Acc. Sc. Torino - Memorie Sc. Fis. xxAtti (2009), Sc. Fis. pag 143 - pag(2009) TEORIA DEI NUMERI TEORIA DEI NUMERI On Diophantine triples from Pell and Pell-Lucas numbers ˇ Nota di ZVONKO CERIN eGIAN MARIO GIANELLA presentata dal Socio corrispondente MARIUS I. STOKA nell’adunanza del 13 maggio 2009 Abstract. This paper presents four examples of infinite sequences of triples of natural numbers with the property that the product of any two of them increased by a fixed integer n is a square of a number. Two sequences are built from Pell numbers for n 4 while the other two sequences are built from Pell-Lucas numbers for n 8. We show many interesting properties of these sequences and various methods how to get complete squares from them. Some geometric invariants for the associ- ated tetrahedra are also computed. Many 3 3 matrices with rows from these sequences of triples have nice determinants and permanents. Keywords: property Dn, Pell numbers, Pell-Lucas numbers, square, determinant, permanent, tetrahedron, volume. Riassunto. Questo articolo presenta quattro esempi di successioni in- finite di terne di numeri naturali con la proprieta` che il prodotto di ogni due di essi aumentato di un numero naturale fisso n e` il quadrato di un numero. Costruiamo due successioni con i numeri di Pell con n 4 ed altre due successioni con i numeri di Pell-Lucas con n 8. Dimostriamo interessanti proprieta` di queste successioni e vari metodi per ottenere quadrati completi da esse. Inoltre sono calcolati degli invarianti geo- metrici per il tetraedro associato. Molte matrici 3 3 con righe ottenute da queste successioni di terne hanno interessanti determinanti e perma- nenti. Parole chiave: proprieta` Dn, numeri di Pell, numeri di Pell-Lucas, quadrato, determinante, permanente, tetraedro, volume. 1. Introduction Recall that for an integer n and a natural number m, the m-tuple a a1 a2 am of natural numbers has the property Dn provided each sum ai a j n is Mathematics Subject Classification 2000: Primary 11B37, 11B39, 11D09. Kopernikova 7, 10010 Zagreb, Hrvatska. E-mail: Dipartimento di Matematica, Universita` di Torino, via Carlo Alberto 10, 10123 Torino, Italia. E-mail: 2 ZVONKO CERINˇ E GIAN MARIO GIANELLA 84 Zvonko Cˇerin e Gian Mario Gianella a square for all indices i j 1 2m with i j. We can also say that a is the Dn-m-tuple. For example, 1 3 8120 is the famous D1-quadruple discovered by Fermat and 2410 is the D 4-triple while 2414 is the D8-triple. The subject of Dn-m-tuples is rather extensive as so far many results have been discovered about them. Here we mention only that it is unknown if there is any Dn-quadruple when n ', where ' is the set 4 3 13581220 (see [4]). Also, there are many examples of infinite sequences of Dn- quadruples that are build from members of various integer sequences like Ho- radam, Fibonacci, Lucas, Pell and Pell-Lucas (see [4], [5]). These examples are mostly for the values of n of the form 2, where is a natural number. In this paper for the values n 4 andn 8 from the set ' we shall con- d d struct pairs of infinite sequences A Ak k0 and B Bk k0 of D 4- d d triples and G Gk k0 and D Dk k0 of D8-triples. The sequences Aand B are built from Pell numbers and the sequences G and D from Pell-Lucas numbers. The Pell and Pell-Lucas sequences of natural numbers Pn and Qn are defined by the recurrence relations P0 0 P1 2 Pn 2Pn 1 Pn 2 for n 2, and Q0 2 Q1 2 Qn 2Qn 1 Qn 2 for n 2. The numbers Qk make the integer sequence A002203 from [6] while the 1 numbers 2 Pk make A000129. The authors studied some sums of these numbers in the recent articles [1], [2] and [3]. The aim of this article is to explore some properties of the sequences A, B, G and D. Since the triples of natural numbers can be used as rows of 3 3 matrices or as coordinates of points in the 3-dimensional Euclidean space we examine what happens when these are from our sequences. In some cases it is possible to compute closed forms of their determinants, permanents, volumes, vector cross-products and to find other interesting relationships. 2. The sequences A and B Let A B : 3 be the functions defined by AnP2n1 2P2n1 P2n3 and BnP2n1 P2n3 2P2n3 Theorem 1. For every natural number n the triples An and Bn have the property D 4. ON DIOPHANTINE TRIPLES 3 On Diophantine triples from Pell and Pell-Lucas numbers 85 Proof. This follows from the following identities: 2 2P2n 1 P2n 3 4 4P2n 3Q2n 2 P2n 1 P2n 3 4 P 2n2 2 2 2P 4 2P2n Q2n 2n1 2 2 2P 4 10P2n 7Q2n 2n3 3 Let S1 S2 S3 : be the basic symmetric functions defined for xa b c by S1xa b c S2xbc ca ab S3xabc The linear expressions involving symmetric functions S1 and S2 of the triples An and Bn are the source of many complete squares as the following theorem shows. Theorem 2. For all integers a and b the following identities hold: 2 2 2 2 abS1An b S2An 4b a 5bP2n 4bQ2n a 2 2 2 2 abS1Bn b S2Bn 4b a 11bP2n 8bQ2n a 1 J j Y j Proof. Let 1 2 and 1 2 . Since P and J Y J j 2 j j 1 n Q j J Y , after the substitutions Y J and A J , the left hand side 2 2 2 abS1An b S2An 4b a becomes 2 57 40 2 7bA4 57b 40b 2 8A2a 5A2a 2 98A4 2 However, this is precisely the right hand side 5bP2n 4bQ2n a . In particular, when b 1 and a 0 and b 1 and a 2 we obtain the fol- lowing statement. Corollary 1. For every natural number n the triples An and Bn have the property that the numbers S2An 4, S2Bn 4 and 2S1An S2An 2S1Bn S2Bn are squares. 4 ZVONKO CERINˇ E GIAN MARIO GIANELLA 86 Zvonko Cˇerin e Gian Mario Gianella Q For natural numbers a, b and c and any sequence Qn of numbers, let Mabc be the 3 3 matrix with rows Qa, Qb and Qc. Since the columns of the A B matrices Mabc and Mabc are linearly dependent it is clear that the determinants A B detMabc and detMabc are equal to zero. In our next theorem we shall compute the permanents of these matrices. a b c 4 Let A 1 2 , B 1 2 and C 1 2 . Let us define 33 4 4 4 4 4 4 4 4 2 A B C and the symmetric expressions 32, 31 (from A , B and C ) and 33 (from A2, B2 and C2) analogously. Theorem 3. For all natural numbers a, b and c the following quotients express A B the permanents perMabc and perMabc: 4 4 4 4 4 4 212333 11332 31 28733 11132 31 2 33 4 4 4 4 4 4 271733 14332 31 250733 13132 31 2 33 a b c 1 Proof. We take A J , B J , C J and Y J . With these substitutions A and a little help from Maple V we conclude that the permanent perMabc has the above form. In particular, when we take three consecutive triples (i. e., a n, b n 1 A and c n 2 for a natural number n), then the permanents perMabc and B perMabc are 34446Q6n 48714P6n 4130Q2n 5810P2n 200766Q6n 283926P6n 12530Q2n 17710P2n Let us introduce three binary operations , and on the set 3 of triples of natural numbers by the rules a b c u v wau bv cw a b c u v wav bw cu and a b c u v waw bu cv The next result shows that the sequences An and Bn are miraculously con- nected with respect to the products , and . For a natural number m, let U 1 2m and V 1 2m. ON DIOPHANTINE TRIPLES 5 On Diophantine triples from Pell and Pell-Lucas numbers 87 Theorem 4. For all natural numbers n, m and u and each product the determinant of the matrix with rows An Bn, An m B n m and An um Bn um is independent of the value n. These determi- nants are equal to M, 0 (zero) and M, where M 64 2 U 4u 1 V 4 V 4u U 4 1 V 4 U 4 Proof. This could be proved by the same method that was used in the proof of Theorem 2. Another nice surprise is that the following analogue of Theorem 2 for the sequence An Bn is true. Its proof is almost the same. Theorem 5. For all integers a and b the following identity holds: 2 2 abS1An Bn b S2An Bn a 51 2 36bP n bQ n 9b a 4 2 4 In particular, when b 1 and a 0 and b 1 and a 1 we obtain the following statement. Corollary 2. For every natural number n the triple An Bn has the property that the numbers S2An Bn, S2An Bn S1An Bn 1 and S2An Bn S1An Bn 1 are squares. 3. The sequences G and D Let G D : 3 be the functions defined by GnQ2n1 2Q2n1 Q2n3 and DnQ2n1 Q2n3 2Q2n3 Theorem 6. For every natural number n the triples Gn and Dn have the property D8.