In this section, we derive our main results.
2.5 Pythagorean triples
It is known that the Pell numbers
\(P_{n}\) have a close connection with square triangular numbers, that is,
$$ \bigl( (P_{k-1}+P_{k})P_{k} \bigr) ^{2}=\frac{(P_{k-1}+P_{k})^{2} ( (P_{k-1}+P_{k})^{2}-(-1)^{k} ) }{2}. $$
(16)
Note that the left side of (
16) describes a square number, whereas the right side describes a triangular number, so the result is a square triangular number (see [
18]). Notice that if a right triangle has integer side lengths
a,
b,
c (necessarily satisfying the Pythagorean theorem
\(a^{2}+b^{2}=c^{2}\)), then
\((a,b,c)\) is known as a Pythagorean triple. As Martin [
19] described, Pell numbers can be used to form Pythagorean triples in which
a and
b are one unit apart, corresponding to right triangles that are nearly isosceles. For instance,
$$\begin{aligned} \bigl(2P_{n}P_{n+1},P_{n+1}^{2}-P_{n}^{2},P_{n+1}^{2}+P_{n}^{2} \bigr) \end{aligned}$$
is a Pythagorean triple. Now we can give the following theorem related to Pythagorean triples.
2.6 The Pell equation
Let
d be any positive nonsquare integer, and let
N be any fixed integer. Then the equation
$$ x^{2}-dy^{2}=\pm N $$
(17)
is known as a Pell-type equation;
\(x^{2}-dy^{2}=N\) is the positive Pell-type equation, and
\(x^{2}-dy^{2}=-N\) is the negative Pell-type equation. It is named after John Pell (1611-1685), a mathematician who searched for integer solutions to equations of this type in the seventeenth century. Ironically, Pell was not the first to work on this problem, nor did he contribute to our knowledge for solving it. Euler (1707-1783), who brought us the
ψ-function, accidentally named the equation after Pell, and the name stuck.
For
\(N=1\), the Pell equation
\(x^{2}-dy^{2}=\pm 1\) is known as the classical Pell equation. The Pell equation
\(x^{2}-dy^{2}=1\) was first studied by Brahmagupta (598-670) and Bhaskara (1114-1185). Its complete theory was worked out by Lagrange (1736-1813), not Pell. It is often said that Euler (1707-1783) mistakenly attributed Brouncker’s (1620-1684) work on this equation to Pell. However, the equation appears in a book by Rahn (1622-1676), which was certainly written with Pell’s help: some say that it is entirely written by Pell. Perhaps Euler knew what he was doing in naming the equation. In 1657, Fermat stated (without giving proof) that the positive Pell equation
\(x^{2}-dy^{2}=1\) has an infinite number of solutions. If
\((m,n) \) is a solution, that is,
\(m^{2}-dn^{2}=1\), then
\((m^{2}+dn^{2},2mn)\) is also a solution since
$$\begin{aligned} \bigl(m^{2}+dn^{2}\bigr)^{2}-d(2mn)^{2}= \bigl(m^{2}-dn^{2}\bigr)^{2}=1. \end{aligned}$$
So the Pell equation
\(x^{2}-dy^{2}=1\) has infinitely many integer solutions. Later, in 1766, Lagrange proved that the Pell equation
\(x^{2}-dy^{2}=1\) has an infinite number of solutions if
d is positive and nonsquare. The first nontrivial solution
\((x_{1},y_{1})\neq (\pm 1,0)\) of this equation is called the fundamental solution from which all others are easily computed since
\(x_{n}+y_{n}\sqrt{d}=(x_{1}+y_{1}\sqrt{d})^{n}\) for
\(n\geq 1\) can be found using, for example, the cyclic method [
20], known in India in the 12th century, or using the slightly less efficient but more regular English method [
20] (17th century). There are other methods to compute this so-called fundamental solution, some of which are based on a continued fraction expansion of the square root of
d given as follows. Let
\(\sqrt{d}=[m_{0};\overline{m_{1},m_{2},\ldots ,m_{l}}]\) denote the continued fraction expansion of period length
l. Set
\(A_{-2}=0\),
\(A_{-1}=1\),
\(A_{k}=m_{k}A_{k-1}+A_{k-2}\) and
\(B_{-2}=1\),
\(B_{-1}=0\),
\(B_{k}=m_{k}B_{k-1}+B_{k-2} \) for nonnegative integers
k. Then
\(C_{k}=\frac{A_{k}}{B_{k}}\) is the
kth convergent of
\(\sqrt{d}\), and the fundamental solution of
\(x^{2}-dy^{2}=1\) is
\((x_{1},y_{1})=(A_{l-1},B_{l-1})\) if
l is even or
\((A_{2l-1},B_{2l-1})\) if
l is odd. Also, if
l is odd, then the the fundamental solution of
\(x^{2}-dy^{2}=-1\) is
\((x_{1},y_{1})=(A_{l-1},B_{l-1})\) (for further details on Pell equations, see [
21‐
23]).
It is known that there is a connection between integer sequences and Pell equations. For instance, Olajas [
9] gave the integer solutions to
\(x^{2}-5y^{2}=\pm 4\) as follows.
For integers
A and
B such that
\(A^{2}-4B\neq 0\) (to exclude a degenerate case),
\(R=\{R_{i}\}_{i=0}^{\infty }=R(A,B,R_{0},R_{1})\) is a second-order linear recurrence if the recurrence relation for
\(i\geq 2\)
$$ R_{i}=AR_{i-1}-BR_{i-2} $$
(18)
holds for its terms and
\(R_{0},R_{1}\) are fixed integers. For the Pell equation
\(x^{2}-8y^{2}=1\), Liptai [
24] proved the following:
Now we can return to our main problem. We consider the integer solutions of the Pell equations
$$\begin{aligned} x^{2}-8y^{2}=1,\qquad x^{2}-2y^{2}=\pm 4,\quad \mbox{and}\quad x^{2}-8y^{2}=\pm 4. \end{aligned}$$
It is known that there are a number of applications of Pell and Fibonacci sequences on the theory of numbers. For instance, Sellers proved in [
25, Thm. 2.1] that the number of domino tilings of the graph
\(W_{4}\times P_{n-1}\) equals the product of the
nth Fibonacci and Pell numbers for all
\(n\geq 2\). Also, there has been a connection between Diophantine quadruples and Fibonacci numbers. Recall that a set of
m positive integers
\(\{a_{1},a_{2},\dots,a_{m}\}\) is called a Diophantine
m-tuble if
\(a_{i}a_{j}+1 \) is a perfect square for all
\(1\leq i< j\leq m\) and is called a
\(D(n)\)-
m-tuble (or a Diophantine
m-tuble with the property
\(D(n)\)) if
\(a_{i}a_{j}+n\) is a perfect square for all
\(1\leq i< j\leq m\).
Cassini’s identity for Fibonacci number is
\(F_{n}F_{n+2}+(-1)^{n}=F_{n+1}^{2} \) and is the basis for the construction of so-called Hoggatt-Bergum’s quadruple. Hoggatt and Bergum [
26] proved that, for any positive integer
k, the set
$$\begin{aligned} \{F_{2k},F_{2k+2},F_{2k+4},4F_{2k+1}F_{2k+2}F_{2k+3} \} \end{aligned}$$
is a Diophantine quadruple. Later Morgado [
27] and Horadam [
28] generalized this result. The identity
$$\begin{aligned} F_{k-3}F_{k-2}F_{k-1}F_{k+1}F_{k+2}F_{k+3}+L_{k}^{2}= \bigl[F_{k}\bigl(2F_{k-1}F_{k+1}-F_{k}^{2} \bigr)\bigr]^{2} \end{aligned}$$
is known as the Morgado identity.
Using Fibonacci numbers, Dujella [
29] defined the elliptic curve (see [
30])
$$\begin{aligned} E_{k}:y^{2}=(F_{2k}x+1) (F_{2k+2}x+1) (F_{2k+4}x+1) \end{aligned}$$
and determined the integer points on it by terms of Fibonacci numbers when the rank of
\(E_{k}(\mathbb{Q})\) is 1.
It is known that there are several identities on Fibonacci and Lucas numbers. Some of them can be given as
\(4F_{k-1}F_{k+1}+F_{k}^{2}=L_{k}^{2}\) and
\(4F_{k-1}F_{k}^{2}F_{k+1}+1=(F_{k}^{2}+F_{k-1}F_{k+1})^{2}\). Using these, Dujella [
31] obtained some quantities on
\(D(F_{k}^{2})\)-quadruples
$$\begin{aligned} &\bigl\{ 2F_{k-1},2F_{k+1},2F_{k}^{3}F_{k+1}F_{k+2},2F_{k+1}F_{k+2}F_{k+3} \bigl(2F_{k+1}^{2}-F_{k}^{2}\bigr)\bigr\} , \\ &\bigl\{ F_{k-1},4F_{k+1},F_{k-2}F_{k-1}F_{k+1} \bigl(2F_{k}^{2}-F_{k-1}^{2} \bigr),F_{k}^{3}F_{k+2}F_{k+3}\bigr\} , \\ &\bigl\{ 4F_{k-1},F_{k+1},F_{k-2}F_{2k-2}F_{2k-1},F_{k}^{3}L_{k}L_{k+1} \bigr\} . \end{aligned}$$
(19)
As in (
19), the set
$$\begin{aligned} \bigl\{ F_{k-3}F_{k-2}F_{k+1},F_{k-1}F_{k+2}F_{k+3},F_{k}L_{k}^{2},4F_{k-1}^{2}F_{k}F_{k+1}^{2} \bigl(2F_{k-1}F_{k+1}-F_{k}^{2}\bigr)\bigr\} \end{aligned}$$
is a
\(D(L_{k}^{2})\)-quadruple.
Dujella and Ramasamy [
32] considered the Fibonacci numbers and
\(D(4)\)-quadruple. They proved that the set
$$\begin{aligned} \{F_{2k},5F_{2k},4F_{2k+2},4L_{2k}F_{4k+2} \} \end{aligned}$$
is a
\(D(4)\)-quadruple. Also, they considered integer solutions of the Pell equations by using a
\(D(4)\)-quadruple.
In the future work, we are planing to study \(D(n)\)-quadruples for the sequences mentioned for some n.