On Diophantine triples from Pell and Pell-Lucas numbers

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1 Acc. Sc. Torino - Memorie Sc. Fis. xx Atti (2009), Sc. Fis. pag pag (2009) TEORIA DEI NUMERI TEORIA DEI NUMERI On Diophantine triples from Pell and Pell-Lucas numbers Nota di ZVONKO ČERIN e GIAN 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 associated 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 infinite di terne di numeri naturali con la proprietà che il prodotto di ogni due di essi aumentato di un numero naturale fisso n è 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 proprietà di queste successioni e vari metodi per ottenere quadrati completi da esse. Inoltre sono calcolati degli invarianti geometrici per il tetraedro associato. Molte matrici 3 3 con righe ottenute da queste successioni di terne hanno interessanti determinanti e permanenti. Parole chiave: proprietà 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 a 1 a 2 a m of natural numbers has the property Dn provided each sum a i a j n is Mathematics Subject Classification 2000: Primary 11B37, 11B39, 11D09. Kopernikova 7, Zagreb, Hrvatska. Dipartimento di Matematica, Università di Torino, via Carlo Alberto 10, Torino, Italia.

2 2 ZVONKO ČERIN E GIAN MARIO GIANELLA 84 Zvonko Čerin e Gian Mario Gianella a square for all indices i j 1 2 m with i j. We can also say that a is the Dn-m-tuple. For example, is the famous D1-quadruple discovered by Fermat and 2410 is the D4-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 (see [4]). Also, there are many examples of infinite sequences of Dnquadruples that are build from members of various integer sequences like Horadam, 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 and n 8 from the set we shall construct pairs of infinite sequences k k0 and k k0 of D4triples and k k0 and k k0 of D8-triples. The sequences and are built from Pell numbers and the sequences and from Pell-Lucas numbers. The Pell and Pell-Lucas sequences of natural numbers P n and Q n are defined by the recurrence relations and P 0 0 P 1 2 P n 2P n1 P n2 for n 2, Q 0 2 Q 1 2 Q n 2Q n1 Q n2 for n 2. The numbers Q k make the integer sequence A from [6] while the numbers 1 2 P k make A 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,, and. 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 and Let : 3 be the functions defined by n P 2n1 2P 2n1 P 2n3 and n P 2n1 P 2n3 2P 2n3 Theorem 1. For every natural number n the triples n and n have the property D4.

3 ON DIOPHANTINE TRIPLES 3 On Diophantine triples from Pell and Pell-Lucas numbers 85 Proof. This follows from the following identities: 2P 2n1 P 2n3 4 4P 2n 3Q 2n 2 P 2n1 P 2n3 4 P2n2 2 2P2n P 2n Q 2n 2 2P2n P 2n 7Q 2n 2 Let : 3 be the basic symmetric functions defined for xa b c by 1 x a b c 2 x bc ca ab 3 x abc The linear expressions involving symmetric functions 1 and 2 of the triples n and n are the source of many complete squares as the following theorem shows. Theorem 2. For all integers a and b the following identities hold: ab 1 n b 2 2 n 4b 2 a 2 5bP 2n 4bQ 2n a 2 ab 1 n b 2 2 n 4b 2 a 2 11bP 2n 8bQ 2n a 2 Proof. Let 1 2 and Since P j j j 2 and Q j j j, after the substitutions 1 and A n, the left hand side ab 1 n b 2 2 n 4b 2 a 2 becomes bA 4 57b 40b 2 8A 2 a 5A 2 a A 4 However, this is precisely the right hand side 5bP 2n 4bQ 2n a 2. In particular, when b 1 and a 0 and b 1 and a 2 we obtain the following statement. Corollary 1. For every natural number n the triples n and n have the property that the numbers 2 n 4, 2 n 4 and 2 1 n 2 n 2 1 n 2 n are squares.

4 4 ZVONKO ČERIN E GIAN MARIO GIANELLA 86 Zvonko Čerin e Gian Mario Gianella For natural numbers a, b and c and any sequence n of numbers, let Mabc be the 3 3 matrix with rows a, b and c. Since the columns of the matrices Mabc and M abc are linearly dependent it is clear that the determinants detmabc and detm abc are equal to zero. In our next theorem we shall compute the permanents of these matrices. Let A 1 2 a, B 1 2 b and C 1 2 c. Let us define 4 3 A 4 B 4 C 4 and the symmetric expressions 4 2, 4 1 (from A4, B 4 and C 4 ) and 2 3 (from A 2, B 2 and C 2 ) analogously. Theorem 3. For all natural numbers a, b and c the following quotients express the permanents permabc and perm abc : Proof. We take A a, B b, C c and 1. With these substitutions 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 and c n 2 for a natural number n), then the permanents permabc and perm abc are 34446Q6n 48714P6n 4130Q2n 5810P2n Q 6n P 6n 12530Q 2n 17710P 2n Let us introduce three binary operations, and on the set 3 of triples of natural numbers by the rules and a b c u v w au bv cw a b c u v w av bw cu a b c u v w aw bu cv The next result shows that the sequences n and n are miraculously connected with respect to the products, and. For a natural number m, let U 1 2 m and V 1 2 m.

5 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 n n, n m n m and n um n um is independent of the value n. These determinants 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 n n is true. Its proof is almost the same. Theorem 5. For all integers a and b the following identity holds: ab 1 n n b 2 2 n n a 2 36bP 4n bq 4n 9b a 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 n n has the property that the numbers 2 n n, 2 n n 1 n n 1 and 2 n n 1 n n 1 are squares. 3. The sequences and Let : 3 be the functions defined by n Q 2n1 2Q 2n1 Q 2n3 and n Q 2n1 Q 2n3 2Q 2n3 Theorem 6. For every natural number n the triples n and n have the property D8.

6 6 ZVONKO ČERIN E GIAN MARIO GIANELLA 88 Zvonko Čerin e Gian Mario Gianella Proof. This follows from the following identities: 2Q 2n1 Q 2n3 8 6P 2n 4Q 2n 2 Q 2n1 Q 2n3 8 4P 2n 3Q 2n 2 2Q 2 2n1 8 2P 2n 2Q 2n 2 2Q 2 2n3 8 14P 2n 10Q 2n 2 The linear expressions involving symmetric functions 1 and 2 of the triples n and n are also the source of many complete squares as the following theorem shows. Its proof is analogous to the proof of Theorem 2. Theorem 7. For all integers a and b the following identities hold: ab 1 n b 2 2 n 8b 2 a 2 8bP 2n 5bQ 2n a 2 ab 1 n b 2 2 n 8b 2 a 2 16bP 2n 11bQ 2n a 2 In particular, when b 1 and a 0 and b 1 and a 1 we obtain the following statement. Corollary 3. For every natural number n the triples n and n have the property that the numbers 2 n 8, 2 n 8 and are squares. 1 n 2 n 9 1 n 2 n 9 Since the columns of the matrices M abc and M abc are also linearly dependent it is clear that the determinants detm abc and detm abc are equal to zero. In our next theorem we shall compute the permanents of these matrices. The notation was introduced earlier and the method of proof already described. Theorem 8. For all natural numbers a, b and c the following quotients express the permanents perm abc and perm abc :

7 ON DIOPHANTINE TRIPLES 7 On Diophantine triples from Pell and Pell-Lucas numbers 89 In particular, when we take three consecutive triples (i. e., a n, b n 1 and c n 2 for a natural number n), then the permanents perm abc and permabc are 97428Q 6n P 6n 11620Q 2n 16520P 2n Q 6n P 6n 35420Q 2n 50120P 2n The next result shows that the sequences n and n are also connected with respect to the products, and. Recall that for a natural number m, we put U 1 2 m and V 1 2 m. Theorem 9. For all natural numbers n, m and u and each product the determinant of the matrix with rows n n, n m n m and n um n um is independent of the value n. These determinants are equal to 8M, 0 (zero) and 8M, where M 64 2 U 4u 1 V 4 V 4u U 4 1 V 4 U 4 We can also consider analogues of Theorems 4 and 9 for the pairs,, and and the products, and. For example, the determinant of the matrix in the first and the fourth combination for the product is zero while in the second and the third the value of the determinant is 3 times the value for the pair in Theorem 4. Here is an analogue of Theorem 5 for the sequence n n. Theorem 10. For all integers a and b the following identity holds: ab 1 n n b 2 2 n n a 2 36bP 4n 51 2 bq 4n 9b a 2 2 In particular, when b 1 and a 0 and b 1 and a 1 we obtain the following statement. Corollary 4. For every natural number n the triple n n has the property that the numbers 2 n n, 2 n n 1 n n 1 and are squares. 2 n n 1 n n 1

8 8 ZVONKO ČERIN E GIAN MARIO GIANELLA 90 Zvonko Čerin e Gian Mario Gianella and We can get many complete squares from the sequences n n n n n n n n n n Theorem 11. For all integers a and b the following identity holds: 8ab 1 n 6b 2 2 n 9a 2 8ab 1 n 6b 2 2 n 9a bP 8n 3462bQ 8n 204b 3a 2 In particular, when b 1 and a 0 and b 1 and a 1 we obtain the following statement. Corollary 5. For every natural number n the triples n and n have the property that the numbers 6 2 n, 6 2 n, are squares. 6 2 n 8 1 n 9 and 6 2 n 8 1 n 9 4. The sequences,, and The sequence determines the sequence where n is the triple 4P 2n 3Q 2n P 2n2 2P 2n Q 2n given in the proof of Theorem 1. In other words, the coordinates of n satisfy the following relations: n 1 n 2 n 3 4 n 2 n 3 n 1 4 n 3 n 1 n 2 4 We can similarly introduce the sequences, and. There are many expressions built from these sequences that are also squares. Here we give only few simple examples. 1 n n 8 5P 2n 4Q 2n n n 3 2 n 3P 2n 3Q 2n 2

9 ON DIOPHANTINE On Diophantine TRIPLES triples from Pell and Pell-Lucas numbers n n 12 16P 2n 11Q 2n n n 11 2 n 72 46P 2n 33Q 2n 2 1 n n 16 8P 2n 5Q 2n n n 2 n 32 14P 2n 11Q 2n n n 24 22P 2n 16Q 2n n n 15 2 n 80 70P 2n 50Q 2n 2 Of course, we can use also the other two products and and also the sequences,, and. Hence, from all these sequences and the three products using linear expressions of the kind exemplified above we can get many squares. This seems quite natural if we remember that these sequences have the properties D4 and D8. 5. The geometry of the eight sequences In this section the members of our eight sequences being triples of (real) numbers are considered as points in the 3-dimensional Euclidean space. Theorem 12. For all natural numbers a, b, c and d, the a b c d a b c d a b c d a b c d are quadruples of coplanar points in the 3-dimensional Euclidean space. Proof. Recall that the points T i x i y i z i (i ) are coplanar if and only if the oriented volume x 1 1 y 1 z 1 1 x 2 y 2 z x 3 y 3 z 3 1 x 4 y 4 z 4 1 of the tetrahedron T 1 T 2 T 3 T 4 is zero. When we substitute the above quadruples of points into this determinant and ask Maple V for help we get zero.

10 10 ZVONKO ČERIN E GIAN MARIO GIANELLA 92 Zvonko Čerin e Gian Mario Gianella For points A, B, C and D in the 3-dimensional Euclidean space E 3, let ABCD denote the oriented volume of the tetrahedron ABCD. Let T denote the tetrahedron nn 1nn 1. We similarly define tetrahedra T, T and T and T, T, T and T. Theorem 13. For every natural number n, the following is true: 2T T 64 3 Q 2n 32P 2n 2T T Q 2n P 2n 2T T 112Q 2n 160P 2n 2T T 64Q 2n P 2n The centroids of the tetrahedra T, T, T and T are coplanar. The centroids of the tetrahedra T, T, T and T are also coplanar. Proof. The proof is once again a direct computation of determinants with standard substitutions for our sequences. The same applies to the proof of the remaining claims in this paper that we leave to the dedicated reader. The following result shows that the tetrahedra n and n whose vertices are the triples n, n, n and n and n, n, n and n have nice oriented volumes. Theorem 14. For every natural number n, the tetrahedron n has the oriented volume 8 3 P 2n while the tetrahedron n has the oriented volume 4 3 P 2n. Theorem 15. For every natural number n, the tetrahedron 1 n with vertices n, n 1, n and n 1 has the oriented volume 8Q 2n 16P 2n while the tetrahedron 2 n with vertices n 1, n, n 1 and n has the oriented volume 8 3 Q 2n 16 3 P 2n. For every natural number n, the tetrahedron 1 n with vertices n, n 1, n and n 1 has the oriented volume 4Q2n 8P 2n while the tetrahedron 2 n with vertices n 1, n, n 1 and n has the oriented volume 4 3 Q 2n 8 3 P 2n. Theorem 16. The tetrahedra n, n, 1 n, 1 n, 2 n, and 2 n have the centroids in the points n Q 2n 3 2 P 2n 4Q 2n 23 4 P 2n 9Q 2n 51 4 P 2n

11 6Q 2n 15 ON DIOPHANTINE On Diophantine TRIPLES triples from Pell and Pell-Lucas numbers n 2 P 2n 3Q 2n 17 4 P 2n 2Q 2n 11 4 P 2n 1 n 7Q 2n 5P 2n 37 2 Q 2n P 2n 38Q 2n P 2n 53 1 n 2 Q 2n 75 2 P 2n 23 2 Q 2n 65 4 P 2n 8Q 2n 45 4 P 2n 2 n 7Q 2n 5P 2n 9Q 2n 51 4 P 2n 47 2 n 2 Q 2n P 2n 29 2 Q 2n 41 2 P 2n 9Q 2n 51 4 P 2n 11 2 Q 2n 31 4 P 2n The triple n has the property D P 4n Q 4n and the triple n has the property D P 4n Q 4n. Theorem 17. For all natural numbers n and all integers m 0, the vector cross-products n n m and n n m are independent from the value n. They are equal to 8Q 2m 4Q 2m 0 and 2Q 2m 4Q 2m 2Q 2m. The vector cross-product n n m is 08Q 2m 4Q 2m and n n m is 2Q 2m 2Q 2m 4Q 2m. Theorem 18. For all natural numbers n, n n n n is equal to the vector cross-product 2240Q 4n 3168P 4n 384Q 4n 544P 4n 32Q 4n 48P 4n The vector cross-product n n n n is and the vector cross-product n n n n is equal to References 384Q 4n 544P 4n 1120Q 4n 1584P 4n 64Q 4n 96P 4n [1] ČERIN Z. AND GIANELLA G.M., On sums of squares of Pell-Lucas numbers, INTEGERS: Electronic Journal of Combinatorial Number Theory 6 (2006), A15. [2] ČERIN Z. AND GIANELLA G.M., Formulas for sums of squares and products of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis. 140 (2006),

12 12 ZVONKO ČERIN E GIAN MARIO GIANELLA 94 Zvonko Čerin e Gian Mario Gianella [3] ČERIN Z. AND GIANELLA G.M., On sums of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis. 141 (2007), [4] DUJELLA A., Generalized Fibonacci numbers and the problem of Diophantus, Fibonacci Quart. 34 (1996), [5] DUJELLA A., Diophantine quadruples for squares of Fibonacci and Lucas numbers, Portugaliae Math. 52 (1995), [6] SLOANE N., On-Line Encyclopedia of Integer Sequences,. Testo definitivo pervenuto in redazione il