# Diophantine Triples and *k*-Generalized Fibonacci Sequences

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## Abstract

We show that if \(k\ge 2\) is an integer and \(\big (F_n^{(k)}\big )_{n\ge 0}\) is the sequence of *k*-generalized Fibonacci numbers, then there are only finitely many triples of positive integers \(1<a<b<c\) such that \(ab+1,~ac+1,~bc+1\) are all members of \(\big \{F_n^{(k)}: n\ge 1\big \}\). This generalizes a previous result where the statement for \(k=3\) was proved. The result is ineffective since it is based on Schmidt’s subspace theorem.

## Keywords

Diophantine triples Generalized Fibonacci numbers Diophantine equations Application of the Subspace theorem## Mathematics Subject Classification

11D72 11B39 11J87## 1 Introduction

*m*-tuples which are sets of

*m*distinct positive integers \(\{a_1,\ldots ,a_m\}\) such that \(a_ia_j+1\) is a square for all \(1\le i<j\le m\) (see [4], for example). A variation of this classical problem is obtained if one changes the set of squares by some different subsets of positive integers like

*k*-powers for some fixed \(k\ge 3\), or perfect powers, or primes, or members of some linearly recurrent sequence, etc. (see [7, 12, 13, 14, 18]). In this paper, we study this problem with the set of values of

*k*-generalized Fibonacci numbers for some integer \(k\ge 2\). Recall that these numbers denoted by \(F_n^{(k)}\) satisfy the recurrence

### Theorem 1

*x*,

*y*,

*z*.

Our result generalizes the results obtained in [8, 11] and [13], where this problem was treated for the cases \(k=2\) and \(k=3\). In [13] it was shown that there does not exist a triple of positive integers *a*, *b*, *c* such that \(ab+1,ac+1,bc+1\) are Fibonacci numbers. In [11] it was shown that there is no Tribonacci Diophantine quadruple, that is a set of four positive integers \(\{a_1,a_2,a_3,a_4\}\) such that \(a_ia_j+1\) is a member of the Tribonacci sequence (3-generalized Fibonacci sequence) for \(1\le i<j\le 4\), and in [8] it was proved that there are only finitely many Tribonacci Diophantine triples. In the current paper, we prove the same result for all such triples having values in the sequence of *k*-generalized Fibonacci numbers.

For the Proof of Theorem 1, we proceed as follows. In Sect. 2, we recall some properties of the *k*-generalized Fibonacci sequence \(F_n^{(k)}\) which we will need and we prove two lemmata. The first lemma shows that any \(k-1\) roots of the characteristic polynomial are multiplicatively independent. In the second lemma, the greatest common divisor of \(F_x^{(k)}-1\) and \(F_y^{(k)}-1\) for \(2<y<x\) is estimated. In Sect. 3, we assume that the Theorem 1 is false and give, using the Subspace theorem, a finite expansion of infinitely many solutions. In Sect. 4, we use a parametrization lemma which is proved by using results about finiteness of the number of non-degenerate solutions to *S*-unit equations. Applying it to the finite expansion, this leads us to a condition on the leading coefficient, which turns out to be wrong. This contradiction is obtained by showing that a certain Diophantine equation has no solutions; this last Diophantine equation has been treated in particular cases in [2] and [16].

## 2 Preliminaries

*k*is even or prime, the Galois group is certainly \(S_k\) (see [16] for these statements). The polynomial \(\varPsi _k(X)\) has only one root, without loss of generality assume that this root is \(\alpha _1>1\), which is outside the unit disk (formally, this root depends also on

*k*, but in what follows we shall omit the dependence on

*k*on this and the other roots of \(\varPsi _k(X)\) in order to avoid notational clutter). Thus,

Furthermore, the following property is of importance; it follows from the fact that there is no non-trivial multiplicative relation between the conjugates of a Pisot number (cf. [15]). Since in our case it is rather easy to verify, we shall present a proof of what we need.

### Lemma 1

Each set of \(k-1\) different roots (e.g., \(\{\alpha _1, \ldots , \alpha _{k-1}\}\)) is multiplicatively independent.

### Proof

We shall prove the statement only for the set \(\{\alpha _1,\ldots ,\alpha _{k-1}\}\). The general statement follows easily by the transitivity of the Galois Group of the irreducible polynomial \(\varPsi _k(X)\).

*d*its degree and by \(G:={\text {Gal}}(\mathbb {L}/\mathbb {Q})\) the Galois group of \(\mathbb {L}\) over \(\mathbb {Q}\). Let \(G=\{\sigma _1,\ldots ,\sigma _d\}\). We consider the map \(\lambda :\mathcal {O}_\mathbb {L}^\times \rightarrow \mathbb {R}^d\) defined by \( x \mapsto (\log \vert \sigma _1(x) \vert ,\log \vert \sigma _2(x)\vert ,\ldots ,\) \(\log \vert \sigma _d(x)\vert )\). Observe that by the product formula we have for \(x\in \mathbb {L}^\times \) that

*v*we have \(\vert x\vert _v=1\). This means that the image of \(\lambda \) lies in the hyperplane defined by \(X_1+\cdots +X_d=0\) of \(\mathbb {R}^d\).

We will use the property that \(\lambda \) is a homomorphism (see e.g., [17]). So if we can prove that the \(k-1\) vectors \(\lambda (\alpha _1),\ldots ,\lambda (\alpha _{k-1})\) are linearly independent, this will prove the statement.

*G*of \(\mathbb {L}\) over \(\mathbb {Q}\) acts transitively on \(\{\alpha _1,\ldots ,\alpha _k\}\), i.e., for each \(i = 1, \dots , k\) there is some Galois automorphism which sends \(\alpha _i\) to \(\alpha _1\); without loss of generality let \(\sigma _1,\ldots ,\sigma _k\) be such that \(\sigma _i(\alpha _i) = \alpha _1\). Observe that \(\sigma _i^{-1}(\alpha _1)=\{\alpha _i\}\) for \(i=1,\ldots ,k\). We have

Finally, we prove the following result, which generalizes Proposition 1 in [8]. Observe that the upper bound depends now on *k*.

### Lemma 2

### Proof

## 3 Parametrizing the Solutions

In order to simplify the notations, we shall from now onwards write \(F_n\) instead of \(F_n^{(k)}\); we still mean the *n*th *k*-generalized Fibonacci number. The arguments in this section follow the arguments from [8]. We will show that if there are infinitely many solutions to (1), then all of them can be parametrized by finitely many expressions as given in (18) for *c* below.

*a*,

*b*,

*c*), we have

*x*and

*z*, we denote \(d_1 := \gcd (F_y - 1, F_z - 1)\) and \(d_2 := \gcd (F_x - 1, F_z - 1)\), such that \(F_z - 1 \mid d_1 d_2\). Then we use Lemma 2 to obtain

*k*), when

*z*is sufficiently large.

*c*which was given by

*T*is some index, which we will specify later. Since \(x < z\) and \(z < x/C_1\), the remainder term can also be written as \(\mathcal {O}\big (\alpha _1^{-T \Vert x\Vert /C_1}\big )\), where \(\Vert x\Vert =\max \{x,y,z\}=z\). Doing the same for

*y*and

*z*likewise and multiplying those expression gives

*n*depends only on

*T*and where

*J*is a finite set, \(d_j\) are non-zero coefficients in \({\mathbb L}:={\mathbb Q}(\alpha _1,\ldots ,\alpha _k)\), and \(M_j\) is a monomial of the form

*k*), such that

*c*involving terms as in (13); the version we are going to use can also be found in Sect. 3 of [10]. For the set-up—in particular the notion of heights—we refer to the mentioned papers.

*S*be the finite set of infinite places (which are normalized so that the Product Formula holds, cf. [5]). Observe that \(\alpha _1,\ldots ,\alpha _k\) are

*S*-units. According to whether \(-x+y+z\) is even or odd, we set \(\epsilon = 0\) or \(\epsilon = 1\) respectively, such that

*n*(depending on

*T*) from above, we now define \(n+1\) linearly independent linear forms in indeterminants \((C, Y_0, \dots , Y_n)\). For the standard infinite place \(\infty \) on \(\mathbb {C}\), we set

*v*in

*S*, we define

*v*. Thus (16) reduces to

*k*! and hence

*T*(and the corresponding

*n*) in such a way that

*x*,

*y*,

*z*) of (16), we now can conclude that all of them lie in finitely many proper linear subspaces. Therefore, there must be at least one proper linear subspace, which contains infinitely many solutions and we see that there exists a finite set

*J*and (new) coefficients \(e_j\) for \(j\in J\) in \({\mathbb {L}}\) such that we have

Likewise, we can find finite expressions of this form for *a* and *b*.

## 4 Proof of the Theorem

We use the following parametrization lemma:

### Lemma 3

*x*,

*y*,

*z*) are of the form (

*x*(

*n*),

*y*(

*n*),

*z*(

*n*)) for some integer

*n*.

### Proof

*c*can be written in the form

*J*is some finite set, \(e_j\) are non-zero coefficients in \(\mathbb {L}\) and \(L_{i,j}\) are linear forms in \(\mathbf{x}\) with integer coefficients.

This is an *S*-unit equation, where *S* is the multiplicative group generated by \(\{\alpha _1, \dots , \alpha _k, -1\}\). We may assume that infinitely many of the solutions \(\mathbf{x}\) are non-degenerate solutions of (20) by replacing the equation by an equation given by a suitable vanishing subsum if necessary.

We may assume that \((L_{1,i}, \dots , L_{k-1,i}) \ne (L_{1,j}, \dots , L_{k-1,j})\) for any \(i \ne j\), because otherwise we could just merge these two terms.

*S*-unit equations (see [6]) yields that the set of

*L*defined over \(\mathbb {Q}\) and some \(c\in \mathbb {Q}\) with \(L(\mathbf{x}) = c\) for infinitely many \(\mathbf{x}\). So there exist rationals \(r_i, s_i, t_i\) for \(i = 1, 2, 3\) such that we can parametrize

*p*,

*q*) we have \(p \equiv p_0 \mod \varDelta \) and \(q \equiv q_0 \mod \varDelta \). Then \(p = p_0 + \varDelta \lambda , q = q_0 + \varDelta \mu \) and

*x*,

*y*,

*z*are all integers, \(r_i p_0 + s_i q_0 + t_i\) are integers as well. Replacing \(r_i\) by \(r_i \varDelta \), \(s_i\) by \(s_i \varDelta \) and \(t_i\) by \(r_i p_0 + s_i q_0 + t_i\), we can indeed assume that all coefficients \(r_i, s_i, t_i\) in our parametrization are integers.

*J*is a finite set of indices, \(e_j'\) are new non-zero coefficients in \(\mathbb {L}\) and \(L'_{i,j}(\mathbf{r})\) are linear forms in \(\mathbf{r}\) with integer coefficients. Again we may assume that we have \((L'_{1,i}(\mathbf{r}), \dots , L'_{k-1,i}(\mathbf{r})) \ne (L'_{1,j}(\mathbf{r}), \dots , L'_{k-1,j}(\mathbf{r}))\) for any \(i \ne j\).

*S*-unit equations once more, we obtain a finite set of numbers \(\varLambda \), such that for some \(i \ne j\), we have

*n*will be in the same residue class modulo \(\varDelta \), which we shall call

*r*. Writing \(n = m \varDelta + r\), we get

*n*by

*m*, \(r_i\) by \(r_i \varDelta \) and \(s_i\) by \(r r_i + s\), we can even assume that \(r_i, s_i\) are integers. So we have

*m*, so we can choose a still infinite subset such that all of them are in the same residue class \(\delta \) modulo 2 and we can write \(m = 2 \ell + \delta \) with fixed \(\delta \in \{0,1\}\). Thus we have

*c*into (23). Furthermore, we use the Binet formula (3) and write \(F_x, F_y, F_z\) as power sums in

*x*,

*y*and

*z*respectively. Using the parametrization \((x,y,z) = (r_1 m + s_1, r_2 m + s_2, r_3 m + s_3)\) with \(m = 2\ell \) or \(m = 2\ell +1\) as above, we have expansions in \(\ell \) on both sides of (3). Since there must be infinitely many solutions in \(\ell \), the largest terms on both sides have to grow with the same rate. In order to find the largest terms, we have to distinguish some cases: If we assume that \(e_0 \ne 0\) for infinitely many of our solutions, then \(e_0 \alpha _1^{(-x+y+z-\epsilon )/2}\) is the largest term in the expansion of

*c*and we have

### Lemma 4

\(\sqrt{f_1} \notin \mathbb {L}\) and \(\sqrt{f_1\alpha _1} \notin \mathbb {L}\).

### Proof

*w*. But this equation has no integer solutions, which is proved in the theorem below. This concludes the proof. \(\square \)

In order to finish the proof, we have the following result, which might be of independent interest since particular cases were considered before in [2] and [16].

### Theorem 2

The Diophantine equation (25) has no positive integer solutions (*k*, *w*) with \(k\ge 2\).

### Proof

*p*divides \(w_1\) and \((k+1)/2\), then

*p*divides the left-hand side of (26). Thus

*p*divides both

*k*and \((k+1)/2\), so also \(k-2((k+1)/2)=-1\), a contradiction. Thus, the right-hand side is a sum of two coprime squares and therefore all odd prime factors of it must be 1 modulo 4 contradicting the fact that in the left-hand side we have \(k\equiv 3\pmod 4\). This finishes the proof of this theorem. \(\square \)

## Notes

### Acknowledgments

Open access funding provided by University of Salzburg. C.F. and C.H. were supported by FWF (Austrian Science Fund) Grant No. P24574 and by the Sparkling Science project EMMA Grant No. SPA 05/172.

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