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 D
n, 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` D
n, 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 D
n 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
2

m with i  j. We can also say that a is the D
n-m-tuple. For example,

1
3
8
120 is the famous D
1-quadruple discovered by Fermat and
2
4
10 is the D
4-triple while
2
4
14 is the D
8-triple.  The subject of D
n-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 D
n-quadruple when n ', where ' is the set 4
3
1
3
5
8
12
20 (see [4]). Also, there are many examples of infinite
  sequences of D
n- 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  A
k k0 and B  B
k k0 of D
4- d
d 
  triples and G  G
k k0 and D  D
k k0 of D
8-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

A
n
P2n1
2P2n1
P2n3 and B
n
P2n1
P2n3
2P2n3

Theorem 1. For every natural number n the triples A
n and B
n 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 x
a
b
c by S1
xa  b  c
S2
xbc ca ab
S3
xabc The linear expressions involving symmetric functions S1 and S2 of the triples A
n and B
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:

2 2 2 2 abS1
A
n  b S2
A
n 4b  a 
5bP2n  4bQ2n  a
 2 2 2 2 abS1
B
n  b S2
B
n 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  abS1
A
n  b S2
A
n 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 A
n and B
n have the property that the numbers S2
A
n 4, S2
B
n 4 and  

2S1
A
n  S2
A
n
2S1
B
n  S2
B
n 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 Q
n of numbers, let Ma
b
c be the 3 3 matrix with rows Q
a, Q
b and Q
c. Since the columns of the  A B matrices Ma
b
c and Ma
b
c are linearly dependent it is clear that the determinants A B det
Ma
b
c and det
Ma
b
c 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 per
Ma
b
c and per
Ma
b
c:

4 4 4 4 4 4 2123
33 113
32 31  287
33  111
32  31   2
33

4 4 4 4 4 4 2717
33 143
32 31  2507
33  131
32  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 per
Ma
b
c 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 per
Ma
b
c and B per
Ma
b
c 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
w
au
bv
cw

a
b
c 
u
v
w
av
bw
cu
and
a
b
c 
u
v
w
aw
bu
cv The next result shows that the sequences A
n and B
n 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 A
n B
n, A
n  m B

n  m and A
n  um B
n  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 A
n  B
n is true. Its proof is almost the same.

Theorem 5. For all integers a and b the following identity holds:

2 2 abS1
A
n  B
n  b S2
A
n  B
n  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 A
n  B
n has the property that the numbers S2
A
n  B
n,

S2
A
n  B
n  S1
A
n  B
n  1 and S2
A
n  B
n S1
A
n  B
n  1  are squares.

3. The sequences G and D

Let G
D :

3 be the functions defined by

G
n
Q2n1
2Q2n1
Q2n3 and D
n
Q2n1
Q2n3
2Q2n3

Theorem 6. For every natural number n the triples G
n and D
n have the property D
8. 6 ZVONKO CERINˇ E GIAN MARIO GIANELLA 88 Zvonko Cˇerin e Gian Mario Gianella

Proof. This follows from the following identities:

2 2Q2n1 Q2n3  8 
6P2n  4Q2n
2 Q2n1 Q2n3  8 
4P2n  3Q2n
2 2 2Q2n1  8 
2P2n  2Q2n
2 2 2Q2n3  8 
14P2n  10Q2n 

The linear expressions involving symmetric functions S1 and S2 of the triples G
n and D
n are also the source of many complete squares as the fol- lowing theorem shows. Its proof is analogous to the proof of Theorem 2. Theorem 7. For all integers a and b the following identities hold:

2 2 2 2 abS1
G
n  b S2
G
n  8b  a 
8bP2n  5bQ2n  a

2 2 2 2 abS1
D
n  b S2
D
n  8b  a 
16bP2n  11bQ2n  a  In particular, when b  1 and a  0 and b  1 and a  1 we obtain the fol- lowing statement. Corollary 3. For every natural number n the triples G
n and D
n have the property that the numbers S2
G
n  8, S2
D
n  8 and

S1
G
n  S2
G
n  9
S1
D
n  S2
D
n  9 are squares.

G D Since the columns of the matrices Ma
b
c and Ma
b
c are also linearly depen- G D dent it is clear that the determinants det
Ma
b
c and det
Ma
b
c 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 G D the permanents per
Ma
b
c and per
Ma
b
c:

4 4 4 4 4 4 4 287
33 1 11
32 31  4123
33  1 13
32  31    2 
33

4 4 4 4 4 4 4 2507
33 1 31
32 31  4717
33  1 43
32  31    2   33 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 G and c  n  2 for a natural number n), then the permanents per
Ma
b
c and D per
Ma
b
c are

97428Q6n  137784P6n 11620Q2n 16520P2n
  567852Q6n  803064P6n 35420Q2n 50120P2n   The next result shows that the sequences G
n and D
n are also connected with respect to the products ,  and . Recall that for a natural number m, we put U 
1  2m and V 
1 2m.  Theorem 9. For all natural numbers n, m and u and each product

 the determinant of the matrix with rows G
n D
n, G
n  m D

n  m and G
n  um D
n  um is independent of the value n. These determi-  nants 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
A
G,
A
D,
B
G and
B
D and the products ,  and . For example, the determi- nant 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
A
B in Theorem 4.  Here is an analogue of Theorem 5 for the sequence G
n  D
n. Theorem 10. For all integers a and b the following identity holds:

2 2 abS1
G
n  D
n  b S2
G
n  D
n  a  51 a 2 36bP4n  bQ4n 9b    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 G
n  D
n has the property that the numbers S2
G
n  D
n,

S2
G
n  D
n  S1
G
n  D
n  1 and S2
G
n  D
n S1
G
n  D
n  1  are squares. 8 ZVONKO CERINˇ E GIAN MARIO GIANELLA 90 Zvonko Cˇerin e Gian Mario Gianella

We can get many complete squares from the sequences

J
n
A
n  B
n 
G
n  D
n and Y
n
A
n  B
n 
G
n  D
n Theorem 11. For all integers a and b the following identity holds:

2 2 8abS1
J
n  6b S2
J
n  9a  2 2 8abS1
Y
n  6b S2
Y
n  9a  2
4896bP8n  3462bQ8n 204b  3a   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 J
n and Y
n have the property that the numbers 6S2
J
n, 6S2
Y
n,

6S2
J
n  8S1
J
n  9 and 6S2
Y
n  8S1
Y
n  9 are squares.

4. The sequences A, B, G and D

The sequence A determines the sequence A where A
n is the triple
4P2n 3Q2n
P2n2
2P2n  Q2n given in the proof of Theorem 1. In other words, the coordinates of A
n satisfy the following relations:

A
n1  A
n2 A
n3 4
 A
n2  A
n3 A
n1 4
 A
n3  A
n1 A
n2 4  We can similarly introduce the sequences B, G and D. There are many expres- sions built from these sequences that are also squares. Here we give only few simple examples.

2 S1
A
n A
n  8 
5P2n  4Q2n 

2 3S1
A
n A
n 3S2
A
n 
3P2n  3Q2n  

ON DIOPHANTINEOn Diophantine TRIPLES triples from Pell and Pell-Lucas numbers 91 9

2 2S1 B
n B
n 12 
16P2n  11Q2n    

2 10S1 B
n B
n  11S2 B
n  72 
46P2n  33Q2n      2 S1
G
n G
n 16 
8P2n  5Q2n   2 3S1
G
n G
n  S2
G
n  32 
14P2n  11Q2n 

2 2 S1 D
n D
n  24 
22P2n  16Q2n   

2 10S1 D
n D
n  15S2 D
n  80 
70P2n  50Q2n      Of course, we can use also the other two products  and  and also the se- quences A, B, G and D. Hence, from all these sequences and the three prod- ucts 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 D
4 and D
8. 

5. The geometry of the eight sequences

In this section the members of our eight sequences being triples of (real) num- bers are considered as points in the 3-dimensional Euclidean space.

Theorem 12. For all natural numbers a, b, c and d, the

A
a
A
b
G
c
G
d

B
a
B
b
D
c
D
d

A
a
A
b
G
c
G
d

B
a
B
b
D
c
D
d are quadruples of coplanar points in the 3-dimensional Euclidean space.

Proof. Recall that the points Ti
xi
yi
zi (i  1
2
3
4) are coplanar if and only if the oriented volume x1 y1 z1 1 1  x2 y2 z2 1    6  x3 y3 z3 1   x y z 1   4 4 4    of the tetrahedron T1T2T3T4 is zero. When we substitute the above quadruples of points into this determinant and ask Maple V for help we get zero. 10 ZVONKO CERINˇ E GIAN MARIO GIANELLA 92 Zvonko Cˇerin e Gian Mario Gianella

For points A, B, C and D in the 3-dimensional Euclidean space E3, let ABCD denote the oriented volume of the tetrahedron ABCD. Let T denote   AB the tetrahedron A
nA
n  1B
nB
n  1. We similarly define tetrahedra TAD, T and T and T , T , T and T . BG GD AB AD BG GD

Theorem 13. For every natural number n, the following is true: 64 2 T  T  Q2n  32P2n
  AB  AB 3 304 416 2 T  T  Q2n  P2n
  AD  AD 3 3

2 T  T  112Q2n  160P2n
 BG  BG 256 2 T  T  64Q2n  P2n  GD  GD 3

The centroids of the tetrahedra TAB,TAD,TBG and TGD are coplanar. The centroids of the tetrahedra T ,T ,T and T are also coplanar. AB AD BG GD Proof. The proof is once again a direct computation of determinants with stan- dard 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 7
n and 7
n whose ver- tices are the triples n , n , n and n and n , n , n and n A
 B
 G
 D
 A
 B
 G
 D
 have nice oriented volumes.

Theorem 14. For every natural number n, the tetrahedron 7
n has the ori- 8 4 ented volume P2n while the tetrahedron 7
n has the oriented volume P2n.  3 3 Theorem 15. For every natural number n, the tetrahedron 71
n with ver- tices A
n, B
n  1, G
n and D
n  1 has the oriented volume 8Q2n 16P2n   while the tetrahedron 72
n with vertices A
n  1, B
n, G
n  1 and D
n has 8 16 the oriented volume 3 Q2n  3 P2n. For every natural number n, the tetrahedron 71
n with vertices A
n, B
n  1, G
n and D
n  1 has the oriented volume 4Q  8P while the 2n 2n tetrahedron 72
n with vertices A
n  1, B
n, G
n  1 and D
n has the ori- 4 8 ented volume 3 Q2n 3 P2n.   Theorem 16. The tetrahedra 7
n, 7
n, 71
n, 71
n, 72
n, and 72
n have the centroids in the points

3 23 51 W
n Q n  P n
4Q n  P n
9Q n  P n
 2 2 2 2 4 2 2 4 2 ON DIOPHANTINEOn Diophantine TRIPLES triples from Pell and Pell-Lucas numbers 93 11

15 17 11 W
n 6Q n  P n
3Q n  P n
2Q n  P n
 2 2 2 2 4 2 2 4 2 37 105 215 W
n 7Q n  5P n
Q n  P n
38Q n  P n
1  2 2 2 2 4 2 2 4 2 53 75 23 65 45 W
n Q n  P n
Q n  P n
8Q n  P n
1  2 2 2 2 2 2 4 2 2 4 2 51 47 133 W
n 7Q n  5P n
9Q n  P n
Q n  P n
2  2 2 2 4 2 2 2 4 2 29 41 51 11 31 W
n Q n  P n
9Q n  P n
Q n  P n  2  2 2 2 2 2 4 2 2 2 4 2 17 3 17 The triple W
n has the property D 16 4 P4n 32 Q4n and the triple W
n 3 181   has the property D 16P4n
Q4n .   8   16
 Theorem 17. For all natural numbers n and all integers m 0, the vector cross-products A
n G
n  m and A
n G
n  m are independent from the   value n. They are equal to
8Q2m
4Q2m
0 and
2Q2m
4Q2m
2Q2m. The   vector cross-product B
n D
n  m is
0
8Q2m
4Q2m and B
n D
n  m    is
2Q2m
2Q2m
4Q2m.   Theorem 18. For all natural numbers n, the vector cross-product
A
n B
n
G
n D
n is equal to 

2240Q4n  3168P4n
384Q4n 544P4n
32Q4n  48P4n   The vector cross-product
A
n  B
n
G
n  D
n is
0
0
0 and the vec- tor cross-product
A
n  B
n
G
n  D
n is equal to 

384Q4n  544P4n
1120Q4n 1584P4n
64Q4n 96P4n     References

[1] Cˇ ERIN Z. AND GIANELLA G.M., On sums of squares of Pell-Lucas num- bers, INTEGERS: Electronic Journal of Combinatorial Number Theory 6 (2006), A15.

[2] Cˇ ERIN Z. AND GIANELLA G.M., Formulas for sums of squares and products of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis. 140 (2006), 113-122. 12 ZVONKO CERINˇ E GIAN MARIO GIANELLA 94 Zvonko Cˇerin e Gian Mario Gianella

[3] Cˇ ERIN Z. AND GIANELLA G.M., On sums of Pell numbers, Acc. Sc. Torino - Atti Sci. Fis. 141 (2007), 23-31.

[4] DUJELLA A., Generalized Fibonacci numbers and the problem of Dio- phantus, Fibonacci Quart. 34 (1996), 164-175.

[5] DUJELLA A., Diophantine quadruples for squares of Fibonacci and Lu- cas numbers, Portugaliae Math. 52 (1995), 305-318.

[6] SLOANE N., On-Line Encyclopedia of Integer Sequences,    

 
.

Testo definitivo pervenuto in redazione il 13.5.2009.