bennett found the solution of the Diophntine eqution n n x 5 [3] In 014, Adellim nd Dni serched the Diophntine eqution x 3 hve integer solutions ( x,, ) is Received: Revised: Accepted: M 09, 017 August 1, 017 Septemer, 017

2 Sci & Tech RMUTT J Vol7 No (017) , 1, if nd hve integer solutions 3,, 3 ( x,, ) if is odd, even [4] In 015, Adellim nd Din chrcteried the solutions of the Diophntine eqution x [5] Thus, this pper ims to stud the Diophntine eqution x where nd re integers is is x,, Mterils nd Experiment Proposition 1: The Diophntine eqution x hs no integer solution where 4m, m, x, nd re integers with is odd Suppose tht ( x,, ) is solution of the Diophntine eqution x 4m with is odd Since 1(mod 4) then 4m (mod 4), these consider into cses s follow: Cse 1: If is odd then is odd Thus, x 1(mod 4) then x 4m 3(mod 4) This is contrdiction with 1(mod 4) x Cse : If x is even then is even Thus, x 0(mod 4) then x 4m (mod 4) This is contrdiction with 0(mod 4) Therefe, cse 1 nd, the Diophntine eqution x hs no integer solution where 4m, m, x, nd is odd re integers with Proposition : The Diophntine eqution x hs the solutions in the fm x,,,, where 4m, m, x, nd re integers with is even Let e solution of the Diophntine eqution x where nd is even We hve, ( x,, ) 4m x x ( x )( x ) Since is even, so tht ( x )( x ) hence, x x, we hve re even re odd, then x nd x, x, x re even, we hve x x, re integers It follows tht, ( x )( x ) 4 4 x x x x x Hence, then x, these consider into cses s follow: x Cse 1: If so tht, there exists n integer such tht x nd let x

3 0 Sci & Tech RMUTT J Vol7 No (017) where is squre numer Since x nd x x nd tht is, x nd Cse : If x so tht, there exists n integer such tht x nd let x where is squre numer Since x nd x x nd tht is, x nd Therefe, cse 1 nd, the solutions of this eqution re in the fm x,,,, where is squre numer 3 Results nd Discussion Theem 3: The Diophntine eqution x hs the solutions in the fm x,,,, where is odd, nd re integers with is odd nd is n odd squre numer Let e solution of the Diophntine eqution x with is odd We hve, ( x,, ) x, x x ( x )( x ) Hence, ( x )( x ) then ( x ) ( x ), these consider into cses s follow: Cse 1: If ( x ) so tht, there exists n integer such tht x nd let x where is n odd squre numer Since x nd x then x nd tht is, x nd Cse : If ( x ) so tht, there exists n integer such tht x nd let x

4 Sci & Tech RMUTT J Vol7 No (017) 03 where is n odd squre numer Since x nd then x nd tht is, x nd Therefe, cse 1 nd, the solutions of this eqution re in the fm x,,,, where is n odd squre numer x Theem 4: The Diophntine eqution x hs the solutions in the fm x,,,, x, where is odd, nd re integers with is even nd is squre numer Let x,, e solution of the Diophntine eqution x with is even nd is squre numer We hve, x x ( x )( x ) Since is even, so tht ( x )( x ) hence, ( x ) ( x ), we hve x, re even xre, odd, then x nd x x x re even, we hve, re integers It follows tht, ( x )( x ) 4 4 x x x x Hence, then x x Cse 1: If x, these consider into cses s follow: so tht, there exists n integer such tht x nd let x where is squre numer Since x nd x x nd tht is, x nd x Cse : If so tht, there exists n integer such tht x nd let x where is squre numer

5 04 Sci & Tech RMUTT J Vol7 No (017) Since x x nd nd x tht is, x nd Therefe, cse 1 nd, the solutions of this eqution re in the fm x,,,, where is squre numer Theem 5: The Diophntine eqution x hs the solutions in the fm x,,,, 4 4 where nd re integers nd is squre numer Let ( x,, ) e solution of the Diophntine eqution x We hve, 4 m, m, x, x x ( x )( x ) 4 m ( x )( x ) So tht 4 ( x )( x ) hence, 4 ( x ) 4 ( x ), we hve re even re odd, then x nd x re even, we hve x x, re integers It follows tht, 4 m ( x )( x ) 4 4 x x m x, x, Hence, x x m then x m Cse 1: If x m, these consider into cses s follow: x m so tht, there exists n integer such tht x m nd let x m m where is squre numer Since x m nd x x m nd m tht is, x nd 4 4 x Cse : If m so tht, there exists n integer such tht x m nd let x m m where is squre numer Since x m nd x x m nd m tht is, x nd 4 4 Therefe, cse 1 nd, the solutions of this eqution re in the fm

6 Sci & Tech RMUTT J Vol7 No (017) 05 x,,,, 4 4 where is squre numer Exmple find ll positive integer solutions of the Diophntine eqution Consider then x 3(5) B theem 3 So tht 5 then so tht ( x, ) (11,14) so tht ( x, ) (5,10) so tht ( x, ) (37,38) This theem hve three solutions B theem [4] So tht then 11, 5 15, 1 This theem hve two solutions (Incomplete) 1, 5 5, 5 x 75 x 75 5, Conclusions It cn e seen tht this pper shown the solution of the Diophntine eqution x hs no integer solution where 4m, m is n integer, is odd nd hve integer solutions ( x,, ),, where 4m,m 1, is squre numer, m, x, nd re integers with is even, we hve seen tht the solution of the Diophntine eqution x is ( x,, ),, where m 1, is n odd squre numer, m, x, nd re integers with is odd Also, nd ( x,, ),, 4 4 where is squre numer, nd re integers 4m, m,, x 5 Acnowledgement I highl pprecite the edits nd referees f his helpful comments nd suggestions to me this stud re me vlule 6 References [1] A Wiles Modul elliptic curves nd Fermt lst theem Annls of Mthemtics 14 (1995): [] N Bruin The Diophntine eqution x nd x Compositio Mthemtic 118 (1999): n n 5 [3] MA Bennett The eqution x Journl de Thé ie des Nomres de Bdeux 18 (006): [4] S Adellim, H Dni The solution of the Diophntine eqution x 3 Interntionl Journl of Alger 8 (014): [5] S Adellim, H Din Chrcterition of the solution of the Diophntine eqution x Gulf Journl of Mthemtics 3 (015): 1-4