THREE CURIOUS DIOPHANTINE PROBLEMS

Transcription

1 JP Joural of Mathematical Scieces Volume 8, Issues &, 0, Pages -0 0 Ishaa Publishig House This paper is available olie at THREE CRIOS DIOPHATIE PROBLEMS A. VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA Departmet of Mathematics atioal College, Trichy-000 Tamiladu, Idia Departmet of Mathematics SIGC, Trichy-000 Tamiladu, Idia Abstract This paper aims at determiig two o-zero distict itegers ad such that (i) p, q, (ii) p, q, (iii) p, q, respectively, i Sectios A, B ad C. Itroductio umber theory, called the Quee of Mathematics, is a broad ad diverse part of mathematics that developed from the study of the itegers. The foudatios for umber theory as a disciplie were laid by the Greek mathematicia Pythagoras ad his disciples (kow as Pythagoreas). Oe of the oldest braches of mathematics 00 Mathematics Subject Classificatio: D0, D5, D99. Keywords ad phrases: Diophatie problem, iteger triples, system of equatios. Received ovember 9, 0

2 A.VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA itself, is the Diophatie equatios sice its origis ca be foud i texts of the aciet Babyloias, Chiese, Egyptias, Greeks ad so o [7-8]. Diophatie problems were first itroduced by Diophatus of Alexadria who studied this topic i the third cetury AD ad he was oe of the first mathematicias to itroduce symbolism to Algebra. The theory of Diophatie equatios is a treasure house i which the search for may hidde relatio ad properties amog umbers form a treasure hut. I fact, Diophatie problems domiated most of the celebrated usolved mathematical problems. Certai Diophatie problems come from physical problems or from immediate mathematical geeralizatios ad others come from geometry i a variety of ways. Certai Diophatie problems are either trivial or difficult to aalyze [-]. Also oe may [9-5]. I this commuicatio, we attempt for obtaiig three o-zero distict itegers ad such that (i) p, q, (ii) p, q, (iii) p, q. Sectio A Method of Aalysis Let ad be ay two o-zero distict itegers such that, we write where a ad b are o-zero distict itegers. ow, 0 a b, 0 b a, p ( 0 ) ( a b) p. ()

3 THREE CRIOS DIOPHATIE PROBLEMS Take p ( 0 ) P. () From () ad (), we have a b ( 0 ) P. () Also, q ( 0 ) ( a b) q 999L 9( a b) q ( factors) 9 ( a b) q, () where L ( factors ). Assume q Q. (5) From () ad (5), we obtai Solvig () ad (), we get Q a b. () a b [( 0 ) P Q ], [( 0 ) P Q ]. For a ad b to be itegers, P ad Q should be of the same parity. Case. Let P, Q. The a ( 0 ), b ( 0 ).

4 A.VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA Thus, the required values of ad are give by ( ), 9 0 ( ). 9 0 Case. Let, P. Q The ( ) ( ) 5 0 a ( ), ( ) ( ) 5 0 b ( ). Hece, the required values of ad are give by ( ) ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( ) ( ) ( )( ) ( ) ( ) ote. I additio to the above two represetatios of ad, we have aother two represetatios of ad are exhibited below: Represetatio. ( ) ( ), 0 0 s r ( ) ( ). 0 0 s r

5 Represetatio. Sectio B THREE CRIOS DIOPHATIE PROBLEMS 5 ( 0 ) ( r r ) ( 0 ) ( s s s ) ( 0 ) 5 ( 0 ) 5 0, ( 0 ) ( r r ) ( 0 ) ( s s s ) ( 0 ) 5 ( 0 ) 5. Let ad be ay two o-zero distict itegers to be determied such that Solvig (7) ad (8), we get p, (7) q. (8) p q, q p. (9) As our iterest is o fidig iteger values for ad, we have to choose p ad q i (9) suitably.. I what follows, we preset the suitable values of p ad q alog with ad Choice. Let p q. From (9), ad are give below: ( ), ( ) ( ).

6 A.VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA umerical examples 0 0 ( ) Choice. Let p q. By substitutig p ad q i (9), ad are obtaied as umerical examples. 9 9, Choice. Let p, q. By applyig p ad q i (9), ad are foud to be: , 9.

7 umerical examples. Sectio C THREE CRIOS DIOPHATIE PROBLEMS 7 0 ( ) Let ad be ay two o-zero distict itegers such that p, (0) q Elimiatig betwee (0) ad (), we get. () p q Treatig this as a quadratic i ad solvig for, we have p p q (takig positive sig). () 0. Let p q which is writte as the system of two equatios as q p, p q. O solvig the above two equatios, we obtai q s, 8s s, () p 8s s. ()

8 8 A.VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA Treatig () as a quadratic i s ad solvig for s, we have s f ad p [ g ], where f g ( 7 ) ( 7 ), ( 7 ) ( 7 ). Also, q s f 8. Thus, i view of () ad (0) we have f, f which represet the required values of ad satisfyig (0) ad (). umerical examples

9 THREE CRIOS DIOPHATIE PROBLEMS 9 Coclusio that I this paper, we have preseted two ozero distict itegers ad such (i) p, q, (ii) p, q, (iii) p, q. As Diophatie problems are rich i variety, oe may attempt to fid other choices of Diophatie problems. Refereces [] Adre Weil, umber Theory: A Approach through History, From Hammurapito to Legedre, Birkahsuser, Bosto, 987. [] Bibhotibhusa Batta ad Avadhesh arayaa Sigh, History of Hidu Mathematics, Asia Publishig House, 98. [] C. B. Boyer, A History of Mathematics, Joh Wiley & Sos, Ic., ew York, 98. [] L. E. Dickso, History of Theory of umbers, Vol., Chelsea Publishig Compay, ew York, 95. [5] Harold Daveport, The Higher Arithmetic: A Itroductio to the Theory of umbers, 7th ed., Cambridge iversity Press, 999. [] Joh Stilwell, Mathematics ad its History, Spriger-Verlag, ew York, 00. [7] M. D. James Matteso, A Collectio of Diophatie Problems with Solutios, Washigto, Artemas Marti, 888. [8] Titu Adreescu ad Dori Adrica, A Itroductio to Diophatie Equatios, GIL Publicatio House, 00. [9] M. A. Gopala, S. Vidhyalakshmi ad. Thiruiraiselvi, O two iterestig Diophatie problems, Impact J. Sci. Tech. 9() (05), [0] M. A. Gopala, S. Vidhyalakshmi ad. T. Hiruiraiselvi, O iterestig iteger pairs, IJMRME () (05), 0-. [] M. A. Gopala, S. Vidhyalakshmi ad J. Shathi, O iterestig Diophatie problem, IJMRME () (05), 8-70.

10 0 A.VIJAYASAKAR, M. A. GOPALA ad V. KRITHIKA [] K. Meea, S. Vidhyalakshmi, J. Shathi ad K. Agalya, A iterestig Diophatie problem, IJREAS 5() (05), [] M. A. Gopala, S. Vidhyalakshmi ad J. Shathi, A iterestig Diophatie problem, Scholar Bull. (7) (05), -8. [] S. Vidhyalakshmi, E. Premalatha, R. Presea ad V. Krithika, Costructio of a special iteger triplet-i, IJPMS (0), -9. [5] A. Vijayasakar, M. A. Gopala ad V. Krithika, O a Diophatie problem, IJMRD (9) (0), 7-9.