††thanks: Joint first author††thanks: Joint first author

# Anti-Diffusion in an Algae-Bacteria Microcosm:
Photosynthesis, Chemotaxis, and Expulsion

Praneet Prakash<EMAIL_ADDRESS>Department of Applied Mathematics and
Theoretical Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Yasa Baig<EMAIL_ADDRESS>Department of Applied Mathematics and Theoretical
Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
Francois J. Peaudecerf<EMAIL_ADDRESS>Institute de
Physique de Rennes, Universite Rennes, UMR 6251, F-35000 Rennes, France
Raymond E. Goldstein<EMAIL_ADDRESS>Department of Applied
Mathematics and Theoretical Physics, Centre for Mathematical Sciences,
University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

###### Abstract

In Nature there are significant relationships known between microorganisms
from two kingdoms of life, as in the supply of vitamin B12 by bacteria to
algae. Such interactions motivate general investigations into the spatio-
temporal dynamics of metabolite exchanges. Here we study by experiment and
theory a model system: a coculture of the bacterium B. subtilis, an obligate
aerobe that is chemotactic to oxygen, and a nonmotile mutant of the alga C.
reinhardtii, which photosynthetically produces oxygen when illuminated.
Strikingly, when a shaft of light illuminates a thin, initially uniform
suspension of the two, the chemotactic influx of bacteria to the
photosynthetically active region leads to expulsion of the algae from that
area. This effect arises from algal transport due to spatially-varying
collective behavior of bacteria, and is mathematically related to the
“turbulent diamagnetism” associated with magnetic flux expulsion in stars.

In the early 1880s the biologist Theodor Engelmann performed experiments that
were perhaps the first to use bacteria as sensors [1, 2, 3]. Several years
prior he made the first observation of bacterial chemotaxis toward oxygen, by
showing that putrefactive bacteria would migrate toward the chloroplasts of
the filamentous alga Spirogyra. He then determined the “action spectrum” of
photosynthesis—the wavelength-dependent rate of photosynthetic activity—by
passing sunlight through a prism and projecting the spectrum onto a
filamentous green algae held in a chamber that contained those self-same
bacteria, which gathered around the algae in proportion to the local oxygen
concentration, providing a direct readout of the oxygen production rate.

Although Engelmann’s system was engineered for a particular purpose, and at
first glance involves a one-way exchange of oxygen for the benefit of
bacteria, there are many examples of mutualistic exchanges between
microorganisms from two distinct kingdoms of life. One of the most significant
is that involving vitamin B12. In a landmark study [4], it was shown that a
significant fraction of green algae that require this vitamin for their
metabolism do not produce it, and as the ambient concentration of B12 in the
aqueous environment is so low, they instead acquire it from a mutualistic
relationship with bacteria, which benefit from a source of carbon.

The study of B12 transfer raises fascinating questions in biological physics
related to the interplay of metabolite production, chemotaxis, and growth [5],
including the issue of how organisms find each other and stay together in the
turbulent environment of the ocean, and how advection by fluid flows arising
from microorganism motility affects such mutualisms. As it is difficult to
control the production of vitamin B12, we sought to construct a system in
which the production of a chemical species needed by one member of an
interacting pair of organisms could be controlled by the experimentalist.
Taking motivation both from Engelmann’s experiments and the B12 system, we
introduce here a coculture [6] in which an obligate aerobic bacterium (one
that requires oxygen) that is chemotactic toward oxygen, coexists with a green
alga whose photosynthetic activity can be turned on and off simply by
controlling the external illumination.

We use the bacterium Bacillus subtilis, whose aerotaxis has been central in
the study of bioconvection [7] and in the discovery of “bacterial turbulence”
[8], the dynamical state of a concentrated suspension with transient,
recurring vortices and jets of collective swimming on scales large compared to
the individual bacteria. The alga species is the well-studied unicellular
Chlamydomonas reinhardtii, a model organism for biological fluid dynamics [9]
with readily available motility mutants useful in probing the role of swimming
in metabolite transfer. Together these define what we term the algae-bacteria-
chemoattractant system (ABC). While coupled population dynamics problems have
been studied from the familiar reaction-diffusion point of view in bacterial
range expansion [10], marine [11] and more general ecological contexts [12],
we show that there are physical effects that go beyond that standard level of
treatment. Chief among them is the way in which collective motion can act as a
“thermal bath” in enhancing the diffusivity of suspended particles [13, 14],
which in turn raises fundamental issues concerning generalizations of Fick’s
law [15, 16].

Figure 1: Experimental setup. (a) Schematic of system to illuminate a
coculture of a fluorescent strain of the bacterium B. subtilis and a flagella-
less mutant of the green alga C. reinhardtii, initially distributed uniformly
within the sample chamber. After a shaft of visible light illuminates the
central region the algae produce oxygen to which the bacteria are attracted.
(b) Algae and bacteria as observed through the brightfield and fluorescent
channels, respectively. (c) Yellow shading indicates the illuminated region,
with dark algae visible in the background.

A disarmingly simple experiment to understand the dynamics of the ABC system
involves a thin, quasi-two-dimensional suspension of non-motile algae and
fluorescently labelled bacteria at initially uniform concentrations. As
depicted in Fig. 1, a shaft of photosynthetically active light is cast on the
suspension, triggering oxygen production by the illuminated algae. As shown in
Figs. 2(a,b), this leads to chemotaxis of bacteria into the illuminated
region, producing a high concentration of bacteria. Remarkably, we find that
algae are then expelled from the illuminated region. Quantitative measurements
of the local bacterial dynamics in the the system show that this expulsion is
associated with a gradient of collective bacterial behavior from its peak at
the center, leading to a outward algal transport. On longer time scales, after
the algal expulsion, the bacterial concentration returns to uniformity as the
bacteria diffusive outward in the absence of chemotactic stimulus. We name
this Type I dynamics. Figures 2(c,d) show that at sufficiently high initial
bacterial concentrations a new, Type II behavior is observed; expulsion of
alga and consumption of oxygen are sufficiently rapid that many bacteria in
the illuminated region become hypoxic, transition to an immotile state, and
are then also expelled into the dark.

We develop here a system of coupled PDEs that describes the ABC system that
provides a quantitative account of the experimental observations. These PDEs
incorporate diffusion, chemotaxis, oxygen production and consumption and algal
transport by collective effects. The expulsion of algae is similar to well-
known processes in magnetohydrodynamics termed “flux expulsion”. First
discovered for magnetic fields in a prescribed constant vortical flow field
[17], in which field lines are expelled from the vortex, it was later
recognized that this expulsion can arise from gradients in the intensity of
turbulence whose random advection of the magnetic vector potential leads to a
gradient in its effective diffusivity and thence to “turbulent diamagnetism”
[18].

Figure 2: Bacterial influx and algal expulsion. (a) Type-I dynamics. Spatio-
temporal evolution of bacterial concentration after illumination is initiated,
where octagonal ring indicates boundary of illuminated region. At the low
initial bacteria concentration ($\sim\\!1\times 10^{8}$ cm-3), bacteria first
move into the illuminated region and then retreat. (b) Algal concentration as
a function of time, demonstrating expulsion from illuminated region. (c) Type
II dynamics. At a higher concentration ($\sim\\!5\times 10^{8}$ cm-3), many
bacteria become non-motile and are expelled into the dark region, forming a
concentrated circular accumulation that acts as a natural boundary. (d) Algae
are also expelled, accumulating inside the confines of the bacterial
accumulation.

Methods.— The coculture used strain 168 of B. subtilis which was genetically
engineered to express yellow fluorescent protein m-Venus with excitation at
$515\,$nm and emission at $528\,$nm [19]. A single bacterial colony was picked
from an agar plate and grown overnight in Terrific Broth (TB) on an orbital
shaker at $240\,$rpm and $30\,^{\circ}$C. The bacteria intended for
experiments were grown from overnight culture until exponential growth phase
in Tris-min medium spiked with TB medium (Tris-min $+$ $0.1\,$ % w/v glycerol
$+$ $5\,$ % w/v TB). The non-motile C. reinhardtii strain CC477 (bld1-1) was
sourced from the Chlamydomonas Resource Center [20] and grown in Tris-min
medium, on an orbital shaker at $240\,$rpm and $20\,^{\circ}$C. The diurnal
cycle was $12\,$h cool white light ($\sim 15\,\mu$mol photons/m2s PAR), and
$12\,$h in the dark.

Prior to experiments, bacteria and alga from their respective exponential
growth phases were mixed in a modified Tris-min medium, with glycerol as a
carbon source and bovine serum albumin to prevent cell adhesion (Tris-min $+$
$0.1\,$% w/v glycerol $+$ $0.01\,$% v/v BSA). The desired number density of
cells was achieved by centrifugation at $4000\,$g. All cell concentrations and
sizes were measured using a Beckman Coulter Counter (Multisizer 4e). The
concentration of algae was fixed at $5\times 10^{6}\,$ cm-3 while varying the
bacteria concentration in the range $(0.5-5)\times 10^{8}$ cm-3. The mixture
of cells for each experiment was transferred to a glass cover slip chamber
with a depth of $300\,\mu$m, separated by double-sided tape, and sealed
airtight using UV glue. The chamber surfaces were passivated with PEG
($M_{w}=5000\,$ g/mol).

Experiments were performed on a Nikon TE2000-U inverted microscope. The
spatio-temporal variation in bacterial concentration was monitored by
epifluorescence illumination with a $\times 20$ objective using a highly
sensitive, back-illuminated camera (Teledyne Prime $\Sigma$ 95B). Movies of
algae cells were recorded through the brightfield channel using a Phantom V311
high-speed camera (Vision Research) at $\times 20$ magnification. The halogen
lamp served as both a brightfield and photosynthetic light source. The size of
a light shaft used to trigger photosynthesis was controlled by the field iris
in the microscope condenser arm, producing an octogonal boundary in the focal
plane with mean radius of $R=220\,\mu$m.

e

Figure 3: Bacterial accumulation and algal expulsion. Yellow shading in the
plots up to $220\,\mu$m indicates illuminated region. (a) Bacterial and (b)
algal concentrations for a coculture with initial concentrations $b=1\times
10^{8}$ cm-3 and $a=5\times 10^{8}$ cm-3, showing Type I dynamics. Shading of
symbols and lines increase with time, with a ten minute interval between data
sets. (c,d) As in (a,b) but with initial bacterial concentration $b=5\times
10^{8}$ cm-3, exhibiting Type II dynamics. (e) MSD of algae versus time for
the case (a,b) at different radii in the illuminated region. Dashed lines
indicate apparent slope at different times. (f) Algal diffusivity determined
in linear regime of MSD versus radial distance at various times after start of
illumination.

Experimental Results.— Figures 2 and 3 summarize the main experimental
observations associated with a homogeneous initial condition in darkness that
is then illuminated with a shaft of light. For an initial bacterial
concentration of $b=1\times 10^{8}\,$cm-3 giving Type I dynamics, we observe
over the course of the first $\sim 10\,$min after the start of illumination
that the concentration of bacteria at the center of the illuminated region
increases dramatically, as visualized in Fig. 1(a) and quantified in Fig.
2(b), reaching a peak enhancement of a factor of $\sim 5$, with a roughly
linear decrease out to the edge. That peak then relaxes away until the
bacterial concentration is again nearly uniform after $\sim 35\,$min. At the
peak of accumulation, after $\sim 15\,$min, there is a clear bacterial
depletion zone just outside the illuminated region, whose width is estimated
to be $\sim\\!150\,\mu$m. During this period, as shown in Fig. 2(b), the algal
concentration becomes strongly depleted in the illuminated region, leading to
a ring of accumulation at the boundary.

We measured the mean squared displacement (MSD) of algae versus time at a
range of radii $r$ from the light shaft center. The MSD in Fig. 3(e) exhibits
systematic upward curvature, with a local exponent of unity at time $t_{1}\sim
6-7\,$s, but faster behavior for $t>t_{1}$. Such superdiffusion for tracers is
a well-known consequence of active turbulence in concentrated bacterial
suspensions [8, 13]. A heuristic illustration of the strong gradient in
collective behavior in the illuminated region is obtained by determining an
effective algal diffusion constant $\tilde{D}_{a}$ from the slope of the MSD
curve at $t_{1}$ (Fig. 2(f)), which is a strongly decreasing function of
distance from the shaft center, that mirrors the decreasing bacterial
concentration. The largest of these diffusivities is $\sim 25$ times the
purely thermal value $D_{\rm th}=k_{B}T/6\pi\mu a\simeq 0.02\,\mu$m2/s, where
$\mu$ is the medium viscosity and $a=10\,\mu$m is twice the algal radius since
most algae exist as pairs (“palmelloids”). Yet, by itself, an effective algal
diffusivity $\tilde{D}_{a}\sim 0.5\,\mu$m2/s implies a diffusive time
$R^{2}/\tilde{D}_{a}\sim 700\,$min for algae at the center to escape purely by
random motion, far longer than the observed time of $30\,$min; the
superdiffusive behavior seen for $t>t_{1}$ in Fig. 3(e) signals a
qualitatively different transport process.

At the higher initial bacterial concentration $b=5\times 10^{8}\,$cm-3 we
observe the distinct Type II behavior: expulsion happens much more rapidly,
the bacterial concentration profile becomes nonmonotic in radius, with a
strong peak just outside the illuminated region, and the peak of the
concentration of expelled algae is narrower. Close microscopic inspection of
the region of high bacterial concentration in the dark region shows that most
of the bacteria there are immotile and the expelled algae reside just inside
the bacterial accumulation ring. We deduce that many bacteria in the
illuminated region have become hypoxic and are expelled from that area through
much the same process that expelled the algae.

Figure 4: Theoretical predictions. (a,b) Type I dynamics. (a) Non-dimensional
bacterial dynamics from experiment (circles) compared to theoretical fit
(solid lines) for first thirty minutes of experiment. Rightmost column in each
row shows the theoretical time evolution over 30 minutes, sampled every $\sim$
3 minute. Shading intensity increases with time. (b) As in (a) but for algae,
showing expulsion. Insets show theoretical oxygen profiles $\chi$. (c,d) Type
II dynamics. (c) Experimental bacterial dynamics (circles) compared to
predictions of the ABCD model. Dashed lines indicate immotile (black) and
active (blue) bacteria, whose sum is orange solid line. Rightmost plot shows
time evolution of the total bacterial concentration. (d) Algal expulsion for
high density experiment. For (a),(b) we took
$\kappa^{2}=1,\epsilon=1,d_{b}=0.1,d_{a}=0.0001,\gamma=1.2,\zeta=0.002$. For
(c), (d), we use
$\gamma=0.35,\zeta=0.005,\kappa^{2}=5,d_{\delta}=10^{-4},\rho=1.0,\chi^{*}=0.1,\zeta_{d}=0.04,\delta^{*}=0.01$
with all other parameter values held over.

The ABC model.— We now turn to a mathematical model for the behavior described
above, focusing first on Type I expulsion. In this case, a minimal description
of the system entails the concentration fields $a({\bf r},t),b({\bf r},t)$ and
$c({\bf r},t)$ for algae, bacteria, and oxygen, respectively. If $D_{c}$ is
the oxygen diffusion constant (assumed independent of all other variables),
$k_{+}$ is the rate of oxygen production per algal cell when illuminated with
a spatially varying light intensity $I(r)$ and the oxygen consumption rate per
bacterium has a Michaelis-Menten form, then

$c_{t}=D_{c}\nabla^{2}c+k_{+}aI(r)-k_{-}b\frac{c}{K_{c}+c},$ (1)

where $K_{c}$ is the Michaelis constant.

Consider first the situation of uniform illumination ($I=1$) and uniform
concentrations $a_{0}$ and $b_{0}$ of algae and bacteria. A steady state
$c^{*}=K_{c}k/(1-k)$, with $k=k_{+}a_{0}/k_{-}b_{0}$, can be reached in which
consumption balances production, provided $k<1$. We assume this inequality is
always satisfied and typically invoke the weak production limit $k\ll 1$, in
which the oxygen consumption can be approximated as $k_{-}bc/K_{c}$.

We explain algal expulsion from the illuminated region through three
intermediate calculations; oxygen production in a uniform suspension;
bacterial chemotaxis in the presence of oxygen production; algal dynamics due
to an inhomogeneous bacterial concentration. Suppose that the light is
constrained to a shaft of radius $R$. Scaling time, space, and concentrations
via $T=tD_{c}/R^{2}$, $\eta=r/R$, $\chi=c/c^{*}$, $\alpha=a/a_{0}$, and
$\beta=b/b_{0}$, letting $\epsilon=k/(1-k)$, and introducing the screening
length

$\lambda=\left(D_{c}\tau_{c}\right)^{1/2},$ (2)

where $\tau_{c}=K_{c}/k_{-}b_{0}$ is the characteristic consumption time of
oxygen, the dynamics (1) takes the form

$\chi_{T}=\nabla^{2}\chi+(1-k)\kappa^{2}\alpha\Theta(1-\eta)-\kappa^{2}\beta\frac{\chi}{1+\epsilon\,\chi},$
(3)

where now $\nabla^{2}=\nabla^{2}_{\eta}$, and $\Theta$ is the Heaviside
function. Here, $\kappa^{2}=R^{2}/\lambda^{2}$ is also the ratio
$\tau_{D}/\tau_{c}$ of the time $\tau_{D}=R^{2}/D_{c}\sim 25\,$s for oxygen to
equilibrate diffusively across the illuminated region to the consumption time.

If we clamp concentrations $\alpha$ and $\beta$ at unity, take $k\ll 1$, and
enforce continuity in $\chi$ and $\chi_{\eta}$ at $\eta=1$, the steady state
of (3) in an unbounded domain is

$\chi(\eta)=\begin{cases}1-\kappa K_{1}(\kappa)I_{0}(\kappa\eta)&\eta\leq
1,\\\ \kappa I_{1}(\kappa)K_{0}(\kappa\eta)&\eta\geq 1,\end{cases}$ (4)

in terms of modified Bessel functions $K_{\nu}$ and $I_{\nu}$. From the inner
solution we see that the oxygen concentration at the centre of the illuminated
region asymptotes to unity ($c^{*}$ in unrescaled units) for large domain size
$\kappa$, but is attenuated strongly as $\kappa$ falls below unity. The outer
solution behaves as $\chi\sim\exp(-(r-R)/\lambda)$, showing that $\lambda$
serves as the characteristic penetration depth of oxygen into the surrounding
bacterial population and sets the depletion zone seen in Fig. 1(b). With
$D_{c}\sim 2\times 10^{3}\,\mu$m2/s and $R=220\,\mu$m, and the estimate
$\tau_{c}\sim 10^{2}\,$s [21], we find $\lambda\sim 400\,\mu$m, and thus
$\kappa\sim 1-2$.

Relaxing the assumption of uniform bacterial concentration, it is clear that
beyond the time $\tau_{D}$ bacteria within a distance $\lambda$ of the edge of
the illuminated region will experience the steepest oxygen gradient and
chemotax most rapidly inwards. This can be described by the simplest
combination of diffusion and chemotaxis as in the Keller-Segel model, [22],
$b_{t}=D_{b}\nabla^{2}b-\bm{\nabla}\cdot\left(gb\bm{\nabla}c\right)$ where the
diffusion constant $D_{b}$ arises from random cellular swimming and the
response coefficient $g$ is taken to be constant. Scaling as above, we obtain

$\beta_{T}=d\nabla^{2}\beta-\gamma\bm{\nabla}\cdot(\beta\bm{\nabla}\chi),$ (5)

where $d=D_{b}/D_{c}$, and $\gamma=gc^{*}/D_{c}$.

A steady state can be reached when the chemotactic flux
$\gamma\beta\bm{\nabla}\chi$ balances the diffusive flux $d\bm{\nabla}\beta$,
yielding

$\beta_{ss}(\eta)=A\exp\left(\dfrac{\gamma}{d}\chi(\eta)\right),$ (6)

where $A$ is a normalization constant. While this steady state profile
captures the observation that the bacterial concentration reaches its maximum
at $\eta=0$, where $\chi$ itself is maximized, the steady state profile (6)
only develops on time scales sufficient for bacteria outside the depletion
zone to move inwards and replenish partially the depletion. This time will be
at least $(R+\lambda)^{2}/D_{b}\gg\tau_{D}$. Prior to this the bacterial
concentration is nonmonotonic, with the depletion zone seen in Fig. 2(a).

The flagella-less algae used in our experiments do not swim; their movement
arises from collective flows driven by the concentrated bacteria. At the very
least this leads to enhanced diffusivity, which, as seen in Fig. 3(f), varies
in space, and by itself, is captured by an algal flux ${\bf
J}_{D}=-\tilde{D}_{a}{\bm{\nabla}}a$. The appearance of “bacterial turbulence”
at higher concentrations implies that the algae are passive scalars in an
inhomogeneous turbulent flow. It is well-known that passive particles will be
expelled into separatrices between regions of high vorticity. When those
boundaries change with time, the effect to transport particles from regions of
high turbulence to low, without requiring any gradients in particle
concentration, akin to chemokinesis, where cells accumulate in regions where
they swim slowly. In a simple approximation, the extent of the turbulent
transport is proportional to the bacterial concentration, giving a
contribution to the algal flux ${\bf J}=-pa{\bm{\nabla}}b$ for some $p>0$.
With the concentration seen in Fig. 3(a), whose radial gradient is negative,
this contribution leads to an outward flux of algae.

Assembling these contributions and rescaling as above, we obtain two
equivalent forms of the dynamics,

$\displaystyle\alpha_{T}$
$\displaystyle=\bm{\nabla}\cdot\left(d_{a}\bm{\nabla}\alpha\right)+\zeta\bm{\nabla}\cdot\left(\alpha\bm{\nabla}\beta\right),$
(7a) $\displaystyle\alpha_{T}$
$\displaystyle-\zeta\bm{\nabla}\beta\cdot\bm{\nabla}\alpha=\bm{\nabla}\cdot\left(d_{a}\bm{\nabla}\alpha\right)+\zeta\alpha\nabla^{2}\beta,$
(7b)

where $d_{a}=\tilde{D}_{a}/D_{c}$ and $\zeta=pb_{0}/D_{c}$. In the first form
(7a), we see a parallel to the bacterial chemotaxis equation (5); algae
exhibit negative “bacteria-taxis”. In symmetrizing the two problems, this
“completion” of the dynamics is loosely analogous to the introduction of the
displacement current in Maxwell’s equations as a parallel to Ampere’s law. In
the second form (7b), we see that there is an explicit advective contribution
and a “reactive” term $\alpha\nabla^{2}\beta$. This latter contribution plays
an important role in the algal expulsion, as the bacterial concentration
within the illuminated region has a large negative second derivative, and this
term thus forces $\alpha$ down there, whereupon it is advected outward. This
is a process of “anti-diffusion”.

Figures 4(a,b) show how the model defined by Eqs. (3), (5) and (7b) provide a
quantitative fit to the observed Type I dynamics. In particular, we see the
prompt accumulation of bacteria in the illuminated region followed by
expulsion of algae. In the model and in experiment, the fraction of algae
ultimately expelled is $\sim 0.5$, so there is still oxygen production within
the illuminated region after $\sim 30\,$min, leading to continued chemotactic
attraction of bacteria inward. This leads to a slower decay of the bacterial
concentration back to its original uniform, low value than seen in experiment.
This may reflect processes such as bacterial adaptation to the oxygen and a
gradual reduction in oxygen production by algae.

The ABCD model.— While the ABC model can account for the essential features of
Type I dynamics, it does not allow for loss of motility of bacteria at low
oxygen concentrations in Type II dynamics. This transformation has been
recognized as important in the context of bioconvection [21, 23], where the
influx of oxygen at the air-water interface of a bacterial suspension competes
with consumption within the fluid, leading to a hypoxic region hundreds of
microns below the surface. Hypoxia-induced motility transitions have also been
observed in the penetration of oxygen into suspensions of E. coli [24].

To account for this transformation we view the “dormant”, non-motile state of
the bacteria, with concentration $d$, as a separate population distinct from
the motile form, so the extended “ABCD” model involves algae, bacteria,
chemoattractant, and dormant bacteria. The interconversion rate as a function
of oxygen concentration $c$ is taken as a simple generalization of the
substrate-dependent growth of the Monod model, $v_{con}bK_{sat}/(c+K_{sat})$,
where $v_{con}$ is the maximum conversion rate to the immotile form and
$K_{sat}$ is the concentration at which half-maximal conversion occurs.

The dynamics of dormant bacteria include generation, expulsion, and a very
small diffusion constant $D_{d}$. Setting $\delta=d/b_{0}$, the rescaled
dynamics takes the form

$\delta_{T}-\zeta_{d}\bm{\nabla}\beta\cdot\bm{\nabla}\delta=d_{d}\nabla^{2}\delta+\zeta_{d}\delta\nabla^{2}\beta+\rho\beta\frac{\chi^{*}}{\chi^{*}+\chi},$
(8)

where $d_{d}=D_{d}/D_{c}$, $\rho=v_{con}R^{2}/D_{c}$ and
$\chi^{*}=K_{sat}/c^{*}$. Accordingly the bacterial dynamics (5) acquires the
corresponding loss term from conversion, becoming

$\beta_{T}=d\nabla^{2}\beta-\gamma\bm{\nabla}\cdot(\beta\bm{\nabla}\chi)-\rho\beta\frac{\chi^{*}}{\chi^{*}+\chi}.$
(9)

A final modification to the ABC model involves steric effects on algal
diffusion that occur when the concentration of dormant bacteria is large. This
is modelled by modifying the rescaled algal diffusivity $d_{a}$ to
$d_{a}(1-\delta/(\delta+\delta^{*}))$, a form that reduces $d_{a}$ at large
$\delta$ but retains its positivity. With these components, Fig. 4(c,d) show
that the ABCD model captures the expulsion of both algae and immotile
bacteria, with a narrow accumulation of dormant bacteria blocking transport of
algae.

Spatially-averaged model.— Insight into the basic process of algal expulsion
can be obtained by constructing a spatially-averaged model which takes as
dynamical degrees of freedom the mean concentrations of algae, bacteria, and
oxygen inside the illuminated region, denoted by $\bar{\alpha}$,
$\bar{\beta}$, and $\bar{\chi}$. As indicated schematically in Fig. 5(a), the
specification of this averaged model requires estimates of the various fluxes
across the boundary of the illuminated region. We anticipate that turbulent
bacterial dynamics will homogenize oxygen inside the illuminated region, save
for the area close to the boundary of width $\lambda$ where the concentration
rapidly decreases. From an integral form of (1) and the divergence theorem we
have $\partial_{t}(\pi R^{2}\bar{c})=-D_{c}\oint ds{\hat{\bf
n}}\cdot\bm{\nabla}c$, where ${\hat{\bf n}}$ is the outward normal to the
illuminated region. If we estimate ${\hat{\bf
n}}\cdot\bm{\nabla}c\sim\bar{c}/\lambda$, then the PDE (1) becomes the ODE

$\frac{d\bar{\chi}}{dT}=-2\kappa\bar{\chi}+(1-k)\kappa^{2}\bar{\alpha}-\kappa^{2}\bar{\beta}\frac{\bar{\chi}}{1+\epsilon\bar{\chi}}.$
(10)

Figure 5: Spatially averaged dynamics. Average concentration of bacteria
(orange) and algae (green) in illuminated region from experiments (circles)
and model (lines) shown in scattered points. For bacteria, simple chemotaxis
is shown by solid orange line, adaptive chemotaxis shown dashed. Inset:
Schematic of fluxes across the illuminated region.

Similar estimates hold for the averaged bacteria dynamics, where the relevant
fluxes are the outward diffusive and inward chemotactic contributions. The
diffusive contribution is precisely analogous to that for oxygen above. Since
on the time scales of the experiment only bacteria within a distance $\lambda$
outside the illuminated region sense the gradient, the inward Keller-Segel
flux is also estimated as above, but with an upper limit on the bacterial
concentration due to steric effects. This leads to the ODE version of (5),

$\frac{d\bar{\beta}}{dT}=-2d\kappa\bar{\beta}+2\gamma\kappa\left(\beta_{\rm
max}-\bar{\beta}\right).$ (11)

Finally, the algae have an active flux $-pa\bm{\nabla}b$ that leads to
expulsion, but there is little back-diffusion into the illuminated region.
Yet, as algae accumulate at the boundary (Fig. 2), they form a thick ring that
inhibits continued expulsion of algae. Introducing a saturation of the algal
flux we obtain the ODE version of (7a),

$\frac{d\bar{\alpha}}{dT}=2\zeta\kappa\bar{\beta}\left(\alpha_{\rm
max}-\bar{\alpha}\right).$ (12)

Using parameter values taken from Fig. 4 and fitting
$\beta_{\text{max}}\approx 5$ and $\alpha_{\text{max}}\approx 0.5$, we obtain
the solid green and orange curves shown in Fig. 5 for mean algal and bacterial
concentrations within the illuminated region. The reduced model captures the
slow exponential decay of the algal concentration, but while it reproduces the
rapid bacterial influx in the first few minutes of the experiments, the
predicted decay of the bacterial concentration, as remarked earlier for the
full ABC model, is far slower than observations. The possibility that this
discrepancy arises from adaptation of the bacteria to elevated oxygen levels
can be explored by assuming a simple linear decay of the chemotactic
coefficient, as $\gamma(T)=\gamma_{0}-T/\tau$, where $\tau$ sets the
adaptation rate. For $\tau=1200$, the bacterial dynamics (dashed orange line
in Fig. 5) matches closely with data, while the algal expulsion (not depicted)
is essentially unchanged from the non-adaptive case.

Discussion.— As our results suggest new phenomenology in active matter
systems, it is instructive to return to the connection between algal expulsion
and flux expulsion in magnetohydrodynamics. In the original case considered
[17], a magnetic field in a fluid with velocity ${\bf u}$ obeys the Maxwell
equation,

${\bf B}_{t}=\bm{\nabla}\times\left({\bf u}\times{\bf
B}\right)+D_{m}\nabla^{2}{\bf B},$ (13)

where $D_{m}$ is the magnetic diffusivity, and both ${\bf B}$ and ${\bf u}$
are taken to confined to the $xy$-plane. The magnitude of the magnetic vector
potential ${\bf A}=A\hat{\bf z}$ then obeys the advection-diffusion equation

$A_{t}+{\bf u}\cdot\bm{\nabla}A=D_{m}\nabla^{2}A.$ (14)

With ${\bf u}$ a prescribed single-vortex velocity field, the initial
condition $A=B_{0}x$, corresponding to the space-filling uniform magnetic
field ${\bf B}=\bm{\nabla}\times{\bf A}=-B_{0}\hat{\bf y}$, homogenizes inside
the vortex, leaving ${\bf B}=0$ zero there but $A$ essentially undisturbed
outside.

In the case of turbulent transport consider later [18], it was shown that on
long time and length scales the effect of the inhomogeneous turbulence is
captured by an equation of motion for a vector potential component $\bar{A}$,
averaged on those scales, of the form

$\bar{A}_{t}=\bm{\nabla}\cdot\left(D^{*}\bm{\nabla}\bar{A}\right)$ (15)

where $D*$ is a turbulent diffusivity tensor. For the particular case in which
the turbulence varies along one direction, say $y$, then the averaged
$x$-component of the magnetic field ${\cal F}=\bar{B}_{x}$ obeys

${\cal F}_{t}-2(\partial_{y}D^{*})\partial_{y}{\cal F}=D^{*}\nabla^{2}{\cal
F}+{\cal F}\partial_{yy}D^{*}.$ (16)

This form has the same structure as our Eq. (7a), with the bacterial
concentration $\beta$ playing the role of the diffusivity tensor component. We
conclude that the main difference between the two problems is the expulsion of
a scalar (concentration) field in the present context compared to the
expulsion of a vector quantity in MHD.

The results described here highlight the rich dynamical behavior that occurs
in mixed active matter systems involving microorganisms from two Kingdoms of
Life. While the dynamics of symbiosis is generally studied on time scales
relevant to population dynamics or evolution [25, 26, 27], our findings
indicate that short-term dynamics on the scale of minutes and hours can have a
considerable impact on the spatial-temporal aspects of association, where
chemotaxis and phenotypic switching dominate. Although we considered the
simple case of light intensity that is piecewise constant, inhomogeneous
activity arises. The treatment of such nonuniform active matter has received
attention only recently, in the context of “invasion” [28] and active gels
[29], but is central to any discussion of realistic ecologies. In this sense,
generalizing the present setup to allow algal motility and phototaxis in the
presence of an inhomogeneous light field may reveal even more striking
dynamics when the oxygen sources are themselves motile.

###### Acknowledgements.

We are grateful to Jim Haseloff for providing the fluorescent strain of B.
subtilis and to Kyriacos Leptos for numerous discussions. This work was
supported in part by the Gordon and Betty Moore Foundation, Grant No. 7523 (PP
& REG) and the U.K. Marshall Aid Commemoration Commission (YB).

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# Narayana numbers as product of three repdigits in base $g$

P. Tiebekabe 1,2, K. R. Kakanou 2, H. Ben Yakkou 3

###### Abstract

In this paper, we show that there are only finitely many Narayana’s numbers
which can be written as product of three repdigits in base $g$ with $g\geq 2$.
Moreover, for $2\leq g\leq 10$, we determine all these numbers.

Keywords and phrases: Narayana numbers, repdigits, linear form in logarithms;
Baker’s method, reduction method.

2020 Mathematics Subject Classification: 11B39, 11J86, 11Y50, 11D61.

##  1 Introduction

The problems of the terms of linear recurrence sequences written as a product
of repdigits in any base have been intensely studied by several researchers
specialized in Number Theory. In this article, we consider the linear
recurrent sequence of third order, The Narayana’s Cows numbers defined as
follows:

$\mathcal{N}_{n}=\mathcal{N}_{n-1}+\mathcal{N}_{n-3}\quad\text{for}\quad n\geq
3\quad\text{with}\quad\mathcal{N}_{0}=0,\mathcal{N}_{1}=\mathcal{N}_{2}=1.$

For more details on the work related to the determination of the terms of
linear recurrent sequences which are repdigits in any base, we refer the
reader to the following recent results [1]–[4].

The concept of Narayana’s cows numbers, derived from Indian mythology and
Hinduism, holds a significant place in mathematics. These numbers have been
extensively studied due to their properties and relationships with other
mathematical sequences, and their important applications in other various
fields such as cryptography, coding theory, and graph theory. In this paper,
we delve into a fascinating aspect of Narayana numbers by examining their
representation as product s of three repdigits in base $g$ with $g\geq 2$.

Repdigits, which consist of repeated digits, have garnered attention for their
mathematical properties and patterns. In a fixed base $g\geq 2$, a repdigit
has the following form,

$\sum_{i=0}^{n-1}d\times g^{i}=d\times\cfrac{g^{n}-1}{g-1},$

where $1\leq d\leq g-1$ and $n$ a positive integer.

The proofs of our main results are based on a double application of Baker’s
method and on a reduction algorithm using computations based on continued
fractions. The method used to determine Narayana numbers, which are products
of three repdigits is similar to that used by Adédji [2] and by Adédji et al.
[3].

The present paper is organized as follows: in Section 2, we present our main
results, Section 3 is devoted to reminding necessary results for the proofs of
our results, and in Section 4, we prove our results.

##  2 Statement of main results

In this section, we state all the main results obtained in this paper.

######  Theorem 1.

Let $g\geq 2$ be an integer. Then the Diophantine equation

$\mathcal{N}_{k}=d_{1}\cfrac{g^{\ell}-1}{g-1}\cdot
d_{2}\cfrac{g^{m}-1}{g-1}\cdot d_{3}\cfrac{g^{n}-1}{g-1}$ (1)

has only finitely many solutions in integers $k,d_{1},d_{2},d_{3},\ell,m,n$
such that $1\leq d_{i}\leq g-1$ for $i=1,2,3$ and $n\geq m\geq\ell\geq 1$.
Further, we have

$n<5.91\times 10^{49}\log^{9}g\quad\text{and}\quad k<4.73\times
10^{50}\log^{10}g.$

Under the notation and assumptions of the Theorem 1, if (1) holds for
$\biggl{(}k,d_{1},d_{2},d_{3},\ell,m,n\biggr{)}$, then we write

$\displaystyle\mathcal{N}_{k}=[a,b,c]_{g}=a\times b\times c,$

where

$a=d_{1}\times\cfrac{g^{\ell}-1}{g-1}=\overline{\underbrace{d_{1}\cdots
d_{1_{g}}}_{\ell-1\ \text{times}}},\quad
b=d_{2}\times\cfrac{g^{m}-1}{g-1}=\overline{\underbrace{d_{2}\cdots
d_{2_{g}}}_{m-1\ \text{times}}},\quad\text{and}\quad
c=d_{3}\times\cfrac{g^{n}-1}{g-1}=\overline{\underbrace{d_{3}\cdots
d_{3_{g}}}_{n-1\ \text{times}}}.$

In the following theorem, we completely and explicitly give all solutions of
the equation (1) corresponding to $2\leq g\leq 10.$

######  Theorem 2.

The only Narayana numbers which are a product of three repdigits in base $g$
with $2\leq g\leq 10$ are

$\\{1,2,3,4,6,9,13,28,60,88,129,189\\}.$

More precisely, we have

Table 1: Narayana numbers which are a product of three repdigits in base $g$, $2\leq g\leq 10.$ $k$ | $\mathcal{N}_{k}$ |  $[a,b,c]_{g}$
---|---|---
$1,2,3$ | $1$ |  $\left[1,1,1\right]_{g}\quad\text{for}\quad g=2,\ldots,10.$
$4$ | $2$ |  $\left[1,1,2\right]_{g}\quad\text{for}\quad g=3,\ldots,10.$
$5$ | $3$ |  $\left[1,1,11\right]_{2}$, $\quad\left[1,1,3\right]_{g}\quad\text{for}\quad g=4,\ldots,10$.
$6$ | $4$ |  $\left[1,1,11\right]_{3}$, $\quad\left[1,1,4\right]_{g}\quad\text{for}\quad g=5,\ldots,10$, $\quad\left[1,2,2\right]_{g}\quad\text{for}\quad g=3,\ldots,10$.
$7$ | $6$ |  $\left[1,1,11\right]_{5}$, $\quad\left[1,2,3\right]_{g}\quad\text{for}\quad g=4,\ldots,10$, $\quad\left[1,1,6\right]_{g}\quad\text{for}\quad g=7,\ldots,10$.
$8$ | $9$ |  $\left[1,11,11\right]_{2}$, $\quad\left[1,1,111\right]_{3}$, $\quad\left[1,1,11\right]_{8}$, $\quad\left[1,1,9\right]_{10}$, $\quad\left[1,3,3\right]_{g}\quad\text{for}\quad g=4,\ldots,10$.
$9$ | $13$ |  $\left[1,1,111\right]_{3}$
$11$ | $28$ |  $\left[1,1,44\right]_{6}$, $\quad\left[1,2,22\right]_{6}$, $\quad\left[1,4,11\right]_{6}$, $\quad\left[2,2,11\right]_{6}$, $\quad\left[1,4,7\right]_{g}\quad\text{for}\quad g=8,\ldots,10$, $\left[2,2,7\right]_{g}\quad\text{for}\quad g=8,\ldots,10$.
$13$ | $60$ |  $\left[2,2,33\right]_{4}$, $\quad\left[2,3,22\right]_{4}$, $\quad\left[1,1,66\right]_{9}$, $\quad\left[1,2,33\right]_{9}$, $\quad\left[1,3,22\right]_{9}$,$\quad\left[1,6,11\right]_{9}$, $\quad\left[2,3,11\right]_{9}$, $\quad\left[2,5,6\right]_{g}\quad\text{for}\quad g=7,\ldots,10$, $\quad\left[3,4,5\right]_{g}\quad\text{for}\quad g=6,\ldots,10$.
$14$ | $88$ |  $\left[1,1,88\right]_{10}$, $\quad\left[1,2,44\right]_{10}$, $\quad\left[1,4,22\right]_{10}$, $\quad\left[1,8,11\right]_{10}$, $\quad\left[2,2,22\right]_{10}$, $\quad\left[2,4,11\right]_{10}$.
$15$ | $129$ |  $\left[1,1,333\right]_{6}$, $\quad\left[1,3,111\right]_{6}$.
$16$ | $189$ |  $\left[1,11,111111\right]_{2}$, $\quad\left[1,3,333\right]_{4}$, $\quad\left[3,3,111\right]_{4}$, $\quad\left[3,3,33\right]_{6}$, $\quad\left[1,3,77\right]_{8}$, $\quad\left[1,7,33\right]_{8}$, $\quad\left[3,7,11\right]_{8}$, $\quad\left[3,7,9\right]_{10}$.

##  3 Preliminary Results

In this section, we give some notation and recall certain definitions and
results required for the proof of our main results.

###  3.1 Some properties of Narayana sequence

Narayana’s cows sequence comes from a problem with cows proposed by Indian
mathematician Narayana in the 14th century. In this problem, we assume that
there is a cow at the beginning and each cow produces a calf every year from
the 4th year. Narayana’s cow problem counts the number of calves produced each
year [5].

The characteristic polynomial of Narayana’s cows sequence
$\\{\mathcal{N}_{n}\\}_{n\geq 0}$ is

$\varphi(x)=x^{3}-x^{2}-1.$

Furthermore, the zeros of $\varphi(x)$ are

$\displaystyle\alpha_{\mathcal{N}}$
$\displaystyle=\cfrac{1}{3}\biggl{(}\sqrt[3]{\cfrac{1}{2}(29-3\sqrt{93})}+\sqrt[3]{\cfrac{1}{2}(3\sqrt{93}+29)}+1\biggr{)},$
$\displaystyle\beta_{\mathcal{N}}$
$\displaystyle=\cfrac{1}{3}-\cfrac{1}{6}\biggl{(}1-i\sqrt{3}\biggr{)}\sqrt[3]{\cfrac{1}{2}(29-3\sqrt{93})}-\cfrac{1}{6}\biggl{(}1+i\sqrt{3}\biggr{)}\sqrt[3]{\cfrac{1}{2}(3\sqrt{93}+29)},$
$\displaystyle\gamma_{\mathcal{N}}$
$\displaystyle=\cfrac{1}{3}-\cfrac{1}{6}\biggl{(}1+i\sqrt{3}\biggr{)}\sqrt[3]{\cfrac{1}{2}(29-3\sqrt{93})}-\cfrac{1}{6}\biggl{(}1-i\sqrt{3}\biggr{)}\sqrt[3]{\cfrac{1}{2}(3\sqrt{93}+29)}.$

Then, the Narayana sequence can be obtained by Binet formula

$\mathcal{N}_{n}=a_{\mathcal{N}}\alpha_{\mathcal{N}}^{n}+b_{\mathcal{N}}\beta_{\mathcal{N}}^{n}+c_{\mathcal{N}}\gamma_{\mathcal{N}}^{n}.$
(2)

From the three initial values of Nayarana sequence, and using Vieta’s theorem,
one has

$a_{\mathcal{N}}=\cfrac{\alpha_{\mathcal{N}}^{2}}{\alpha_{\mathcal{N}}^{3}+2},\quad
b_{\mathcal{N}}=\cfrac{\beta_{\mathcal{N}}^{2}}{\beta_{\mathcal{N}}^{3}+2},\quad\text{and}\quad
c_{\mathcal{N}}=\cfrac{\gamma_{\mathcal{N}}^{2}}{\gamma_{\mathcal{N}}^{3}+2}.$
(3)

The minimal polynomial of $a_{\mathcal{N}}$ over $\mathbb{Z}$ is
$31x^{3}-3x-1$.

Setting
$\Pi(n)=\mathcal{N}_{n}-a_{\mathcal{N}}\alpha_{\mathcal{N}}^{n}=b_{\mathcal{N}}\beta_{\mathcal{N}}^{n}+c_{\mathcal{N}}\gamma_{\mathcal{N}}^{n}$,
we notice that

$\biggl{|}\Pi(n)\biggr{|}<\cfrac{1}{\alpha_{\mathcal{N}}^{n/2}}\quad\text{for
all}\ n\geq 1.$ (4)

We note that the characteristic polynomial has a real zero
$\alpha_{\mathcal{N}}(>1)$ and two complex conjugate zeros
$\beta_{\mathcal{N}}$ and $\gamma_{\mathcal{N}}$ with
$|\beta_{\mathcal{N}}|=|\gamma_{\mathcal{N}}|<1$. In fact,
$\alpha_{\mathcal{N}}\approx 1.46557$ . We also have the following property of
$(\mathcal{N}_{n})_{n\geq 0}$.

######  Lemma 1.

For the sequence $(\mathcal{N}_{n})_{n\geq 0}$, we have,

$\alpha_{\mathcal{N}}^{n-2}\leq\mathcal{N}_{n}\leq\alpha_{\mathcal{N}}^{n-1},\quad\text{for}\quad
n\geq 1.$

######  Proof 1.

One can easily prove Lemma 1 using induction on $n$.

Let
$\mathbb{K}_{\varphi}:=\mathbb{Q}(\alpha_{\mathcal{N}},\beta_{\mathcal{N}})$
be the splitting field of the polynomial $\varphi$ over $\mathbb{Q}$. Then,
$[\mathbb{K}_{\varphi},\mathbb{Q}]=6$. Furthermore,
$[\mathbb{Q}(\alpha_{\mathcal{N}}):\mathbb{Q}]=3$. The Galois group of
$\mathbb{K}$ over $\mathbb{Q}$ is given by

$\mathcal{G}_{\varphi}:=\text{Gal}(\mathbb{K/Q})\cong\\{(1),(\alpha_{\mathcal{N}}\beta_{\mathcal{N}}),(\alpha_{\mathcal{N}}\gamma_{\mathcal{N}}),(\beta_{\mathcal{N}}\gamma_{\mathcal{N}}),(\alpha_{\mathcal{N}}\beta_{\mathcal{N}}\gamma_{\mathcal{N}}),(\alpha_{\mathcal{N}}\gamma_{\mathcal{N}}\beta_{\mathcal{N}})\\}\cong
S_{3}.$

Thus, we identify the automorphisms of $\mathcal{G}_{\varphi}$ with the
permutations of the zeros of the polynomial $\varphi$. For example, the
permutation $(\alpha_{\mathcal{N}}\beta_{\mathcal{N}})$ corresponds to the
automorphisms
$\sigma_{\varphi}:\alpha_{\mathcal{N}}\to\beta_{\mathcal{N}},\leavevmode\nobreak\
\beta_{\mathcal{N}}\to\alpha_{\mathcal{N}},\leavevmode\nobreak\
\gamma_{\mathcal{N}}\to\gamma_{\mathcal{N}}$.

###  3.2 Linear forms in logarithms

We begin this subsection with a few reminders about the logarithmic height of
an algebraic number. Let $\eta$ be an algebraic number of degree $d$,
$a_{0}>0$ be the leading coefficient of its minimal polynomial over
$\mathbb{Z}$ and let $\eta=\eta^{(1)},\ldots,\eta^{(d)}$ denote its
conjugates. The quantity defined by

$h(\eta)=\cfrac{1}{d}\biggl{(}\log|a_{0}|+\sum_{j=1}^{d}\log\max(1,|\eta^{(j)}|)\biggr{)}$

is called the logarithmic height of $\eta$. Some properties of height are as
follows. For $\eta_{1},\eta_{2}$ algebraic numbers and $m\in\mathbb{Z}$, we
have

$\displaystyle h(\eta_{1}\pm\eta_{2})$ $\displaystyle\leq
h(\eta_{1})+h(\eta_{2})+\log 2,$ $\displaystyle h(\eta_{1}\eta_{2}^{\pm 1})$
$\displaystyle\leq h(\eta_{1})+h(\eta_{2}),$ $\displaystyle h(\eta_{1}^{m})$
$\displaystyle=|m|h(\eta_{1}).$

In particular, if $\eta=p/q\in\mathbb{Q}$ is a rational number in its reduced
form with $q>0$, then $h(\eta)=\log(\max\\{|p|,q\\})$.

We can now present the famous Matveev’s result used in this study. Let
$\mathbb{L}$ be a real number field of degree $d_{\mathbb{L}}$,
$\eta_{1},\ldots,\eta_{s}\in\mathbb{L}$ and
$b_{1},\ldots,b_{s}\in\mathbb{Z}\setminus\\{0\\}$. Let
$B\geq\max\\{|b_{1}|,\ldots,|b_{s}|\\}$ and

$\Lambda=\eta_{1}^{b_{1}}\cdots\eta_{s}^{b_{s}}-1.$

Let $A_{1},\ldots,A_{s}$ be real numbers such that

$A_{i}\geq\max\\{d_{\mathbb{L}}h(\eta_{i}),|\log\eta_{i}|,0.16\\},\quad
i=1,\ldots,s.$

With the above notation, Matveev [6] proved the following result.

######  Theorem 3.

Assume that $\Lambda\neq 0$. Then

$\log|\Lambda|>-1.4\cdot 30^{s+3}\cdot s^{4.5}\cdot
d_{\mathbb{L}}^{2}\cdot(1+\log d_{\mathbb{L}})\cdot(1+\log B)\cdot A_{1}\cdots
A_{s}.$

We also need the following result from Sanchez and Luca [7].

######  Lemma 2.

Let $r\geq 1$ and $H>0$ be such that $H>(4r^{2})^{r}$ and $H>L/(\log L)^{r}$.
Then

$L<2^{r}H(\log H)^{r}.$

###  3.3 Reduction method

The bounds on the variables obtained via Baker’s theory [8] are too large for
any computational purposes. To reduce the bounds, we use the reduction method
due to Dujella and Pethő [9, Lemma 5a]. For a real number $X$,
$|X|:=\min\\{|X-n|:n\in\mathbb{Z}\\}$ stands for the distance of $X$ to the
nearest integer.

######  Lemma 3.

Let $M$ be a positive integer, $p/q$ be a convergent of the continued fraction
expansion of an irrational number $\tau$ such that $q>6M$, and $A,B,\mu$ be
some real numbers with $A>0$ and $B>1$. Furthermore, let

$\varepsilon:=|\mu q|-M\cdot|\tau q|.$

If $\varepsilon>0$, then there is no solution to the inequality

$0<|u\tau-v+\mu|<AB^{-w}$ (5)

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

$u\leq M\;\mbox{and}\;w\geq\cfrac{\log(Aq/\varepsilon)}{\log B}.$

##  4 Proofs of main results

###  4.1 Proof of Theorem 1

To prove Theorem 1, we will use the following lemma which provides a relation
on the size of $k$ versus $n$ and $g$.

######  Lemma 4.

All solutions of the Diophantine equation (1) satisfy

$\displaystyle k<8n\log g.$

######  Proof 2.

From (1), we have

$\alpha_{\mathcal{N}}^{k-2}\leq\mathcal{N}_{k}=d_{1}\cfrac{g^{\ell}-1}{g-1}\cdot
d_{2}\cfrac{g^{m}-1}{g-1}\cdot
d_{3}\cfrac{g^{n}-1}{g-1}\leq(g^{n}-1)^{3}<g^{3n}.$

Taking logarithm on both sides, we get $(k-2)\log\alpha_{\mathcal{N}}<3n\log
g$. Since $n\geq 2$ and $g\geq 2$, we obtain the desired inequality. This ends
the proof.

######  Proof 3 (Proof of Theorem 1).

If $n=1$, then $\ell=m=1$. So, equation (1) becomes

$\mathcal{N}_{k}=d_{1}d_{2}d_{3}$

which implies

$\alpha_{\mathcal{N}}^{k-2}\leq\biggl{(}g-1\biggr{)}^{3}$

which leads to

$k<2+3\cfrac{\log g}{\log\alpha_{\mathcal{N}}}.$

Now, suppose $n\geq 2$. From (1) and (2), we have

$\mathcal{N}_{k}=a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}+b_{\mathcal{N}}\beta_{\mathcal{N}}^{k}+c_{\mathcal{N}}\gamma_{\mathcal{N}}^{k}=d_{1}\cfrac{g^{\ell}-1}{g-1}\cdot
d_{2}\cfrac{g^{m}-1}{g-1}\cdot d_{3}\cfrac{g^{n}-1}{g-1}.$

Which implies

$\displaystyle
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}-\cfrac{d_{1}d_{2}d_{3}g^{\ell+m+n}}{(g-1)^{3}}$
$\displaystyle=-\cfrac{d_{1}d_{2}d_{3}\biggl{(}g^{\ell+m}+g^{\ell+n}+g^{m+n}\biggr{)}}{(g-1)^{3}}$
(6)
$\displaystyle+\cfrac{d_{1}d_{2}d_{3}\biggl{(}g^{l}+g^{m}+g^{n}\biggr{)}}{(g-1)^{3}}-\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}-\Pi(k).$

Taking the absolute values of both sides of (6) and using (4), we get

$\displaystyle\biggl{|}a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}-\cfrac{d_{1}d_{2}d_{3}g^{\ell+m+n}}{(g-1)^{3}}\biggr{|}$
$\displaystyle<\cfrac{d_{1}d_{2}d_{3}\biggl{(}g^{\ell+m}+g^{\ell+n}+g^{m+n}\biggr{)}}{(g-1)^{3}}$
(7)
$\displaystyle+\cfrac{d_{1}d_{2}d_{3}\biggl{(}g^{l}+g^{m}+g^{n}\biggr{)}}{(g-1)^{3}}+\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}+\cfrac{1}{\alpha_{\mathcal{N}}^{k/2}}.$

Multiplying both sides of (7) by
$\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}g^{\ell+n+m}}$ and noticing the fact that
$\ell\leq m\leq n$, we get the inequality

$\displaystyle\biggl{|}\cfrac{(g-1)^{3}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}\cdot
g^{-(\ell+n+m)}}{d_{1}d_{2}d_{3}}-1\biggr{|}$
$\displaystyle<\cfrac{1}{g^{\ell}}+\cfrac{1}{g^{m}}+\cfrac{1}{g^{n}}+\cfrac{1}{g^{\ell+m}}+\cfrac{1}{g^{\ell+n}}$
$\displaystyle+\cfrac{1}{g^{m+n}}+\cfrac{1}{g^{\ell+m+n}}+\cfrac{(g-1)^{3}}{\alpha_{\mathcal{N}}^{k/2}d_{1}d_{2}d_{3}g^{\ell+n+m}}$
$\displaystyle<8\cdot g^{-\ell}.$

So, we get

$\displaystyle\biggl{|}\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}}\cdot\alpha_{\mathcal{N}}^{k}\cdot
g^{-(\ell+n+m)}-1\biggr{|}<8\cdot g^{-\ell}.$ (8)

We put

$\Gamma_{1}:=\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}}\cdot\alpha_{\mathcal{N}}^{k}\cdot
g^{-(\ell+n+m)}-1.$

Let us show $\Gamma_{1}\neq 0$. We proceed by the contrary. Assume that
$\Gamma_{1}=0$. Then

$a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}=\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}\cdot
g^{\ell+m+n}$

which implies

$\sigma_{\varphi}\biggl{(}a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}\
\biggr{)}=b_{\mathcal{N}}\beta_{\mathcal{N}}^{k}=\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}\cdot
g^{\ell+m+n}.$

Taking the absolute value, we get

$\biggl{|}b_{\mathcal{N}}\beta_{\mathcal{N}}^{k}\biggr{|}=\biggl{|}\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}\cdot
g^{\ell+m+n}\biggr{|}.$

We have $\biggl{|}b_{\mathcal{N}}\beta_{\mathcal{N}}^{k}\biggr{|}<1$ instead
$\biggl{|}\cfrac{d_{1}d_{2}d_{3}}{(g-1)^{3}}\cdot g^{\ell+m+n}\biggr{|}>1$
since $1\leq\ell\leq m\leq n$, which leads to a contradiction. Hence
$\Gamma_{1}\neq 0$.

In order to apply Matveev’s result to $\Gamma_{1}$, set

$\displaystyle
t:=3,\quad\eta_{1}:=\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}},\quad\eta_{2}:=\alpha_{\mathcal{N}},\quad\eta_{3}:=g,$
$\displaystyle\quad b_{1}:=1,\quad b_{2}:=k,\quad b_{3}:=-(\ell+m+n),$

and
${\mathbb{K}}:={\mathbb{Q}}(\eta_{1},\eta_{2},\eta_{3})={\mathbb{Q}}(\alpha_{\mathcal{N}})$
which is a real number field of degree $d_{\mathbb{K}}=3$.

Using properties of the logarithmic height, we get

$h(\eta_{2})=h(\alpha_{\mathcal{N}})=\cfrac{\log\alpha_{\mathcal{N}}}{3},\quad
h(\eta_{3})=h(g)=\log g$

and

$\displaystyle h(\eta_{1})$
$\displaystyle=h\biggl{(}\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}}\biggr{)}$
$\displaystyle\leq
h(a_{\mathcal{N}})+h\biggl{(}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}}\biggr{)}$
$\displaystyle\leq\cfrac{1}{3}\log
23+\log\biggl{(}\max\biggl{\\{}(g-1)^{3},d_{1}d_{2}d_{3}\biggr{\\}}\biggr{)}$
$\displaystyle<3\log(g)+2<6\log g\quad\text{since}\ g\geq 2.$

Thus, we can take

$A_{1}=18\log(g),\quad A_{2}=\log\alpha_{\mathcal{N}},\quad\text{and}\
A_{3}=3\log g.$

By Lemma 4, we have $k<8n\log g$, so we put $B=8n\log g$.
Using Theorem 3, we see that

$\displaystyle\log|\Gamma_{1}|>$ $\displaystyle-1.4\times 30^{6}\times
3^{4.5}\times 3^{2}(1+\log 3)(1+\log(8n\log g))$
$\displaystyle\qquad\times(18\log(g))(3\log g\log\alpha_{\mathcal{N}})$
$\displaystyle>-5.6\times 10^{13}(1+\log(8n\log g))(\log^{2}g).$

Comparing the above inequality with (8), we obtain that

$\displaystyle\ell\log g-\log 8<5.6\times 10^{13}(1+\log(8n\log
g))(\log^{2}g).$

Since $g\geq 2$ and $n\geq 2$, we have

$1+\log(8n\log g)<8\log(n\log g)$

so we get

$\ell<4.5\times 10^{14}\log n\log^{2}g.$

Rewrite (1), we get

$\cfrac{a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}(g-1)}{d_{1}(g^{\ell}-1)}+\cfrac{\Pi(k)(g-1)}{d_{1}(g^{\ell}-1)}=\cfrac{d_{2}d_{3}}{(g-1)^{2}}\biggl{(}g^{n+m}-g^{m}-g^{n}+1\biggr{)},$

which implies

$\displaystyle\cfrac{a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}(g-1)}{d_{1}(g^{\ell}-1)}-\cfrac{d_{2}d_{3}g^{n+m}}{(g-1)^{2}}=-\cfrac{\Pi(k)(g-1)}{d_{1}(g^{\ell}-1)}-\cfrac{d_{2}d_{3}g^{m}}{(g-1)^{2}}-\cfrac{d_{2}d_{3}g^{n}}{(g-1)^{2}}+\cfrac{d_{2}d_{3}}{(g-1)^{2}}.$
(9)

Taking the absolute values of both sides of (9), we have

$\biggl{|}\cfrac{a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}(g-1)}{d_{1}(g^{\ell}-1)}-\cfrac{d_{2}d_{3}g^{n+m}}{(g-1)^{2}}\biggr{|}<\cfrac{(g-1)}{d_{1}(g^{\ell}-1)\alpha_{\mathcal{N}}^{k/2}}+\cfrac{d_{2}d_{3}g^{m}}{(g-1)^{2}}+\cfrac{d_{2}d_{3}g^{n}}{(g-1)^{2}}+\cfrac{d_{2}d_{3}}{(g-1)^{2}}.$

Dividing both sides of the inequality above by
$\cfrac{d_{2}d_{3}g^{n+m}}{(g-1)^{2}}$ and using the fact that $n\geq 2$, we
see that

$\displaystyle\biggl{|}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}\cdot g^{-(n+m)}-1\biggr{|}$
$\displaystyle\leq\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)\alpha_{\mathcal{N}}^{k/2}g^{n+m}}+\cfrac{1}{g^{n}}+\cfrac{1}{g^{m}}+\cfrac{1}{g^{n+m}}$
$\displaystyle<4\cdot g^{-m}.$

Then, we have

$\displaystyle\biggl{|}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}\cdot
g^{-(n+m)}-1\biggr{|}<\cfrac{4}{g^{m}}.$ (10)

We put

$\Gamma_{2}=\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}\cdot g^{-(n+m)}-1.$

One can check that $\Gamma_{2}\neq 0$, proceeding as we do for $\Gamma_{1}$.
Let us apply Matveev’s result for $\Gamma_{2}$. Let

$\displaystyle
t:=3,\quad\eta_{1}:=\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\cdot
a_{\mathcal{N}},\quad\eta_{2}:=\alpha_{\mathcal{N}},\quad\eta_{3}:=g,$
$\displaystyle\quad b_{1}:=1,\quad b_{2}:=k,\quad b_{3}:=-(m+n),$

and
${\mathbb{K}}:={\mathbb{Q}}(\eta_{1},\eta_{2},\eta_{3})={\mathbb{Q}}(\alpha_{\mathcal{N}})$
of degree $d_{\mathbb{K}}=3$. By Lemma 4, we have $k<8n\log g$, so we put
$B=8n\log g$. We have

$h(\eta_{2})=h(\alpha_{\mathcal{N}})=\cfrac{\log\alpha_{\mathcal{N}}}{3},\quad
h(\eta_{3})=h(g)=\log g,$

and

$\displaystyle h(\eta_{1})$
$\displaystyle=h\biggl{(}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\cdot
a_{\mathcal{N}}\biggr{)}$ $\displaystyle\leq
h(a_{\mathcal{N}})+h\biggl{(}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)}\biggr{)}$
$\displaystyle\leq\cfrac{1}{3}\log
23+\log\biggl{(}\max\biggl{\\{}(g-1)^{3},d_{1}d_{2}d_{3}\biggr{\\}}\biggr{)}+h\biggl{(}\cfrac{1}{g^{\ell}-1}\biggr{)}$
$\displaystyle<2+3\log(g-1)+\log(g^{\ell}-1)$
$\displaystyle<(3+\ell)\log(g)+2$ $\displaystyle<(6+\ell)\log
g\quad\text{since}\ g\geq 2.$

Thus, we can take

$A_{1}=(18+3\ell)\log(g),\quad A_{2}=\log\alpha_{\mathcal{N}},\quad\text{and}\
A_{3}=3\log g.$

Using Theorem 3, we see that

$\displaystyle\log|\Gamma_{2}|>$ $\displaystyle-1.4\times 30^{6}\times
3^{4.5}\times 3^{2}(1+\log 3)(1+\log(8n\log g))$
$\displaystyle\qquad\times((18+3\ell)\log(g))(3\log
g\log\alpha_{\mathcal{N}})$ $\displaystyle>-3.1\times 10^{12}(1+\log(8n\log
g))(\log^{2}g)(18+3\ell).$

Comparing with (10), we get

$m\log g-\log 4<3.1\times 10^{12}(1+\log(8n\log g))(\log^{2}g)(18+3\ell).$

We have

$1+\log(8n\log g)<8\log n\log g\quad\text{and}\quad\ell<4.5\times 10^{14}\log
n\log^{2}g.$

So,

$m<3.8\times 10^{28}\log^{2}n\log^{4}g.$

Reorganizing (1), we get

$\cfrac{d_{3}g^{n}}{g-1}-\cfrac{(g-1)^{2}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}}{d_{1}d_{2}(g^{\ell}-1)(g^{m}-1)}=\cfrac{d_{3}}{g-1}+\cfrac{\Pi(k)(g-1)^{2}}{d_{1}d_{2}(g^{\ell}-1)(g^{m}-1)}.$

We have

$\biggl{|}\cfrac{d_{3}g^{n}}{g-1}-\cfrac{(g-1)^{2}\cdot
a_{\mathcal{N}}\alpha_{\mathcal{N}}^{k}}{d_{1}d_{2}(g^{\ell}-1)(g^{m}-1)}\biggr{|}<\cfrac{d_{3}}{g-1}+\cfrac{(g-1)^{2}}{\alpha_{\mathcal{N}}^{k/2}d_{1}d_{2}(g^{\ell}-1)(g^{m}-1)}$

by taking the absolute values of both sides of (9). Dividing both sides of the
above inequality by $(d_{3}g^{n})/(g-1)$ and using the fact that $n\geq 2$, we
see that

$\displaystyle\biggl{|}1-\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)}\cdot
g^{-n}\cdot\alpha_{\mathcal{N}}^{k}\biggr{|}$
$\displaystyle<\cfrac{1}{g^{n}}+\cfrac{1}{g^{n-1}}<\cfrac{2}{g^{n-1}}.$
$\displaystyle<2\cdot g^{1-n}.$

Then, we have

$\displaystyle\biggl{|}\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)}\cdot
g^{-n}\cdot\alpha_{\mathcal{N}}^{k}-1\biggr{|}<2\cdot g^{1-n}.$ (11)

We put

$\Gamma_{3}=\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)}\cdot
g^{-n}\cdot\alpha_{\mathcal{N}}^{k}-1.$

One can verify that $\Gamma_{3}\neq 0$. Let us analyze Matveev’s result for
$\Gamma_{3}$. Let

$\displaystyle
t:=3,\quad\eta_{1}:=\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)},\quad\eta_{2}:=\alpha_{\mathcal{N}},\quad\eta_{3}:=g,$
$\displaystyle\quad b_{1}:=1,\quad b_{2}:=k,\quad b_{3}:=-n,$

and
${\mathbb{K}}:={\mathbb{Q}}(\eta_{1},\eta_{2},\eta_{3})={\mathbb{Q}}(\alpha_{\mathcal{N}})$
of degree $d_{\mathbb{K}}=3$. By Lemma 4, we have $k<8n\log g$, so we put
$B=8n\log g$. We have

$h(\eta_{2})=h(\alpha_{\mathcal{N}})=\cfrac{\log\alpha_{\mathcal{N}}}{3},\quad
h(\eta_{3})=h(g)=\log g,$

and

$\displaystyle h(\eta_{1})$
$\displaystyle=h\biggl{(}\cfrac{a_{\mathcal{N}}(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)}\biggr{)}$
$\displaystyle\leq
h(a_{\mathcal{N}})+h\biggl{(}\cfrac{(g-1)^{3}}{d_{1}d_{2}d_{3}(g^{\ell}-1)(g^{m}-1)}\biggr{)}$
$\displaystyle\leq\cfrac{1}{3}\log
23+\log\biggl{(}\max\\{(g-1)^{3},d_{1}d_{2}d_{3}\\}\biggr{)}+h\biggl{(}\cfrac{1}{g^{\ell}-1}\biggr{)}+h\biggl{(}\cfrac{1}{g^{m}-1}\biggr{)}$
$\displaystyle<2+3\log\biggl{(}g-1\biggr{)}+\log\biggl{(}g^{\ell}-1\biggr{)}+\log\biggl{(}g^{m}-1\biggr{)}$
$\displaystyle<(3+\ell+m)\log(g)+2$ $\displaystyle<(6+\ell+m)\log
g\quad\text{since}\ g\geq 2.$

Thus, we can take

$A_{1}=3(6+\ell+m)\log\biggl{(}g\biggr{)},\quad
A_{2}=\log\alpha_{\mathcal{N}},\quad\text{and}\ A_{3}=3\log g.$

Using Theorem 3, we see that

$\displaystyle\log\biggl{|}\Gamma_{3}\biggr{|}$ $\displaystyle>-1.4\times
30^{6}\times 3^{4.5}\times 3^{3}(1+\log 3)(1+\log(8n\log g))$
$\displaystyle\qquad\times((6+\ell+m)\log\biggl{(}g\biggr{)})(3\log
g\log\alpha_{\mathcal{N}})$ $\displaystyle>-9.31\times 10^{12}(1+\log(8n\log
g))(\log^{2}g)(6+\ell+m).$

Comparing with (11), we get

$(n-1)\log g-\log 2<9.31\times 10^{12}(1+\log(8n\log g))(\log^{2}g)(6+\ell+m)$

We have

$\displaystyle 1+\log(8n\log g)<8\log n\log g,\quad m<3.8\times
10^{28}\times\log^{2}n\log^{4}g$ $\displaystyle\text{and}\quad\ell<4.5\times
10^{14}\log n\log^{2}g.$

Thus

$\displaystyle 6+\ell+m$ $\displaystyle<4.51\times 10^{14}\log
n\log^{2}g+3.8\times 10^{28}\times\log^{2}n\log^{4}g$ $\displaystyle<4\times
10^{28}\times\log^{2}n\log^{4}g.$

So, we have

$n<3\times 10^{42}\log^{3}n\log^{6}g.$

Now we apply Lemma 2, by setting

$r:=3,\quad L:=n\quad\text{and}\quad H:=3\times 10^{42}\cdot\log^{6}g,$

we get

$\displaystyle n$ $\displaystyle<2^{3}\cdot 3\times
10^{42}\cdot\log^{6}g\times\log^{3}\biggl{(}3\times
10^{42}\cdot\log^{6}g\biggr{)}$ $\displaystyle<2.4\times
10^{43}\cdot\log^{6}g\cdot(95.6+6\log\log g)^{3}$ $\displaystyle<5.91\times
10^{49}\log^{9}g.$

Notice that we have used the inequality $95.6+6\log\log g<135\log g$ which
holds since $g\geq 2$.

###  4.2 Proof of Theorem 2

Since $2\leq g\leq 10$, according to Theorem 1, we have

$\ell\leq m\leq n<1.08\times 10^{53}\quad\text{and}\quad k<1.99\times
10^{54}.$

Consequently, the next step is to reduce the upper bounds above in order to
identify the set of the interval in which the possible solutions of (1) lie.
To do this, we proceed in three steps.

####  Step 1.

Using (8), let

$\Lambda_{1}:=-\log(\Gamma_{1}+1)=(\ell+m+n)\log
g-k\log\alpha_{\mathcal{N}}-\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}}\biggr{)}.$

Notice that (8) can be rewritten as

$\biggl{|}e^{-\Lambda_{1}}-1\biggr{|}<\cfrac{8}{g^{\ell}}.$

Observe that $\Lambda_{1}\neq 0$, since $e^{-\Lambda_{1}}-1=\Gamma_{1}\neq 0$.
Assume that $\ell\geq 5$. Then

$\biggl{|}e^{-\Lambda_{1}}-1\biggr{|}<\cfrac{8}{g^{\ell}}<\cfrac{1}{2}.$

Since $\biggl{|}x\biggr{|}<2\biggl{|}\mathrm{e}^{x}-1\biggr{|}$, if
$\biggl{|}x\biggr{|}<\cfrac{1}{2}$ holds, then

$\biggl{|}\Lambda_{1}\biggr{|}<\cfrac{16}{g^{\ell}}.$

Substituting $\Lambda_{1}$ in the above inequality with its value and dividing
through by $\log\alpha_{\mathcal{N}}$, we get

$\biggl{|}(\ell+m+n)\biggl{(}\cfrac{\log
g}{\log\alpha_{\mathcal{N}}}\biggr{)}-k+\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}}\biggr{)}}{\log\alpha_{\mathcal{N}}}\biggr{|}<\cfrac{16}{\log\alpha_{\mathcal{N}}g^{\ell}}.$

Then, we can apply Lemma 3 with the data

$\displaystyle\tau:=\cfrac{\log
g}{\log\alpha_{\mathcal{N}}},\quad\mu:=\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}}\biggr{)}}{\log\alpha_{\mathcal{N}}},\quad
A:=\cfrac{16}{\log\alpha_{\mathcal{N}}},\quad$ $\displaystyle
B:=g,\quad\text{and}\quad w:=\ell\quad\text{with}\quad 1\leq d_{1}\leq
d_{2}\leq d_{3}\leq g-1.$

We can take $M:=1.99\times 10^{54}$, since $k<8n\log g<1.99\times 10^{54}$.
So, for the remaining proof, we use Mathematica to apply Lemma 3. For the
computations, if the first convergent $q_{t}$ is such that $q_{t}>6M$ does not
satisfy the condition $\varepsilon>0$, then we use the next convergent until
we find the one that satisfies the conditions. Thus, we have that

Table 2: Upper bound on $\ell$ $g$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{118}$ | $q_{100}$ | $q_{110}$ | $q_{115}$ | $q_{90}$ | $q_{106}$ | $q_{112}$ | $q_{102}$ | $q_{96}$
$\varepsilon\geq$ | 0.36 | 0.26 | 0.03 | 0.01 | 0.06 | 0.001 | 0.0019 | 0.005 | 0.01
$\ell\leq$ | 194 | 121 | 99 | 87 | 76 | 72 | 67 | 62 | 59

Therefore

$1\leq\ell\leq\cfrac{\log((16/\log\alpha_{\mathcal{N}})\cdot
q_{118}/0.36)}{\log 2}\leq 194.$

#### Step 2.

In this step, we have to reduce the upper bound on $m$. To do this, let us
consider

$\Lambda_{2}:=-\log(\Gamma_{2}+1)=(m+n)\log
g-k\log\alpha_{\mathcal{N}}+\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)}\biggr{)}.$

Thus inequality (10) becomes

$\biggl{|}e^{-\Lambda_{2}}-1\biggr{|}<\cfrac{4}{g^{m}}<\cfrac{1}{2},$

which holds for $m\geq 4$. It follows that

$\displaystyle\biggl{|}(m+n)\cfrac{\log
g}{\log\alpha_{\mathcal{N}}}-k+\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)}\biggr{)}}{\log\alpha_{\mathcal{N}}}\biggr{|}<\cfrac{8}{g^{m}\log\alpha_{\mathcal{N}}}.$
(12)

So, the conditions of Lemma 3 are satisfied. Applying this lemma to Inequality
(12) with the following data

$\displaystyle\tau:=\cfrac{\log
g}{\log\alpha_{\mathcal{N}}},\quad\mu:=\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)}\biggr{)}}{\log\alpha_{\mathcal{N}}},\quad
A:=\cfrac{8}{\log\alpha_{\mathcal{N}}},\quad B:=g,\quad\text{and}\quad w:=m$

with $\quad 1\leq d_{1}\leq d_{2}\leq d_{3}\leq g-1$ and $1\leq\ell\leq 194$.
As $k<8n\log g<1.99\times 10^{54}$, we can take $M:=1.99\times 10^{54}$. With
Mathematica we get the following results :

Table 3: Upper bound on $m$ $g$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{118}$ | $q_{100}$ | $q_{110}$ | $q_{115}$ | $q_{90}$ | $q_{106}$ | $q_{112}$ | $q_{102}$ | $q_{96}$
$\varepsilon\geq$ | 0.004 | 0.0007 | 0.0003 | 0.001 | 0.0002 | 0.0005 | 0.0001 | 0.005 | 0.001
$m\leq$ | 199 | 125 | 102 | 88 | 78 | 72 | 68 | 62 | 60

In all cases, we can conclude that

$1\leq m\leq\cfrac{\log((8/\log\alpha_{\mathcal{N}})\cdot
q_{115}/0.0009)}{\log 2}\leq 200.$

#### Step 3.

Finally, to reduce the bound on $n$ we have to choose

$\Lambda_{3}:=\log(\Gamma_{3}+1)=(n)\log
g-k\log\alpha_{\mathcal{N}}+\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)(g^{m}-1)}\biggr{)}.$

We have,

$\biggl{|}e^{-\Lambda_{3}}-1\biggr{|}<\cfrac{2}{g^{n-1}}<\cfrac{1}{2},$

which is valid for $n\geq 4$ and $g\geq 2$. It follows that

$\displaystyle\Biggl{|}m\cfrac{\log
g}{\log\alpha_{\mathcal{N}}}-k+\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)(g^{m}-1)}\biggr{)}}{\log\alpha_{\mathcal{N}}}\Biggr{|}<\cfrac{4}{g^{n-1}\alpha_{\mathcal{N}}}.$
(13)

Now we have to apply Lemma 3 to (13) by taking the following parameters

$\displaystyle\tau:=\cfrac{\log
g}{\log\alpha_{\mathcal{N}}},\quad\mu:=\cfrac{\log\biggl{(}\cfrac{(g-1)^{3}a_{\mathcal{N}}}{d_{1}d_{2}d_{3}\cdot(g^{\ell}-1)(g^{m}-1)}\biggr{)}}{\log\alpha_{\mathcal{N}}},\quad
A:=\cfrac{8}{\log\alpha_{\mathcal{N}}},\quad$ $\displaystyle
B:=g,\quad\text{and}\quad w:=n-1$

with $\quad 1\leq d_{1}\leq d_{2}\leq d_{3}\leq g-1$, $\quad 1\leq\ell\leq
194$ and $1\leq m\leq 183$.

Using the fact that $k<8n\log g<1.99\times 10^{54}$, we can take
$M:=1.99\times 10^{54}$, and we get

Table 4: Upper bound on $n$ $g$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10
---|---|---|---|---|---|---|---|---|---
$q_{t}$ | $q_{118}$ | $q_{99}$ | $q_{110}$ | $q_{115}$ | $q_{90}$ | $q_{106}$ | $q_{112}$ | $q_{102}$ | $q_{96}$
$\varepsilon\geq$ | 0.00006 | 0.002 | 0.007 | 0.0002 | 0.0008 | 0.002 | 0.0005 | 0.02 | 0.009
$n\leq$ | 204 | 124 | 99 | 89 | 77 | 71 | 67 | 61 | 59

It follows from the above table that

$1\leq n\leq\cfrac{\log((4/\log\alpha_{\mathcal{N}})\cdot
q_{118}/10^{-6})}{\log 2}\leq 205,$

which is valid for all $g$ such that $2\leq g\leq 10$. In light of the above
results, we need to check the equation (1) in the cases $2\leq g\leq 10$ for
$1\leq d_{1},d_{2},d_{3}\leq 9$, $1\leq n\leq 205$, $1\leq m\leq 200$,
$1\leq\ell\leq 194$ and $1\leq k\leq 11500$. A quick inspection using Sagemath
reveals that the Diophantine equation (1) in the range $2\leq g\leq 10$ has
only the solution listed in the statement of Theorem 2. This completes the
proof of Theorem 2.

##  5 Discussions

In addition to Baker’s method and linear forms in logarithms, there are other
so-called “classical” methods and techniques for solving exponential
Diophantine equations. These include the modular arithmetic method, $p$-adic
analysis, Fermat’s method of infinite descent, the factorization method,
solving using inequalities, the mathematical induction method, the parametric
method, and so on. It would be interesting to treat the same problems
approached in this article with other methods than those of the linear forms
in logarithms. The modular arithmetic method could be used to determine
Narayana numbers, which are products of three repdigits in base $g$ with
$g\geq 2$ due to the interesting divisibility properties possessed by the
repdigits.

##  6  Acknowledgments

The authors express their gratitude to the anonymous reviewer for the
instructive suggestions. The first author is partially supported by Université
de Kara (Togo) and the second author is supported by UCAD, the Université
Cheikh Anta Diop de Dakar. This project was initiated when the first author
visited UCAD on a research stay. He thanks the authorities for the warm
hospitality and the working environment.

## References

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## Addresses

1 Faculté des Sciences et Techniques (FST), Département de Mathématiques,
Universié de Kara, Togo. 2 Laboratoire d’Algèbre, de Cryptologie, de Géométrie
Algébrique et Applications (LACGAA), Université Cheikh Anta Diop de Dakar
(UCAD), Sénégal.
Email<EMAIL_ADDRESS>
2 Laboratoire d’Algèbre, de Cryptologie, de Géométrie Algébrique et
Applications (LACGAA), Université Cheikh Anta Diop de Dakar (UCAD), Sénégal

Email<EMAIL_ADDRESS>
3 Faculty of Sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah University,
Atlas-Fez, Morocco

Email<EMAIL_ADDRESS>