Mskolc Mathematcal Notes HU e-issn 787-43 Vol. 8 (7), No., pp. 779 786 DOI:.854/MMN.7.536 ON A DIOPHANTINE EUATION ON TRIANGULAR NUMBERS ABDELKADER HAMTAT AND DJILALI BEHLOUL Receved 6 February, 5 Abstract. A number N s a trangular number f t can be wrtten as N D C C C n for some natural number n. We study the problem of fndng all nonnegatve nteger solutons of the Dophantne equaton. C C C X/ C. C C C Y / D. C C C Z/. Usng ths equaton, some new and curous ncreasng nteger sequences are bult. Mathematcs Subject Classfcaton: D5; D4 Keywords: Dophantne equaton, trangular numbers. INTRODUCTION A trangular number s a number of the form T n D n P kd k D n.n C /=, where n s a natural number. So the frst few trangular numbers are ; ; 3; 6; ; 5; ; 8;... (sequence A7 n [6]). A well known fact about the trangular numbers s that X s a trangular number f and only f 8X C s a perfect square. Trangular numbers can be thought of as the numbers of dots needed to make a trangle. We are concerned by the Dophantne equaton of the form T X C T Y D T Z (.) Papers [ 5] gave many nterestng results concernng the problem of solvablty of Dophantne equatons related to trangular numbers. The am of ths paper s twofold: on the one hand, t gves all solutons of the Dophantne equaton (.), and on the other hand, t gves us a method of fndng explctly (and quckly) nfnte famles of solutons of (.) n where many new nteger sequences are derved.. GENERAL SOLUTION Let us start wth some techncal lemmas. Lemma. Let a;b;c;d N. Then ab D cd f and only f there exst p;q;m;n N such that a D mn; b D pq; c D mp and d D nq. c 7 Mskolc Unversty Press
78 ABDELKADER HAMTAT AND DJILALI BEHLOUL Proof. Let a D AN p and b D N and B are fnte sets. The fact cd D A A p B B qˇ and d D ˇ ;ˇ ˇ wth D C qˇ ; n D B B A A pq; c D mp and d D nq: A A qˇ BN p where p ;q are prmes, ;ˇ mples that c D AN B B p BN qˇ and ˇ D ˇ C ˇ : If we put m D p and q D B B qˇ qˇ, where ; ; p A A ; p D then we obtan, a D mn; b D Lemma. Equaton (.) s equvalent to X.X C / D.Z Y /.Z C Y C /. Proof. Replacng T X by equaton (.) becomes X.X C /.e., Thus X.X C /, T Y by C Y.Y C / D Y.Y C / Z.Z C / ; and T Z by Z.Z C / X C X C Y C Y D Z C Z: (.) X.X C / D.Z Y /.Z C Y C /: Now, we are able to establsh the followng theorem. Theorem 8. All nonnegatve nteger solutons of the equaton T X CT Y D T Z are < X D mn gven by Y D :.nq mp / W m;n;p;q N Z D.nq C mp / where pq mn D and nq mp N: Proof. Accordng to Lemma, (.) s equvalent to X.X C / D.Z Y C /, thus thanks to Lemma, one has Therefore we get X D mn X C D pq Z Y D mp Z C Y C D nq where m;n;p;q N: X D mn Y D.nq mp / Y /.Z C
ON A DIOPHANTINE EUATION ON TRIANGULAR NUMBERS 78 Z D.nq C mp / where m;n;p;q N such that pq mn D and nq mp N: Example. p D 4; q D 7; m D 3 and n D 9. Clearly, X D 7; Y D 5 and Z D 37: 3. SOLUTIONS FAMILIES Frst of all, note that t s not easy to fnd ntegers m;n;p;q such that pq mn D and nq mp N: The usage of matrces enables us to obtan a large number of solutons of (.). In ths secton, we wll construct several nfnte many famles solutons of our trangular equaton. By completng squares n (.), we obtan Multplyng both sdes by 4, we get.x C / C.Y C / D.Z C / C 4 :.X C / C.Y C / D.Z C / C : By puttng, x D X C ; y D Y C and D Z C, we obtan x C y D C : (3.) Snce the proof of the followng proposton can be seen easly we omt t. Proposton. The two famles A are solutons of (3.) for all nteger a. @ a a @ a a Theorem. There exst some 33 matrces M; wth elements n f;;3g; such that f @ x y A s a soluton of (3.) then M@ x y A s also a soluton. Proof. Let M D It s clear that @ A B C E F G L M N @ A B C E F G L M N A Now, f the followng system holds A and let @ x y A D @ x y A a soluton of (3.). @ Ax C By C C Fy C G C Ex Lx C My C N A: A L C E D (3.) B C F M D C C G N D
78 ABDELKADER HAMTAT AND DJILALI BEHLOUL then AB LM C EF D AC LN C EG D BC MN C F G D.Ax C By C C / C.Fy C G C Ex /.Lx C My C N / D : It follows from ths, @ Ax C By C C Fy C G C Ex A s a soluton of (3.). Lx C My C N We wll explore, n fve steps, all solutons of the system.s/ are n the space f;;3g 9. Case : A D L D E D : B C F M D (3.3) C C G N D B C F D M (3.4) C C G D N BC MN C F G D : From (3.4), we obtan F D or B D ; then (3.3) mples B D M or F D M then F D or B D whch s mpossble. Case : A D ; L D E D : B C F M D C C G N D B M C F D (3.5) C N C G D (3.6) BC MN C F G D : Equatons (3.5) and (3.6) mply that B and C are even. Then from M F D 3 (3.7) N G D 5 (3.8) M F D N G D MN F G D 4: Equaton (3.7) gves M D ;F D and (3.8) gves N D 3;G D ;
then M D ON A DIOPHANTINE EUATION ON TRIANGULAR NUMBERS 783 @ A B C E F G L M N Case 3 : A D ;L D E D 3: A D @ 3 A: B C F M D C C G N D B 3M C 3F D (3.9) C 3N C 3G D (3.) BC MN C F G D : Equatons (3.9) and (3.) gve us B D C D 3; then F M D 8 G N D (3.) M C F D N C G D MN C F G D 9: It s easy to see that equaton (3.) s mpossble n f;;3g Case 4 : A D : Equaton 3. mples that L D ; E D : So, we get B C F M D C C G N D B M C F D (3.) C N C G D (3.3) BC MN C F G D : Equatons (3.) and (3.3) gve us F D G D ; thus M B D 3 (3.4) N C D 5 (3.5) M B D N C D BC MN C 4 D : Equaton (3.4) gves us M D ;B D and (3.5) gves N D 3;C D :
784 ABDELKADER HAMTAT AND DJILALI BEHLOUL We obtan M D @ A B C E F G L M N A D @ 3 A: Case 5 : A D 3: Equaton ( ) n.s/ mples that : L D 3;E D : Now B C F M D C C G N D 3B 3M C F D (3.6) 3C 3N C G D (3.7) BC MN C F G D : Equatons (3.6) and (3.7) gve us F D G D 3; thus M B D 8 N C D (3.8) B M C D C N C D BC MN C 9 D : It s easy to see that equaton (3.8) s mpossble n f;;3g : Corollary. The equaton (.) admts nfnte many solutons. Proof. Let M D M and @ x y A D @ A a soluton of (3.) then M n @ A s a soluton of (3.) for all nteger n : 4. NEW INTEGER SEUENCES Usng the on-lne Encyclopeda of nteger sequences (see [6]), we can easly check that the followng two nteger sequences formed by solutons X of trangular equaton (.) : ;5;35;3;79;699;439;3545;375;79973::: and ;8;54;3; 884; 988; 645; 37338; 75864; 6887;::: are new. 4.. Constructons : We start wth @ 3 a soluton of (3.), thanks to theorem, by multplyng recursvely by M, we 3 obtan
@ 3 7 @ 4783 47 833 665 857 ON A DIOPHANTINE EUATION ON TRIANGULAR NUMBERS 785 @ 7 69 99 @ 47 49 577 @ 744 744 9 388 899 @ 379 377 3363 @ 599447 599449 69537 @ 3859 386 96 @ 8783 878 4 43 ::: are also solutons of (3.). We get, x D 3;;7;47;379;3859;8783;4783;744;599447;::: But x D X C ; so X D ;5;35;3;79;699;439;3545;375;79973;::: s a new sequence of odd ntegers formed by solutons X of our trangular equaton (.). : We start wth @ 5 A a soluton of (3.), thanks to theorem, by multplyng recursvely by M, we obtan 5 @ 7 @ 9 3 @ 645 649 @ 3769 3773 @ 977 98 @ 8 8 5 7 57 95 5333 383 8 65 @ 746637 746 64 55 97 @ 43579 435 733 654 77 @ 5363745 5363749 35869755 ::: are also solutons of (3.). We get, x D ;7;9;645;3769;977;8;746637;43579;5363745;::: But x D X C ; then X D ;8;54;3;884;988;645;37338;75864;6887;::: s a new sequence of even ntegers formed by solutons X of our trangular equaton (. ). REFERENCES [] M. A. Bennett, A queston of Serpnsk on trangular numbers. Integers, vol. 5, no., pp. A5,, 5. [] J. A. Ewell, On sums of three trangular numbers. Fbonacc uart., vol. 6, no. 4, pp. 33 335, 988. [3] I. Joshua and J. Lenny, Arthmetc progressons nvolvng trangular numbers and squares. J. Comb. Number Theory, vol. 5, no. 3, pp. 65 79, 3. [4] R. Keskn and O. Karaatl, Some new propertes of balancng numbers and square trangular numbers. J. Integer Seq., vol. 5, no., pp. Artcle..4, 3,. [5] W. Serpnsk, Theorem on trangular numbers. Elem. Math., vol. 3, pp. 3 3, 968. [6] N. Sloane, The On-Lne Encyclopeda of Integer Sequences., https://oes.org/.
786 ABDELKADER HAMTAT AND DJILALI BEHLOUL Authors addresses Abdelkader Hamtat A.T.N. Laboratory, Faculty of Mathematcs, USTHB BP 3, El Ala, 6, Bab Ezzouar, Algers ALGERIA E-mal address: ahamttat@gmal.com Djlal Behloul A.T.N. Laboratory, Faculty of Mathematcs, USTHB BP 3, El Ala, 6, Bab Ezzouar, Algers ALGERIA E-mal address: dbehloul@yahoo.fr