0 Research

# Graph Models for Biological Pathway Visualization:
State of the Art and Future Challenges

Hsiang-Yun Wu
TU Wien e-mail<EMAIL_ADDRESS>Austria  Martin Nöllenburg
TU Wien e-mail<EMAIL_ADDRESS>Austria  Ivan Viola
King Abdullah University of Science and Technology (KAUST), Saudi Arabia
e-mail<EMAIL_ADDRESS>
###### Abstract

The concept of multilayer networks has become recently integrated into complex
systems modeling since it encapsulates a very general concept of complex
relationships. Biological pathways are an example of complex real-world
networks, where vertices represent biological entities, and edges indicate the
underlying connectivity. For this reason, using multilayer networks to model
biological knowledge allows us to formally cover essential properties and
theories in the field, which also raises challenges in visualization. This is
because, in the early days of pathway visualization research, only restricted
types of graphs, such as simple graphs, clustered graphs, and others were
adopted. In this paper, we revisit a heterogeneous definition of biological
networks and aim to provide an overview to see the gaps between data modeling
and visual representation. The contribution will, therefore, lie in providing
guidelines and challenges of using multilayer networks as a unified data
structure for the biological pathway visualization.

###### keywords:

Graph drawing, multilayer network, biological pathway

Introduction

Over the past decade, several new scenarios from sciences or everyday life
have benefited from formulating a relationship between entities as a graph.
For example, social networks [26], which describe person-to-person
relationships, are developed since 20th century. The scientific discipline of
complexity science is concerned with studying huge sets of relationships among
entities, to make sense of the complex interrelationships within a particular
structure or phenomenon. Due to the rapid development of data measurements,
simple graph representation is no longer sufficient to capture a real-world
complex system, because the number of nodes grows prohibitively large, as well
as the density of relationships in the system. Often, fortunately, the
phenomenon can be structured along particular domain semantics that organize
the graph into certain higher level structures. In social networks, for
example, these could relate to geographical location, place of study, work, or
hobbies. Such additional structure, gives rise to the concept of a multilayer
network [21] to generalize the modeling of a complex relationship through a
series of layers.

In addition to social networks, another well-known application for complexity
science is analysis of network structures in biology. The life functions are
prominently organized in a relationship of interacting elements and chemical
compounds, forming super complex networks of reactions carried out all over
the entire life form. These networks are large and heavily interconnected.
Biological pathways, which form a graph containing dozens of chemical elements
representing a particular element of life, are the simplest relational entity
of such networks. To semantically manipulate such a network of interconnected
pathways, one can segment it into multiple sub-networks, such as _pathways_
arranged with different functionalities. Semantics in pathways can be
understood in several ways. One aspect could be the functionalities among the
data, and another aspect could be species associated with the data. According
to the definition by Kivelä et al. [21], the aforementioned aspects can be
formulated as groups of layers of different types.

Unfortunately, the full set of pathways is too large to be easily handled in
high quality with common graph visualization techniques. Several research
studies in visualization tend to solve the problem, by organizing aspects into
clusters, such as _subsystems_ in biological ontology to reflect the
underlying structures in the datasets. The definition of the multilayer
network facilitates us to revisit the modeling and visualization of biological
pathways in a whole. In this paper, we investigate the underlying graph
modeling and visualization of conventional biological pathways, and further
research their common and different properties due to the heterogeneity in the
state of the art. Our contribution is to provide a summary to show the similar
and dissimilar properties of the underlying graph models used in both pathway
modeling and visualization. This is done by a study on the collected
literature and the classification through types of underlying graph data
structures not only from the well-known pathway databases but also the pathway
visualization methodologies.

The remainder of this paper is structured as follows: In Section 1, we define
the underlying graph structures often used in modeling biological pathways.
Next, we investigate the visualization of biological pathway diagrams by
researching hand-crafted diagrams and machine-computed layouts in Section 2.
In Section 3, we discuss and summarize some future research directions and
challenges of pathway visualization, followed by a conclusion in Section 4.

##  1 Underlying Structures of Biological Pathways

A complex phenomenon is often formulated using a _graph_ to manipulate a
relationship mathematically.

Definition: A simplest graph model can be described as a tuple $G=(V,E)$,
consisting of a set of vertices $V=\\{v_{1},v_{2},...,v_{n}\\}$ representing
individual entities, and the mutual connectivity is represented by the edges
$E=\\{e_{1},e_{2},...,e_{m}\\}\subseteq V\times V$. This is a common
definition that allows us to describe the simplest relationship between
entities.

The term used for describing more nested relationships between entities is
often called multiplex [3], while other terminologies that describe similar
concept are also used in conventional research. Also reported by Kivela et al.
[21] originally, those terms include _multiplex_ , _multilevel_ ,
_multivariate_ [28], _multidimensional_ , _multirelational_ , _network of
networks_ can be reframed and encapsulated by the definition of _multilayer
networks_. In other words, one can use multilayer networks as an umbrella term
to cover conventional formulation, nonetheless to say in the field of
biological networks [14].

This concept is not fully captured in the early research of biological
pathways, in which researchers tend to formulate the network as
straightforward as possible based on their needs. According to our
investigation, the biological relationship is often formulated as _Substrate
Graph_ , _Bipartite Graph_ , _Hypergraph_ , _Reaction Graph_ , or
_Stoichiometric Matrix_ in biological pathways. In this section, we summarize
different graph structures that are frequently used in modelling biological
pathways, and in comparison to the formulation of multilayer networks. This
will give readers an overview of the models and further understand the power
of using multilayer network in the future. We have described the simplest
graph definition and will explain how it is expended to model biological
pathways.

###  1.1 Substrate Graph

The development of computational tools in systems biology allows us to analyze
the collected datasets through the power of computers [41]. The underlying
data structure is important in the sense that it requires efficient access to
the data so that the algorithms can perform efficiently. The substrate graph
is one of the pioneering graph structures.

Definition: A substrate graph is essentially equal to the simple graph
introduced previously. Nonetheless, since enzymes binding with chemical
reactants are called substrates ($m_{i}$ in the figure), for each $v\in V$ in
a substrate graph represent multiple reactants together with enzymes as a
single vertex. For each $e\in E$ in a substrate graph describes a reaction
between the substrates.

###  1.2 K-partite Graph

A K-partite graph is a graph whose vertices can be partitioned into $K$
different disjoint sets. Bipartite graphs are specific types of K-partite
graphs, where $K=2$, and are common representation for biological pathways. By
definition, each vertex in a bipartite graph can be either categorized as a
metabolite vertex ($m_{i}$) or a reaction vertex ($R_{i}$), but not both.

Definition: A graph $G$ is bipartite if and only if there exists a vertex
partition, where for each $v\in V$, $V=P_{1}\cup P_{2}$ and $P_{1}\cap
P_{2}=\emptyset$. Each edge $e=(v_{i},v_{j})\in E$, $v_{i}\in P_{1}$ and
$v_{j}\in P_{2}$ so that it is guaranteed that the end vertices of an edge do
not belong to the same vertex set.

###  1.3 Hypergraph

In principle, a hypergraph is a more intuitive and direct representation of
biological pathways. A hyperedge in a hypergraph can refer to a single
reaction, in which participating metabolites are involved. Although
hypergraphs can be converted into bipartite graphs, and vice-versa, due to the
difficulty of management, only a few tools support hypergraph representations
[41].

Definition: A hypergraph $G=(V,E)$ includes a set of vertices, while $E$ is a
set of non-empty subsets of $V$, namely hyperedges. Formally, $E$ is a subset
of $P$, where $P$ is the power set of $V$.

The advantage of hypergraph representation is that users can immediately
understand if enzymes are involved in this reaction. While with the substrate
graph representation, it is hard to discriminate if the metabolite is a
metabolite or a specific type of enzymes accelerating the corresponding
reaction.

###  1.4 Reaction Graph

A reaction graph is also a simple graph structurally, but contains different
semantics in comparison to a substrate graph. Vertices here are distinct
reactions, and the edges are metabolites involved in the network.

Definition: Vertices $v\in V$ in a reaction graph $G$ represent reactions in
biological networks, edges $e\in E$ stand for metabolites involved.

A reaction graph is predominantly used for topological analysis, such as
shortest path analysis or centrality analysis, so that users can rank graphs
according to important concepts in the field [15].

###  1.5 Clustered Graph

As explained by Kivelä et al. [21], one can consider multilayer networks as a
superset for complex networks. _Clustered graphs_ is one of the interesting
subsets.

Definition: A clustered graph is a simple graph $G$ with additional grouping
information. Each $v\in V$ in a clustered graph belong to one or more clusters
$c\in C=\\{c_{1},c_{2},...,c_{k}\\}$. In other words, clusters can also form a
hierarchy using a cluster tree.

In some definitions of clustered graphs, all clusters in $C$ are disjoint and
form a partition of the vertex set $V$ [7]. Nonetheless, in the context of
biological pathways, clusters $c$ are not necessarily disjoint. For example,
ATP, a universal energy molecule occurring in mitochondria and cytoplasm, is
often used to drive several biological reactions. If we consider these
compartments as clusters, they are overlapping since ATP can be transported
from the mitochondria to the cytoplasm. Another example would be the
relationships of ATP in the biological ontology. Since ATP occurs in many
categories of biochemical reactions, including _Citric Acid Cycle_ and _Urea
Cycle_ , its representation should be covered by multiple clusters in the
model. In some cases, to simplify the visual complexity of clustered graphs,
biologists duplicate unimportant vertices (e.g., vertices with high degree) to
create a specific type of clustered graph, where aliases of an identical
vertex only belong to a corresponding cluster. In other words, clusters become
disjoint in this case.

###  1.6 Multilayer Graph

A _multilayer graph_ is a simple graph $G$ with additional layer information
to describe real-world properties of the network in a whole. Layers in
multilayer networks are used to describe the corresponding relationships,
where each of which records the property of the corresponding relationships.
In this paper, we follow the formal definition by McGee et al. [24].

Definition: Since each $v\in V$ can belong to several layers, we can consider
vertices as pairs $V_{M}\subseteq V\times L$, where $L$ is the set of
associated layers. Edges $E_{M}\subseteq V_{M}\times V_{M}$ indicate the
connectivity of pairs $(v_{i},l_{p})$, $(v_{j},l_{q})$. An edge is considered
as an intra-layer edge when $l_{p}=l_{q}$ or an inter-layer edge when
$l_{p}\neq l_{q}$, respectively. In biological networks, we would have
$L=\\{l_{1},l_{2},l_{3},...,l_{p}\\}$, where $l_{1}$ could be metabolites
occurring in mitochondria and $l_{2}$ could be metabolites existing in
cytoplasm, and so on. Note that some metabolites, such as $H_{2}O$, which
occurs in both mitochondria and cytoplasm, can be connected using an
interlayer edge. This formulation becomes powerful in the sense that it covers
existing concepts and can be further used as an intermediate form to transform
one concept to another, not only model wise but also visually [21, 13].

In practice, we can use multilayer graphs as a unified graph structure because
the graphs described in Sections 1.1-1.5 are specific subsets of multilayer
graphs. Thus, researchers can always convert the aforementioned graphs to
multilayer graphs and again convert to the target graph data structures. This
scheme allows us to perform a systematically consistent conversion to
different graph representations, as well as using the multilayer graph as a
standard diagram when comparing to other visual representation.

###  1.7 Stoichiometric Matrix (S Matrix)

Last but not least, in addition to using graphs, the aforementioned
relationship can be modelled using a stoichiometric matrix, namely _S Matrix_.
With this formulation, researchers can conduct some stoichiometry-based
pathway search methods which operate on the S matrix.

Definition: In a stoichiometric matrix, each data element (row) is a
metabolite and each dimension (column) corresponds to biological reactions.
The value can be positive or negative, which indicate the stoichiometric
consumption of reactants and products, respectively. Unlike a directed graph
representation that inherits the reversibility information of a reaction, this
concept needs to be defined along with additional constraints, for the
Stoichiometric Matrix representations [41]. Graphs and Matrices are sibling
representations of relationship models. Each of which has advantages and
disadvantages, and can be inter-converted as well.

##  2 Biological Pathway Visualization

The purpose of developing pathway visualization is to convey the underlying
knowledge of the data models to the users effectively. Ideally, the
visualization is expected to reflect this information intuitively.
Unfortunately, pathway visualization is still a challenging problem due to the
size and the complexity of biological models. Moreover, the heterogeneity of
biological networks makes the problem even worse, since it becomes difficult
to develop a unified framework that supports various types of information. In
this section, we aim to summarize the existing visualization techniques in
comparison to the models described in the previous section. The focus includes
both the overview diagram from the public biological databases and the
machine-generated pathway diagrams.

###  2.1 Diagrams from Biological Databases

Biological databases enable researchers to store, analyze, and share
biological information systematically [43, 33]. Metabolic pathway and protein
function databases often give an overview map [29] to firstly demonstrate the
high-level structures of the organized biological information. _BioCyc
Database Collection_ is a Pathway/Genome Database, which collects metabolic
and genome pathways of organisms. Its overview diagram, _Pathway Collage_
[31], contains user-specified pathways for an organism and can be arranged and
customized on their user interfaces. This is a semi-automatic approach [18],
where smaller graphs are automatically computed [19], followed by a manual
arrangement to avoid overlaps [31]. Vertices here are small-molecule
metabolites or proteins, and edges biochemical or transport reactions. The
underlying graph here is a multi-faceted clustered graph.

_BRENDA_ [4] is an information system, which allows the users to search,
filter, and retrieve enzyme functions and where a molecule is involved in the
enzyme functions. Since 2017, the team introduced an overview map of the
metabolic pathways, to visualize the enzyme and ligand information. The map
has been created manually drawn using the network editor Cytoscape [35], in
order to create a map which is familiar to biochemists and similar to the maps
in biological textbooks. The visualization is in a top-down fashion, where
they show the category (approx. seven categories) using color boxes, and
expanding the detailed pathways when clicking on the boxes. The current map
contains ligand and enzyme as vertices and edges as reactions; therefore, it
is a combination of clustered graph and bipartite graph representations.

_HMDB_ [42] contains detailed information about small molecule metabolites
found in the human body. The visualization suite, SMPDB [8], provides an
interactive interface to small human metabolic pathways, but the team did not
integrate them into an overview map. The pathways in the database are hand-
drawn pathways specific to humans. A vertex can be a metabolite or enzyme, and
edges are reactions in the biological processes. The difference here is that
the designer added background icons such as mitochondria or endoplasmic
reticulum, to describe the spatial location of the reactions in the physical
world. The visualization uses a bipartite graph representation with annotated
image labels.

_KEGG PATHWAY Database_ [17] collects manual-drawn pathway maps, which
represent our knowledge on the molecular reactions. The vertices in the
overview map are metabolites and edges are reactions, while the involved
enzymes are integrated in a singe edge representation. The diagram is clean
with only category information. The users need to click on a vertex or an edge
to explore further details due to the complexity of the diagram. This overview
diagram is a combination of clustered graph representation together with a
user interface. _MANET database_ [20] is a database that allows users to trace
the evolution of protein architecture in biomolecular networks. The interface
supports scripts, query, and statistical analysis. The researchers utilize the
pathway maps created from the KEGG database [17], and thus do not have their
own specification.

_Reactome_ [36] is the pioneering database, which incorporates fully automatic
layout algorithms to assist navigation of human metabolic pathways. The system
first provides a hierarchical visualization to show the relationship of
biological pathway categories. Once the user selects a category, the system
will zoom in and show a detailed pathway diagram. The collection of pathways
is organized to show the relationship of pathway categories, then the detailed
pathways, unlike the other databases, which tend to create a big overview map
based on species. Vertices represent the participants of reactions, which are
categorized into ten types, including complex, protein, and RNA. Edges in the
pathway diagrams are reactions. The database uses a hypergraph representation
for the information.

_ReconMap_ [29] is developed based on the genome-scale reconstruction of human
metabolism, and provides an interface to access its content and associated
omics data and simulation results. It is known as the largest hand-crafted
human metabolic pathway map that facilitates an overview of the major
biological processes in the human body. The vertices in the diagram are
metabolites which are categorized based on their semantics, and edges
represent biological processes. This overview diagram is arranged in a way
that reactions happen in the same compartment in a cell are strictly placed
within a certain representative region. To achieve this, some high-degree
vertices are duplicated. This diagram is a combination of a clustered graph
with disjoint clusters and a hypergraph representation.

_WikiPathways_ [37] is another collection of knowledge concerning biological
pathways. Similar to HMDB, WikiPathways collects many promising pathway
information but does not visually integrate them. Vertices here are
metabolites and edges are reactions. The diagrams are using hypergraph
representation.

Other well-known pathway diagrams (which are not databases) such as _Roche
Biochemical Pathways_ [25], even embed chemical structures to guarantee the
high quality of the pathway map. The diagram is considered as a hypergraph
representation with large number of vertex annotations.

###  2.2 Software-Assisted Pathway Diagrams

Since new biological pathways are unceasingly introduced and added to pathway
databases, pathway visualization has developed a series of variations to
support pathway analysis. Recently, Murray et al. [27] proposed a
visualization task taxonomy for the analysis of biological pathway data. They
summarize that relationship tasks are considered as the most common and
essential tasks in their investigation, while they also point out that
existing network visualization tools are not that fully suitable for
biological pathways due to their limited capability [39, 30]. The more
constraints added to the layout, the higher running time is needed. Since the
layout problem is still resource-consuming, most of the techniques can handle
graphs with up to $500$ vertices.

For this reason, pathway visualization is commonly categorized into two types,
which are an interactive exploration scheme on a static map, and a fully
automatic layout approach. For the first case, the static map must be created
in advance. Scientists need to either use pathway editors, such as
CellDesigner [9], SBGN-ED [22], or Newt [34] to create the pathway diagrams,
or adopt conventional network analysis tools, such as Gephi [1] and Cytoscape
[35] to apply standard layout approaches. While the maps are often of better
visual quality, creating static maps often requires a lengthy trial and error
process, until the design is finally satisfactory. With a nice hand-crafted
map, biologists can rely on powerful user interactions with the maps to
support a comprehensive understanding of the datasets. For example, basic
interactions such as semantic zooming [10] and visual aggregation [5, 6] have
been investigated to analyze large networks. Note that interaction has been
proved useful for large network visualization, but it may also generate extra
overhead for doing simple connectivity tasks [45]. Nevertheless, interaction
provides us an alternative to expand and collapse the visualization in order
to navigate and access our target of interests.

On the other hand, the second type requires sophisticated layout algorithms by
introducing several pathway-specific constraints. Several research works focus
on the visually pleasing and well readable layout of small biological
networks. Gerasch et al. rebuild small KEGG pathway maps [12] by removing
unnecessary elements in the diagram, to conserve the original KEGG layout. The
approach is built on top of hierarchical sub-networks to support user
interaction. Jenson and Papi present MetDraw, an approach that generates
pathway diagrams using a familiar graph library GraphViz, together with
overlaying omics data [16]. Li and Kurata formulate the pathway diagram using
an annealing procedure [23].

The idea is to arrange vertices on a 2D square grid to exhibit cluster
information and produce a compact diagram. Bachmaier et al. [2] also summarize
several pathway layout approaches extended from the well-known force-directed
technique. Wu et al. [44] introduced a new design for biological pathways,
where an urban map metaphor is used to maintain the readability of low-level
and high-level relationship for larger networks, such as human metabolic
pathways. Several pathway visualization toolkits are also investigated in the
state-of-the-art literature [38, 32, 11, 2]. Most automatic layout approaches
rely on constrained layout techniques, which often assume the input graph is
simple and thus restricts the visual representation of biological pathways.
Since the multilayer network concept is considered as a framework for the
graph formulation of a complex system, it thus can serve as an intermediate
structure to convert between different graph models and visualizations.

##  3 Future Challenges of Pathway Visualization

After investigating the commonly used pathway databases, we summarize the
challenges as follows:

  * Structural Motifs: Structural network motifs are simplified and representative patterns that show the most common structures in the biological networks. Besides, the motifs that abstract the shapes of metabolites can provide insight during simulation and analysis [25]. One interesting aspect of these motifs is that they provide a visual summary for the structures so that the visualization is clean and readers can intuitively understand the knowledge behind it [40]. Technically, this introduces another challenge since the screen space reserved for the motifs should be balanced with the network layout, which implies a difficult layout challenge for multilayer networks.

  * Global vs. Local Relationship for Biological Tasks: As reported by Murray et al. [27], detailed relationship information, as well as the corresponding aspects, are expected to be clearly presented for analysis or educational purposes in a pathway visualization. This implies that any well-organized layer structures should be compatible with the high-level and detailed-level connectivity of biological pathways. Aside from pathway modeling, a visual hierarchy in multilayer networks should be explicitly developed in order to maximally reduce the cognitive work load of the visualization users.

  * Scalability and Interactivity: Scalability and interactivity are key factors for all of the visualization techniques. This is especially significant for pathway analysis because the new findings impact the field frequently. A typical example is the study of _glucose_ , which was considered as a fast supply of energy initially but later on is also found to have a strong activation of cancer metabolism. This changes the topology of the graph along time. We, therefore, have a strong need for dynamic layout approaches, and this must be applied to large graphs since there are thousands of metabolites and reactions in human metabolism.

  * Application-Oriented Approach: In the past decades, scientists often appreciate more general visualization techniques for networks so that they can be easily applied to different research fields. Nonetheless, since the field in biology is becoming more and more complex, and several properties in the domain may not exist in other domains, a domain-specific approach is required. This is because a general approach is not specialized enough to support the regular analysis, which also generates a lot of overhead for domain scientists.

  * Dynamics and Time-dependency: Time is another important factor. How to incorporate time as an axis in multilayer networks is still an open research question because it adds another complexity in both modeling and visualization.

  * Uncertainty: The current biological network is deterministic based on experiment settings. However, in the real-world case, some pathway interactions can be significant, while some are not. This information could provide more insight into the effectiveness of drugs during development and should be considered as one primary direction in the next research generation.

  * Comparative vis, translation between layout and visualization: Since the data collected in biology is heterogeneous, researchers would like to compare similar and dissimilar pathways across different organisms, cell compartments, and species, to investigate potential solutions for the interesting research questions. Multilayer networks facilitate this opportunity since they integrate the heterogeneity into a standard scheme, and therefore enable researchers to retrieve and provide a framework for the application of advanced visualization techniques.

##  4 Conclusion and Future Work

In this paper, we summarize the roles and challenges of various graph
structures used in biological pathway modeling and visualization. Meanwhile,
we collect frequently used hand-crafted and machine-generated pathway maps, to
give an overview on how the graph, as well as multilayer networks, can be
employed in the domain. Finally, we summarize the potential research
challenges and directions for multilayer network visualization, in the context
of biological pathways.

In the future, we attempt to investigate, step-by-step, the aforementioned
challenges, to further develop a unified biological pathway visualization
technology. Our first task is to conduct a general framework that integrates
various graph models by using multilayer networks as central data structures.
This can be an exciting research direction in the sense that scientists can
explicitly select and switch the corresponding graph visualization based on
their preferences. This also facilitates the possibility to convert or
translate among different network visualizations to cognitively support data
analysis for various purposes.

###### Acknowledgements.

The project has received funding from the EU Horizon 2020 research and
innovation programme under the MSCA grant No. 747985, from the Vienna Science
and Technology Fund (WWTF) grant No. VRG11-010, from the Austrian Science Fund
(FWF) grant No. P31119, and from King Abdullah University of Science and
Technology (KAUST) through award BAS/1/1680-01-01.

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# Products of Three $k-$Generalized Lucas Numbers as Repdigits

Alaa ALTASSAN and Murat ALAN
Department of Mathematics, King Abdulaziz University,
21589, Jeddah, Saudi Arabia<EMAIL_ADDRESS>
Department of Mathematics, Yildiz Technical University,
34210, Istanbul, Turkey<EMAIL_ADDRESS>
###### Abstract

Let $k\geq 2$ and let $(L_{n}^{(k)})_{n\geq 2-k}$ be the $k-$generalized Lucas
sequence with certain initial $k$ terms and each term afterward is the sum of
the $k$ preceding terms. In this paper, we find all repdigits which are
products of arbitrary three terms of $k-$generalized Lucas sequences. Thus, we
find all non negative integer solutions of Diophantine equation
$L_{n}^{(k)}L_{m}^{(k)}L_{l}^{(k)}=a\left(\dfrac{10^{d}-1}{9}\right)$ where
$n\geq m\geq l\geq 0$ and $1\leq a\leq 9.$

Key Words: $k-$generalized Lucas sequences; $k-$Lucas numbers; repdigits;
linear forms in logarithms.
2010 Mathematics Subject Classification: 11B37, 11B39, 11J86.

##  1 Introduction

Let $k\geq 2$ be an integer. The $k-$generalized Lucas sequence or, for
simplicity, the $k-$Lucas sequence is defined by the recurrence relation as

$L_{n}^{(k)}=L_{n-1}^{(k)}+\cdots+L_{n-k}^{(k)}\quad\text{for all}\quad n\geq
2$

with the initial condition $L_{0}^{(k)}=2$, $L_{1}^{(k)}=1$ for all $k\geq 2$
and $L_{2-k}^{(k)}=\cdots=L_{-1}^{(k)}=0$ for all $k\geq 3.$ For $k=2,$ this
sequence is the classical Lucas sequence, and in this case we omit the
superscript $(k)$ in the notation.

Recall that, a positive integer is called a repdigit if it has only one
distinct digit in its decimal expansion. These are the numbers of the form
$a{(10^{d}-1)/9}$ for some $d\geq 1$ and $1\leq a\leq 9.$ In recent years,
many authors have worked on problems involving relations between terms of some
binary recurrence sequences and repdigits [9, 10, 13, 17, 22]. Some authors
extended these problems to the case involving order $k$ generalization of
these binary recurrence sequences [1, 3, 4, 6, 8, 18, 24, 23]. In fact, in
[16], Luca found all repdigits in Lucas sequence whereas in [7], the authors
extend this result to the $k-$Lucas sequences. Recently, in [12], the authors
determined all repdigits which are products of two Lucas numbers. In this
paper, we continue to search in this line and extend some previous works, by
examining the product of arbitrary three $k-$Lucas numbers and repdigits. More
precisely, we find all repdigits which are product of three $k-$Lucas numbers.
Our main result is stated in the following theorem.

###### Theorem 1

If the Diophantine equation

$L_{n}^{(k)}L_{m}^{(k)}L_{l}^{(k)}=a\left(\dfrac{10^{d}-1}{9}\right),\quad
d\geq 2\quad n\geq m\geq l\geq 0\quad\text{and}\quad 1\leq a\leq 9$ (1)

has solutions in non-negative integers, then

$\displaystyle(k,n,m,l,a,d)\in$
$\displaystyle\\{(2,5,0,0,4,2),(2,5,1,0,2,2),(2,5,1,1,1,2),(2,5,2,0,6,2),$
$\displaystyle(2,5,2,1,3,2),(2,5,2,2,9,2),(2,5,3,0,8,2),(2,5,3,1,4,2),$
$\displaystyle(2,5,4,1,7,2),(4,5,0,0,8,2),(4,5,1,0,4,2),(4,5,1,1,2,2),$
$\displaystyle(4,5,2,1,6,2)\\}.$

###### Corollary 2

If the product of any two $k-$Lucas numbers is a repdigit with at least two
digits then this product is one of the following up to order of the factors:

$\displaystyle L_{5}^{(2)}L_{0}^{(2)}=11\cdot 2=22,\quad$ $\displaystyle\quad
L_{5}^{(2)}L_{1}^{(2)}=11\cdot 1=11,\quad$ $\displaystyle\quad
L_{5}^{(2)}L_{2}^{(2)}=11\cdot 3=33,$ $\displaystyle
L_{5}^{(2)}L_{3}^{(2)}=11\cdot 4=44,\quad$ $\displaystyle\quad
L_{5}^{(2)}L_{4}^{(2)}=11\cdot 7=77,\quad$ $\displaystyle\quad
L_{5}^{(4)}L_{0}^{(4)}=22\cdot 2=44,$ $\displaystyle
L_{5}^{(4)}L_{1}^{(4)}=22\cdot 1=22,\quad$ $\displaystyle\quad
L_{5}^{(4)}L_{2}^{(4)}=22\cdot 3=66.$

In particular, we state the following result for the Lucas sequence.

###### Corollary 3

The only repdigits that can be written as products of three Lucas numbers are
2, 1, 3, 4, 7, 11, 22, 33, 44, 66, 77, 88 and 99 .

The proof depends on some techniques on lower bounds for linear forms in
logarithms of algebraic numbers. Also, we use a version of reduction algorithm
due to Dujella and Pethő [11], which is in fact originally introduced by Baker
and Davenport in [2].

##  2 The Tools

Let $\theta$ be an algebraic number, and let

$c_{0}x^{d}+c_{1}x^{d-1}+\cdots+c_{d}=c_{0}\prod_{i=1}^{d}(x-\theta^{(i)})$

be its minimal polynomial over $\mathbb{Z},$ with degree $d,$ where the
$c_{i}$’s are relatively prime integers with $c_{0}>0,$ and the
$\theta^{(i)}$’s are the conjugates of $\theta$.

The logarithmic height of $\theta$ is defined by

$h(\theta)=\dfrac{1}{d}\left(\log
c_{0}+\sum_{i=1}^{d}\log\left(\max\\{\lvert\theta^{(i)}\rvert,1\\}\right)\right).$

In particular, the logarithmic height of a rational number $\theta=r/s,$ where
$r$ and $s$ are relatively prime integers and $s>0,$ is
$h(r/s)=\log\max\\{\lvert r\rvert,s\\}$. The following properties are very
useful in calculation of a logarithmic height :

  * $\bullet$

$h(\theta_{1}\pm\theta_{2})\leq h(\theta_{1})+h(\theta_{2})+\log 2$.

  * $\bullet$

$h(\theta_{1}\theta_{2}^{\pm 1})\leq h(\theta_{1})+h(\theta_{2})$.

  * $\bullet$

$h(\theta^{s})=\lvert s\rvert h(\theta),$ $s\in\mathbb{Z}$.

######  Theorem 4 (Matveev’s Theorem, [19])

Assume that $\alpha_{1},\ldots,\alpha_{t}$ are positive real algebraic numbers
in a real algebraic number field $\mathbb{K}$ of degree $d_{\mathbb{K}},$ and
let $b_{1},\ldots,b_{t}$ be rational integers, such that

$\Lambda:=\alpha_{1}^{b_{1}}\cdots\alpha_{t}^{b_{t}}-1,$

is not zero. Then

$\lvert\Lambda\rvert>\exp\left(C(t)d_{\mathbb{K}}^{2}(1+\log
d_{\mathbb{K}})(1+\log B)A_{1}\cdots A_{t}\right),$

where

$C(t):=-1.4\cdot 30^{t+3}\cdot t^{4.5}\quad\text{,}\quad B\geq\max\\{\lvert
b_{1}\rvert,\ldots,\lvert b_{t}\rvert\\},$

and

$A_{i}\geq\max\\{d_{\mathbb{K}}h(\alpha_{i}),\lvert\log\alpha_{i}\rvert,0.16\\},\quad\text{for
all}\quad i=1,\ldots,t.$

For a real number $\theta,$ we put
$\lvert\lvert\theta\rvert\rvert=\min\\{\lvert\theta-n\rvert:n\in\mathbb{Z}\\},$
which represents the distance from $\theta$ to the nearest integer. Now, we
cite the following lemma which we will be used to reduce some upper bounds on
the variables.

###### Lemma 5

[5, Lemma 1] Let $M$ be a positive integer, and let $p/q$ be a convergent of
the continued fraction of the irrational number $\tau$ such that $q>6M.$ Let
$A,B$ and $\mu$ be some real numbers with $A>0$ and $B>1$. If
$\epsilon:=\lvert\lvert\mu q\rvert\rvert-M\lvert\lvert\tau q\rvert\rvert>0$,
then there is no solution to the inequality

$0<\lvert u\tau-v+\mu\rvert<AB^{-w},$

in positive integers $u,v$ and $w$ with

$u\leq M\quad\text{and}\quad w\geq\frac{\log(Aq/\epsilon)}{\log B}.$

##  3 Properties of $k-$Lucas Numbers

The characteristic polynomial of the $k-$generalized Lucas sequence is

$\Psi_{k}(x)=x^{k}-x^{k-1}-\cdots-x-1,$

which is an irreducible polynomial over $\mathbb{Q}[x].$ The polynomial
$\Psi_{k}(x)$ has exactly one real distinguished root $\alpha(k)$ outside the
unit circle [20, 21, 25]. The other roots of $\Psi_{k}(x)$ are strictly inside
the unit circle [21]. This root $\alpha(k),$ say for simplicity $\alpha,$ is
located in the interval

$2(1-2^{-k})<\alpha<2\quad\text{for all}\quad k\geq 2.$

Let

$f_{k}(x)=\dfrac{x-1}{2+(k+1)(x-2)}.$

It is known that the inequalities

$1/2<f_{k}(\alpha)<3/4\quad\text{and}\quad\left\lvert
f_{k}(\alpha_{i})\right\rvert<1,\quad 2\leq i\leq k$ (2)

are hold, where $\alpha:=\alpha_{1},\cdots,\alpha_{k}$ are all the roots of
$\Psi_{k}(x)$ [5, Lemma 2]. In particular, we deduce that $f_{k}(\alpha)$ is
not an algebraic integer. In the same lemma, it is also proved that

$h(f_{k}(\alpha))<3\log{k}$ (3)

holds for all $k\geq 2,$ which will be useful in our study.

In [7], Bravo and Luca showed that

$L_{n}^{(k)}=\sum_{i=1}^{k}(2\alpha_{i}-1)f_{k}(\alpha_{i})(\alpha_{i})^{n-1}\quad\text{and}\quad\left\lvert
L_{n}^{(k)}-f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}\right\rvert<3/2,$

for all $k\geq 2.$ Thus,

$L_{n}^{(k)}=f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n),$ (4)

where $\lvert e_{k}(n)\rvert<3/2.$ As in the classical case where $k=2$, we
have the bounds

$\alpha^{n-1}\leq L_{n}^{(k)}\leq 2\alpha^{n},$ (5)

for all $n\geq 1$ and $k\geq 2$ [7]. It is known that

$L_{i}^{(k)}=3\cdot 2^{i-2},\quad\text{ for all }\quad 2\leq i\leq k.$ (6)

##  4 Proof of Theorem 1

First, we may directly derive some useful relations among the variables in
(1). Indeed, from the fact that

$10^{d-1}<a\left(\dfrac{10^{d}-1}{9}\right)<10^{d},$

together with (5), we find the following two relations

$\left(\dfrac{1+\sqrt{5}}{2}\right)^{n-3}\leq\alpha^{n+m+l-3}<10^{d}\Rightarrow\dfrac{n-3}{5}<d,$
(7)

and

$10^{d-1}<8\alpha^{n+m+l}<2^{n+m+l+3}\Rightarrow d<n+2\leq 2n,$ (8)

for all $n\geq 2.$

Before further calculations, by a computer search, we checked all repdigits
having at least two digits and belonging to the products
$L_{n}^{(k)}L_{m}^{(k)}L_{l}^{(k)}$ in the range $2\leq k\leq 25$ and $0\leq
l\leq m\leq n\leq 25,$ to find the variables given in Theorem 1. So from now
on, we will take $\max\\{k,n\\}>25.$

###  4.1 The Case $n\leq k.$

By substituting (6) in (1), we get

$3^{5}\cdot 2^{n+m+l-6}=a({10^{d}-1}).$

Then, as ${10^{d}-1}$ is odd, we find $n+m+l-6\leq 3,$ and hence we get $n\leq
9.$ Since $L_{9}^{(k)}\leq 384$ for all $k\geq 2,$ by a computer search, we
found that there is no solution of (1) when $n\leq k.$

Now assume that $n\geq k+1.$

###  4.2 The Case $n\geq k+1.$

Note that, since $\max\\{k,n\\}=n>25,$ from (7), we may take $d>4.$ First, by
(4), we rewrite (1) as

$\left(f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n)\right)\left(f_{k}(\alpha)(2\alpha-1)\alpha^{m-1}+e_{k}(m)\right)\\\
\left(f_{k}(\alpha)(2\alpha-1)\alpha^{l-1}+e_{k}(l)\right)=a10^{d}/9-a/9.$

Thus, we may write

$f_{k}(\alpha)^{3}(2\alpha-1)^{3}\alpha^{n+m+l-3}-a10^{d}/9=U_{1}-a/9,$

where

$\displaystyle U_{1}=$ $\displaystyle
f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{-2}\left(\alpha^{n+m}e_{k}(l)+\alpha^{n+l}e_{k}(m)+\alpha^{m+l}e_{k}(n)\right)+$
$\displaystyle
f_{k}(\alpha)(2\alpha-1)\alpha^{-1}\left(\alpha^{n}e_{k}(l)e_{k}(m)+\alpha^{m}e_{k}(n)e_{k}(l)+\alpha^{l}e_{k}(n)e_{k}(m)\right)+$
$\displaystyle e_{k}(n)e_{k}(m)e_{k}(l).$

Then, by dividing both sides by
$f_{k}(\alpha)^{3}(2\alpha-1)^{3}\alpha^{n+m+l-3},$ and taking the absolute
values, we get

$\left\lvert\Lambda_{1}\right\rvert<\dfrac{15}{\alpha^{l-3}},$ (9)

where

$\Lambda_{1}:=\alpha^{-(n+m+l-3)}10^{d}f_{k}(\alpha)^{-3}(2\alpha-1)^{-3}\dfrac{a}{9}-1.$

To get the above result, we used the facts that

$\dfrac{\left\lvert
U_{1}\right\rvert}{f_{k}(\alpha)^{3}(2\alpha-1)^{3}\alpha^{n+m+l-3}}\leq
3\dfrac{3/2}{\alpha^{l-1}}+3\dfrac{9/4}{\alpha^{2l-2}}+\dfrac{27/8}{\alpha^{n+m+l-3}}\leq\dfrac{107/8}{\alpha^{l-3}}$

and

$\dfrac{a}{9}\dfrac{1}{f_{k}(\alpha)^{3}(2\alpha-1)^{3}\alpha^{n+m+l-3}}\leq\dfrac{1}{\alpha^{l-3}}.$

Also, we were taking into account that $1<f_{k}(\alpha)(2\alpha-1).$

We turn back to (1), and write it as

$\left(f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n)\right)\left(f_{k}(\alpha)(2\alpha-1)\alpha^{m-1}+e_{k}(m)\right)L_{l}^{(k)}=a10^{d}/9-a/9.$

Thus, we may write

$f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{n+m-2}L_{l}^{(k)}-a10^{d}/9=U_{2}L_{l}^{(k)}-a/9,$

where

$U_{2}=f_{k}(\alpha)(2\alpha-1)\alpha^{-1}\left(\alpha^{n}e_{k}(m)+\alpha^{m}e_{k}(n)\right)+\\\
e_{k}(n)e_{k}(m).$

This time, we divide both sides by
$f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{n+m-2}L_{l}^{(k)},$ and get

$\left\lvert\Lambda_{2}\right\rvert<\dfrac{25/4}{\alpha^{m-2}},$ (10)

where

$\Lambda_{2}:=\alpha^{-(n+m-2)}10^{d}f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}{L_{l}^{(k)}}^{-1}\dfrac{a}{9}-1.$

To get the above result, we used the facts that

$\dfrac{\lvert
U_{2}\rvert}{f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{n+m-2}}\leq
2\dfrac{3/2}{\alpha^{m-1}}+\dfrac{9/4}{\alpha^{n+m-2}}\leq\dfrac{21/4}{\alpha^{m-2}}$

and

$\dfrac{a}{9}\dfrac{1}{f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{n+m-2}L_{l}^{(k)}}\leq\dfrac{1}{\alpha^{m-2}}.$

We rearrange (1) again by rewriting it as

$\left(f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}+e_{k}(n)\right)L_{m}^{(k)}L_{l}^{(k)}=a10^{d}/9-a/9.$

So, we have that

$f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}L_{m}^{(k)}L_{l}^{(k)}-a10^{d}/9=-e_{k}(n)L_{m}^{(k)}L_{l}^{(k)}-a/9.$

Dividing by $f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}L_{m}^{(k)}L_{l}^{(k)}$ and
taking the absolute values, we get that

$\left\lvert\Lambda_{3}\right\rvert<\dfrac{5/2}{\alpha^{n-1}},$ (11)

where

$\Lambda_{3}:=\alpha^{-(n-1)}10^{d}f_{k}(\alpha)^{-1}(2\alpha-1)^{-1}{L_{m}^{(k)}}^{-1}{L_{l}^{(k)}}^{-1}\dfrac{a}{9}-1.$

To get the above result, we used only the fact that

$\dfrac{\lvert
e_{k}(n)\rvert}{f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}}+\dfrac{a}{9}\dfrac{1}{f_{k}(\alpha)(2\alpha-1)\alpha^{n-1}L_{m}^{(k)}L_{l}^{(k)}}\leq\dfrac{3/2+1}{\alpha^{n-1}}=\dfrac{5/2}{\alpha^{n-1}}.$

For each $\Lambda_{1},$ $\Lambda_{2}$ and $\Lambda_{3},$ we will apply Theorem
4 by taking $t=3$. In each case, we take $\eta_{1}:=\alpha,$ $\eta_{2}:=10$,
$b_{2}:=d,$ $b_{3}:=1,$ $\mathbb{K}=\mathbb{Q}(\alpha)$ which is a real number
field with degree $d_{\mathbb{K}}=k.$ Also, $(\lvert
b_{1}\rvert,\eta_{3})=(n+m+l-3,f_{k}(\alpha)^{-3}(2\alpha-1)^{-3}(a/9))$ for
$\Lambda_{1},$ $(\lvert
b_{1}\rvert,\eta_{3})=(n+m-2,f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}(a/9){L_{l}^{(k)}}^{-1})$
for $\Lambda_{2}$ and $(\lvert
b_{1}\rvert,\eta_{3})=(n-1,f_{k}(\alpha)^{-1}(2\alpha-1)^{-1}(a/9){L_{m}^{(k)}}^{-1}{L_{l}^{(k)}}^{-1})$
for $\Lambda_{3}.$ So, from (8), we will take $B$ as $3n,2n$ and $n+2$ for
$\Lambda_{1},$ $\Lambda_{2}$ and $\Lambda_{3},$ respectively.

Since $h(\alpha)=({1}/{k})\log{\alpha}$ and $h(10)=\log{10},$ we take
$A_{1}=\log\alpha$ and $A_{2}=k\log{10}$ for each $\Lambda_{i}.$ But the
parameter $A_{3}$ will be different for each $\Lambda_{i}$ and therefore we
calculate it for each $\Lambda_{i}$ separately.

To calculate the logarithmic height of $\eta_{3},$ we use the two facts,
$h(f_{k}(\alpha))<3\log{k}$ from (3), and $h(2\alpha-1)<\log{3}$ from [7, page
147]. Thus, we find

$\displaystyle h((a/9)f_{k}(\alpha)^{-3}(2\alpha-1)^{-3})$ $\displaystyle\leq
h(a/9)+3h(f_{k}(\alpha))+3h(2\alpha-1)$
$\displaystyle\leq\log{9}+9\log{k}+3\log{3}$
$\displaystyle\leq\log{243}+9\log{k}<17\log{k}.$

So, we take $A_{3}:=17k\log{k}$ for $\Lambda_{1}.$ Now we will show that
$\Lambda_{1}\neq 0.$ Assume that $\Lambda_{1}=0.$ Then, we get

$(a/9)10^{d}=f_{k}(\alpha)^{3}(2\alpha-1)^{3}\alpha^{(n+m+l-3)}.$

Now, we take the image of both sides of this relation by applying any one of
the automorphisms $\sigma_{i}:\alpha\rightarrow\alpha_{i}$ for any $i\geq 2.$
Then, by (2), we take the absolute values to get

$10^{3}\leq(a/9)10^{d}\leq\lvert f_{k}(\alpha_{i})\rvert^{3}\lvert
2\alpha_{i}-1\rvert^{3}\lvert\alpha_{i}\rvert^{(n+m+l-3)}\leq 27,$

which is clearly false. So, $\Lambda_{1}\neq 0$ and hence we may apply Theorem
4 to (9) which gives us a bound for $\Lambda_{1}.$ On the other hand, from
(9), we directly see that

$\log{\lvert\Lambda_{1}\rvert}<\log{15}-(l-3)\log{\alpha}.$

We combine these two results and get the following:

$(l-3)\log{\alpha}-\log{15}<-1.4\cdot 30^{6}\cdot 3^{4.5}\cdot
k^{2}(1+\log{k})(1+\log{3n})\log\alpha\cdot k\log{10}\cdot 17k\log{k}.$

Note that

$1+\log{k}<(5/2)\log{k}\quad\text{and}\quad 1+\log{3n}<1.6\log{3n}$

hold for all $k\geq 2$ and $n\geq 2.$ By taking into account these two facts,
we obtain that

$l<2.3\cdot 10^{13}\cdot k^{4}\log^{2}{k}\log{3n}.$ (12)

For the next step, we need to know that $\Lambda_{2}\neq 0.$ Assume
$\Lambda_{2}=0.$ Then, we get

$(a/9)10^{d}=f_{k}(\alpha)^{2}(2\alpha-1)^{2}\alpha^{(n+m-2)}{L_{l}^{(k)}}.$

Similarly for $\Lambda_{1},$ we get that

$(a/9)10^{d}\leq\lvert f_{k}(\alpha_{i})\rvert^{2}\lvert
2\alpha_{i}-1\rvert^{2}{L_{l}^{(k)}}\leq 9{L_{l}^{(k)}}.$

This inequality together with (1), implies that

$L_{n}^{(k)}L_{m}^{(k)}L_{l}^{(k)}\leq 9{L_{l}^{(k)}}-(a/9)<9{L_{l}^{(k)}}.$

Thus, we find that $L_{n}^{(k)}\leq 9,$ which is a contradiction to the choice
of $n>25.$ So $\Lambda_{2}\neq 0.$ Since

$\displaystyle h((a/9)f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}{L_{l}^{(k)}}^{-1})$
$\displaystyle\leq
h(a/9)+2h(f_{k}(\alpha))+2h(2\alpha-1)+h\left(L_{l}^{(k)}\right)$
$\displaystyle\leq\log{9}+6\log{k}+2\log{3}+\log{2\alpha^{l}}$
$\displaystyle<\log{162}+6\log{k}+l\log{\alpha}<14\log{k}+l\log{2},$

by using the upper bound of $l$ given in (12), we find that

$h((a/9)f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}{L_{l}^{(k)}}^{-1})\leq 1.6\cdot
10^{13}\cdot k^{4}\log^{2}{k}\log{3n}.$

Thus, we may take

$A_{3}:=1.6\cdot 10^{13}\cdot k^{5}\log^{2}{k}\log{3n}$

for $\Lambda_{2}.$ Now, we are ready to apply Theorem 4 to (10). By using the
bound

$\log{\lvert\Lambda_{2}\rvert}<\log{(25/4)}-(m-2)\log{\alpha},$

we obtain

$m<2.2\cdot 10^{25}\cdot k^{8}\log^{3}{k}\log^{2}{3n}.$ (13)

To find the value of $A_{3}$ for $\Lambda_{3},$ we need the following
logarithmic height

$\displaystyle
h((\dfrac{a}{9})f_{k}(\alpha)^{-1}(2\alpha-1)^{-1}{L_{m}^{(k)}}^{-1}{L_{l}^{(k)}}^{-1})$
$\displaystyle\leq h(\dfrac{a}{9})+h(f_{k}(\alpha))+h(2\alpha-1)$
$\displaystyle+h\left(L_{l}^{(k)}L_{m}^{(k)}\right)$
$\displaystyle\leq\log{9}+3\log{k}+\log{3}+\log{4\alpha^{l+m}}$
$\displaystyle\leq\log{108}+3\log{k}+(l+m)\log{2}$
$\displaystyle<10\log{k}+2m\log{2}.$

Therefore, by using the bound of $m$ obtained in (13), we take

$A_{3}:=3.1\cdot 10^{25}\cdot k^{9}\log^{3}{k}\log{3n},$

for $\Lambda_{3}.$ The similar argument to the one that we used for
$\Lambda_{2},$ shows that $\Lambda_{3}\neq 0.$ Thus, by applying Theorem 4 to
$\Lambda_{3},$ together with the inequality (11), we obtain that

$3n<1.3\cdot 10^{38}\cdot k^{12}\log^{4}{k}\log^{3}{3n}.$

We need the following lemma from [14] to get a bound of $n$ depending on $k.$

###### Lemma 6

Let $s\geq 1$ and $T>(4s^{2})^{s}.$ If

$\dfrac{x}{(\log{x})^{s}}<T,\quad\text{then}\quad x<2^{s}T(\log{T})^{s}.$

We take $s=3$ and $T:=1.3\cdot 10^{38}\cdot k^{12}\log^{4}{k}.$ Then,

$\displaystyle(\log(T))^{3}$ $\displaystyle<(\log(1.3)+38\log
10+12\log{k}+4\log(\log{k}))^{3}$ $\displaystyle<170^{3}\log^{3}{k}.$

Thus, we get the following bound :

$n<1.8\cdot 10^{45}k^{12}\log^{7}{k}.$ (14)

Now, we treat the cases $k\leq 650$ and $k>650$ separately.

###  4.3 The Case $k\leq 650$

Let $2\leq k\leq 650.$ Then, from (14), $n$ is also bounded. Let

$\Gamma_{1}:=-(n+m+l-3)\log\alpha+d\log
10+\log(f_{k}(\alpha)^{-3}(2\alpha-1)^{-3}\cdot(a/9)).$

Then

$\lvert\Lambda_{1}\rvert:=\left\lvert\exp(\Gamma_{1})-1\right\rvert<15/\alpha^{l-3}.$

We claim that ${l}\leq 300.$ Suppose that this is not the case. Then,
$15/\alpha^{l-3}<1/2$ and therefore $\lvert\Gamma_{1}\rvert<30/\alpha^{l-3}.$
So, we have

$0<\left\lvert(n+m+l-3)\dfrac{\log\alpha}{\log
10}-d+\mu_{(k,a)}\right\rvert<30/\alpha^{l-3}\log 10.$ (15)

where

$\mu_{(k,a)}:=-\dfrac{\log(f_{k}(\alpha)^{-3}(2\alpha-1)^{-3}\cdot(a/9))}{\log
10}.$

For each $2\leq k\leq 650$, we take $M_{k}:=5.4\cdot
10^{45}k^{12}\log^{7}{k}>3n>n+m+l-3$ and $\tau_{k}=\dfrac{\log{\alpha}}{\log
10}.$ Also, for each $k$, we find a convergent $p_{i}/q_{i}$ of the continued
fraction of irrational number $\tau_{k},$ such that $q_{i}>6M_{k}.$ Then, we
calculate

$\epsilon_{(k,a)}:=\lvert\lvert\mu_{(k,a)}q_{i}\rvert\rvert-
M_{k}\lvert\lvert\tau_{k}q_{i}\rvert\rvert$

for each $a\in\\{1,2,\ldots,9\\}.$ If $\epsilon_{(k,a)}<0,$ then we repeat the
same calculations for $q_{i+1}.$ For each $k,$ except for $(k,a)=(2,9),$ we
found an appropriate $q_{i},$ such that $\epsilon_{(k,a)}>0.$ In fact,
$0.000308<\epsilon_{(k,a)}.$

Thus, from Lemma 5, we find an upper bound on $l-3$ for each $2\leq k\leq 650$
such that none of these bounds is greater than $297.$ So, when
$(k,a)\neq(2,9),$ we conclude that $l\leq 300,$ as we claimed.

If $(k,a)=(2,9),$ then by taking into account $\alpha^{2}=\alpha+1$ for $k=2,$
we see that $3\log{\alpha}+\log(f_{2}(\alpha)^{-3}(2\alpha-1)^{-3}=0,$ and
hence $\epsilon_{(2,9)}=0$ for all $q_{i}.$ Thus, Lemma 5 can not be
applicable in this case. Now, we will use directly the properties of
continuous fractions. From (15), we get

$\left\lvert\dfrac{\log\alpha}{\log
10}-\dfrac{d}{n+m+l}\right\rvert<\dfrac{30}{(n+m+l)\alpha^{l-3}\log 10}.$ (16)

If $\dfrac{30}{(n+m+l)\alpha^{l-3}\log 10}<\dfrac{1}{2(n+m+l)^{2}},$ then
$\dfrac{d}{n+m+l}$ is a convergent of continued fraction expansion of the
irrational number $\dfrac{\log\alpha}{\log 10},$ say $\dfrac{p_{i}}{q_{i}}.$
Since $p_{i}$ and $q_{i}$ are relatively prime, we deduce that $q_{i}\leq
n+m+l\leq 3n<3\cdot 5.7\cdot 10^{47}<1.8\cdot 10^{48}.$

By Maple, we see that $i<101.$ Let
$[a_{0},a_{1},a_{2},a_{3},a_{4},\ldots]=[0,4,1,3,1,1,1,6,\ldots]$ be the
continued fraction expansion of $\dfrac{\log\alpha}{\log 10}.$ Then,
$\max\\{a_{i}\\}=106$ for $i=0,1,2,\ldots,101.$ Thus, from the well-known
property of continued fractions, see for example [15, Theorem 1.1.(iv)], we
write

$\dfrac{1}{106\cdot(n+m+l)^{2}}\leq\dfrac{30}{(n+m+l)\alpha^{l-3}\log 10}.$

Therefore, from the inequality

$\alpha^{l-3}<\dfrac{106\cdot 30\cdot(n+m+l)}{\log 10}<2.5\cdot 10^{52},$

we find that $l<250.$ This bound is valid also in the case

$\dfrac{30}{(n+m+l)\alpha^{l-3}\log 10}\geq\dfrac{1}{2(n+m+l)^{2}}.$

So, whether $(k,a)=(2,9)$ or not, we have that $l\leq 300.$

Let

$\Gamma_{2}:=-(n+m-2)\log\alpha+d\log
10+\log(f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}{L_{l}^{(k)}}^{-1}(a/9)).$

Then,

$\lvert\Lambda_{2}\rvert:=\left\lvert\exp(\Gamma_{2})-1\right\rvert<(25/4)/\alpha^{m-2}.$

Next, we show that there is a bound for $m.$ Suppose that $m>10.$ Then,
$(25/4)/\alpha^{m-2}<1/2$ and therefore
$\lvert\Gamma_{2}\rvert<(25/2)/\alpha^{m-2}.$ So, we have

$0<\left\lvert(n+m-2)\dfrac{\log\alpha}{\log
10}-d+\mu_{(k,l,a)}\right\rvert<25/(2\alpha^{m-2}\log 10),$ (17)

where

$\mu_{(k,l,a)}=-\dfrac{\log(f_{k}(\alpha)^{-2}(2\alpha-1)^{-2}{L_{l}^{(k)}}^{-1}(a/9))}{\log
10}.$

For each $2\leq k\leq 650$, we take $M_{k}:=3.6\cdot
10^{45}k^{12}\log^{7}{k}>2n>n+m-2$ and $\tau_{k}=\dfrac{\log{\alpha}}{\log
10}.$ Also, for each $k,$ like in the previous calculations, we find an
appropriate $q_{i}>6M_{k}.$ Then, we calculate
$\epsilon_{(k,l,a)}:=\lvert\lvert\mu_{(k,l,a)}q_{i}\rvert\rvert-
M_{k}\lvert\lvert\tau_{k}q_{i}\rvert\rvert$ for each $1\leq a\leq 9$ and
$0\leq l\leq 300.$ We find that $\epsilon_{(k,l,a)}>0.132192\cdot 10^{-6}$ for
all $3\leq k\leq 650.$

Thus, from Lemma 5, we find an upper bound on $m-2,$ for each $3\leq k\leq
650,$ and none of these bounds is greater than $298.$

Except for $(k,l,a)=(2,1,9),$ if $k=2,$ then $\epsilon_{(2,l,a)}>0.327364\cdot
10^{-38},$ and hence from Lemma 5, $m\leq 606.$

If $(k,l,a)=(2,1,9),$ then
$2\log{\alpha}+\log(f_{2}(\alpha)^{-2}(2\alpha-1)^{-2})=0$ and hence, from
(17) we find that

$\left\lvert\dfrac{\log\alpha}{\log
10}-\dfrac{d}{n+m}\right\rvert<\dfrac{(25/2)}{(n+m)\alpha^{m-2}\log 10}.$

Now, we follow the same argument as we did for (16) to show that $m<250.$ So,
we have that $m\leq 300$ if $3\leq k\leq 650,$ and $m\leq 606$ if $k=2.$

Let

$\Gamma_{3}:=-(n-1)\log\alpha+d\log
10+\log(f_{k}(\alpha)^{-1}(2\alpha-1)^{-1}{L_{l}^{(k)}}^{-1}{L_{m}^{(k)}}^{-1}(a/9)).$

Thus,

$\lvert\Lambda_{3}\rvert:=\left\lvert\exp(\Gamma_{3})-1\right\rvert<5/(2\alpha^{n-1})<(1/2).$

Then $\lvert\Gamma_{3}\rvert<5/\alpha^{n-1},$ and we have

$0<\left\lvert(n-1)\dfrac{\log\alpha}{\log
10}-d\right\rvert<5/(\alpha^{n-1}\log 10),$ (18)

where

$\mu_{(k,l,m,a)}=-\dfrac{\log(f_{k}(\alpha)^{-1}(2\alpha-1)^{-1}{L_{l}^{(k)}}^{-1}{L_{m}^{(k)}}^{-1}(a/9))}{\log
10}.$

This time, we take $M_{k}:=1.8\cdot 10^{45}k^{12}\log^{7}{k}>n-1$ and
$\tau_{k}=\dfrac{\log{\alpha}}{\log 10}.$ For each $k,$ like in the previous
calculations, we find an appropriate $q_{i}>6M_{k}.$ Then, we calculate
$\epsilon_{(k,l,m,a)}:=\lvert\lvert\mu_{(k,l,m,a)}q_{i}\rvert\rvert-
M_{k}\lvert\lvert\tau_{k}q_{i}\rvert\rvert$ for each $1\leq a\leq 9$ and
$0\leq l\leq m\leq 300.$ This is done by taking into account the facts that

$v_{2}\left(L_{l}^{(k)}L_{m}^{(k)}\right)\leq v_{2}(a)\quad\text{and}\quad
v_{5}\left(L_{l}^{(k)}L_{m}^{(k)}\right)\leq v_{5}(a)$ (19)

to reduce the unnecessary computations. Except for $(k,l,m,a)=(2,1,1,9),$ we
find $\varepsilon_{k}:=\min\\{\epsilon_{(k,l,m,a)}\\}$ for each $k.$ Thus,
from Lemma 5, we obtain an upper bound for $n-1$, and hence for $n,$ say
$n(k),$ for each $2\leq k\leq 650.$ Indeed, none of these bounds are greater
than $297.5$ when $k\geq 3.$ Some of these bounds are
$n(3)<230.93,n(100)<263.5$ $n(200)<273.95,$ $n(300)<284.7,$ $n(400)<281.1,$
$n(500)<284.9,$ $n(600)<293.97,$ $n(605)<297.4$ and $n(650)<290.3.$ Whereas
for $k=2,$ $n(2)<1216.$ In fact, even though $q_{101}>6M_{k}:=6M_{2},$ we have
to take $q_{305}$ for the condition $\epsilon_{(2,l,m,a)}>0$ and we find that
$\epsilon_{(2,l,m,a)}>0.134539\cdot 10^{-91}.$

If $(k,l,m,a)=(2,1,1,9),$ then
$\log{\alpha}+\log(f_{2}(\alpha)^{-1}(2\alpha-1)^{-1})=0$ and hence, from
(18), we find that

$0<\left\lvert\dfrac{\log\alpha}{\log
10}-\dfrac{d}{n}\right\rvert<\dfrac{5}{n\alpha^{n-1}\log 10}.$

Now, we follow the same calculations as we did for (16) to show that the bound
$n(2)<1216$ is valid.

Finally, we write a short computer programme to check that the variables
$n,m,k,l,a$ and $d$ are satisfying (1) by using the bounds $n(k)$ for $2\leq
k\leq 650$ together with (19) and (8). As a result, we find that there is no
new solution of (1) except for those that were given in Theorem 1.

###  4.4 The Case $k>650$

We cite the following lemma from [24, Lemma 2.6].

###### Lemma 7

If $k\geq 2$ and $n\geq k+1$ then

$L_{n}^{(k)}=3\cdot
2^{n-2}(1+\zeta(n,k)),\quad\text{where}\quad\lvert\zeta(n,k)\rvert<\dfrac{1}{2^{k/2}}.$

For $k>650,$ from (14), the inequality $n<2^{k/2}$ holds and hence from Lemma
7, we have

$L_{n}^{(k)}L_{m}^{(k)}L_{l}^{(k)}=27\cdot 2^{n+m+l-6}(1+\zeta(n,k))^{3}.$

Thus, by (1), we get

$(a/9)10^{d}-27\cdot 2^{n+m+l-6}=27\cdot
2^{n+m+l-6}(3\zeta(n,k)+3\zeta(n,k)^{2}+\zeta(n,k)^{3})+(a/9).$

Hence,

$0<\lvert\Lambda_{4}\rvert:=\left\lvert
2^{-(n+m+l-6)}10^{d}3^{-5}a-1\right\rvert<\dfrac{1}{2^{k/2-3}},$

since $(3\zeta(n,k)+3\zeta(n,k)^{2}+\zeta(n,k)^{3})<7/2^{k/2}.$

Now, we apply Theorem 4 to $\Lambda_{4}$ by taking
$(\eta_{1},b_{1})=(2,{-(n+m+l-6)}),$ $(\eta_{2},b_{2})=(10,d),$
$(\eta_{3},b_{3})=(3^{-5}a,1),$ $\mathbb{K}=\mathbb{Q}$ and $B:=3n.$ Thus, we
find that

$(\dfrac{k}{2}-3)\log{2}<1.4\cdot 30^{6}\cdot
3^{4.5}\cdot(1+\log{3n})\log{2}\cdot\log{10}\cdot 5\log{3}.$

From (14), we may take

$\displaystyle 1+\log{3n}$
$\displaystyle<1+\log{5.4}+23\log{10^{2}}+12\log{k}+7\log{\log{k}}$
$\displaystyle<45\log{k},$

and hence we get that $k<1.63\cdot 10^{14}\log{k},$ which implies that

$k<6\cdot 10^{15}.$

So, by (14), we have

$n<3.3\cdot 10^{245}.$

###  4.5 Reducing the Bound on k

Let

$\Gamma_{4}:=-(n+m+l-6)\log 2+d\log 10+\log(a/{3^{5}}).$ (20)

Then
$|\Lambda_{4}\rvert:=\left\lvert\exp(\Gamma_{4})-1\right\rvert<\dfrac{1}{2^{k/2-3}}<\dfrac{1}{2}.$
Hence, we have that $\lvert\Gamma_{4}\rvert<\dfrac{2}{2^{k/2-3}}.$ From (20),
we write

$0<\left\lvert(n+m+l-6)\dfrac{\log 2}{\log 10}-d-\dfrac{\log(a/{3^{5}})}{\log
10}\right\rvert<\dfrac{1}{2^{k/2-4}\log{10}}.$

Let $M:=10^{246}>3n>n+m+l-6,$ $\tau=\dfrac{\log{2}}{\log 10}$ and
$\mu_{a}:=-\dfrac{\log(a/{3^{5}})}{\log 10}.$

Then, the denominator of the $504th$ convergent of $\tau,$ say $q_{504},$
exceeds $6M$ and

$\epsilon_{a}:=\lvert\lvert\mu_{a}q_{505}\rvert\rvert-M\lvert\lvert\tau
q_{505}\rvert\rvert>0.05055,$

for each $a\in\\{1,2,\ldots,9\\}.$ Thus, by applying Lemma 5 with the
parameters $A:=\dfrac{1}{\log{10}},$ $B:=2$ and $w:={k/2-4}$, we find that
${k/2-4}<825.$ That is $k<1660,$ and thus, by (14), $n<9.8\cdot 10^{89}.$

We repeat the same reduction steps by taking $M:=2.94\cdot
10^{90}>3n>n+m+l-6.$ Then, $q_{184}>6M$ and we find that

$\epsilon_{a}:=\lvert\lvert\mu_{a}q_{184}\rvert\rvert-M\lvert\lvert\tau
q_{184}\rvert\rvert>0.009382.$

By Lemma 5, we find that ${k/2-4}<310.$ That is $k<630,$ which contradicts the
fact $k>650.$ So, we conclude that Equation (1) has no solution when $k>650.$
This completes the proof.

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